text:
Hierarchical Introspective Logics
This research was initially stimulated by the announcement in
June 1993 of a proof of the FLT conjecture ("Fermat's Last Theorem").
Once again a mathematical question that might conceivably have been
unanswerable was being answered by the efforts of humans to find an
acceptable proof of a standing conjecture.
Earlier the famous 4 Color Conjecture had yielded to an attack
aided by computers.
So the question that arose in my mind was that of whether or not
there actually exists any mathematically interesting true mathematical
assertion (such as, for example, perhaps the Riemann Hypothesis) which
cannot be proved and which must ultimately be accepted as a new axiom
if mathematics is to proceed in terms of assertions and proofs (Satz;
Beweis).
I was of course aware of Goedel's establishment of the existence
of assertions concerning the natural integers that are not provable
modulo the proof rules of specific given formal systems and also of
the general results on undecidability which also imply the existence
of such assertions that are true but not provable in the context of
a given formal system.
The specific sort of formula that is produced by the construction
of Goedel, or by Rosser's variation on Goedel's construction, is true
but not provable within the formal system on the basis of which it was
constructed. However an acceptable mathematical proof, in verbal form,
was given by Goedel of the truth of his constructed formula provided
that it could be assumed that the formal system at issue was ideally
Thus the Goedel formula or assertion is not really of itself
mysterious; it is provably true, on a common-sense basis, and it shows
therefore that the formal system studied must have been inadequate to
achieve the ideal goal of completeness. (It can be remarked that
logical systems of a more limited application, specifically systems
not adequate for the proof of general propositions of "number theory",
for example the "calculus of sentences", can be complete. For systems
with a narrower scope of applicability completeness does not introduce
paradoxes.)
If a Goedel or Goedel-Rosser assertion is added to one of these
incomplete systems as an axiom then one obtains a new logical system
with similar characteristics to the original. It is thus again
incomplete and again a Goedel assertion can be found within it, a
formula that is true but that is not provable by the formal machinery
of the system.
But better than adding the Goedel or the Goedel-Rosser assertion
to the initial system as an axiom one can instead add to it an axiom
of consistency. That is one can add an axiom stating that the initial
system was formally consistent. This does not also say that the new
system including the added axiom is itself consistent.
The axiom asserting formal consistency could be expressed with
the use of a scheme of Goedel numbering for the formulae expressible
in the initial system and also for proofs possible in that system.
Then it simply asserts that no such proof can result in a formal proof
of falsehood (which is represented by the second truth symbol of the
pair of T and F).
It is known that adding such an axiom of formal consistency to the
initial system makes it then possible, using the added axiom, to prove
either the Goedel assertion or the Rosser assertion for that system
(the initial system).
Extended Systems
This sort of supplementary axiom, which serves naturally to deal
with the incompleteness of a logical system, this supplementation
produces a natural extension of the original system. This type of
extension process was studied in the mid- and later 30's by Turing
in a project that became his Ph.D. thesis at Princeton.
One thing that is notable is that when an axiom is added that
asserts formal consistency of an initial system that this added axiom
is not itself included in relation to the consistency issue; so the
extended system formed by the addition of that axiom is not itself
asserted to be consistent by the added axiom. And thus, for the
newly formed extended system to be itself axiomatically stated to
be consistent requires the introduction of a second added axiom in
a parallel proceeding.
Turing considered this process in his paper (the thesis) "Systems
of logic based on ordinals" published in 1939 in Proc. Lond. Math.
Soc. But Turing did not construct his systems, his towers of
extension, on a basis of "ideal ordinals", corresponding to the
mathematician's traditional way of thinking of mathematical objects,
but rather he worked in terms of arrays of ordinals which were indexed
somehow by ordinary integers and described by a table of ordering by
which the order relation of any two of the indexed entities was
specified. And this order table was required to be specified by
recursive functions.
When I began to think about the same general theme, that of
axiomatically extending an initially specified logical system so as
to "correct" or fill in for the incompleteness of the type discovered
by Goedel, it occurred to me that Turing's concept of how ordinals
could be used may have been intrinsically too limiting and that
it would be desirable to explore what might be done with a less
restrictive concept of ordinal numbers than that of the recursively
presented sets of them used by Turing.
There are many logical "paradoxes" that can arise if we speak
loosely about ordinals and their definability. On the other hand I do
think that Turing's concept of usable ordinals was too restrictive to
allow the related extension scheme to proceed to really interesting
results relating to incompleteness.
Another Form of Incompleteness
The discovery of incompleteness by Goedel was supplemented in
the 30's by the work of Turing, Church, and others who showed that
there was "undecidability" with regard to the totality of assertions
possible within a formal system of the same general category as those
for which Goedel's discovery applied. This showed that it was not
possible, systematically, to decide which assertions were true and
which false. And consequently to that there could not exist proofs
for all of the true assertions since that would make possible the
systematic decision process. Thus a true but unprovable assertion
MUST EXIST although this route, via "undecidability", did not produce
any specific example.
But much more recently a paper was published by Paris and
Harrington showing that a popular form of Peano arithmetic called PA-
(upper suffix -) was incomplete because of the lack of a sufficiently
strong "axiom of infinity". This result was very interesting since it
showed that incompleteness could have this sort of a connection with
"knowledge of infinity" and with the possibilities for infinite
arguments such as proofs by induction.
Of course there are many long known paradoxes, or potential
logical traps, that arise from talking about infinity. One of these
can be illustrated by imagining two mathematical logicians named
Adams and Bentley. So Bentley says "Let beta be the limit ordinal
(number) which is the least ordinal that is larger than any ordinal
alpha definable in the formal language of the system of Adams. This
illustrates the paradoxes that inherently arise because we seek
to "talk about" infinite concepts but we have only finite resources
to use for the actual purpose of communication in words. Any book
written by humans has a text that contains only a finite amount
of information, a certain number of "bytes" as it were. And this
illustrative paradox relates to the classical "Burali-Forti paradox"
of logic and set theory.
Dangers of Paradoxes, Personal Caution
I am speaking about a research project that is not fully complete
since I have not yet written up and submitted for publication any
paper or papers describing the work. Also the details of what axioms
to use and how to select the basic set theory underlying the
hierarchical extension to be constructed are not fully crystallized.
I have also a great fear of possible error in studying topics in
this area. It is not rare, historically, for systems to be proposed
that are either inconsistent or that have unexpected weaknesses. So
I feel that I must be cautious and proceed without rushing to a goal.
And this psychology of fear has also inhibited me from consulting
other persons expert in logic before I could feel that I had gotten
my own ideas into good shape.
The Overview Concept
It is notable that Goedel's construction of an unprovable, yet
true, proposition in a generic sort of formal system is accompanied
also by a proof, under reasonable hypotheses, of the truth of that
"unprovable" proposition. How is this achieved? Well, of course, the
proof does NOT occur WITHIN the formal system for which the Goedel
proposition was constructed, rather it is given verbally in the normal
style of argument for published mathematical papers.
But it can be observed that if another logical system were
constructed with the specific goal of studying the first system
and the proofs possible within it and if this "overview" system
also included axioms to the effect that the machinery of procedure
of the first system is truth-preserving and that the axioms of
the first system are true then it would become provable, in this
overview system, that the first system is consistent and also
"Omega-consistent". And from these proofs would follow, in the
overview system, the proof of the truth of a Goedel assertion for
the first system.
These observations relate to what Turing studied in his paper
"Systems of logic based on ordinals". In hindsight one could say that
Turing's concept of extension was (italicize) THE UNIQUE AND NATURAL
CONCEPT so far as the range of the finite ordinal numbers is
concerned.
This results in an ascending ladder of systems in which each successor
is more complete than any of its predecessors but this ladder ascends
only through the finite ordinal numbers, 0th, 1st, 2nd, 3rd, 4th,
5th, .... nth, ... and does not reach any transfinite levels.
Of course Turing did actually concern himself with transfinite
ordinal levels but he did not have a way of uniquely describing them.
(This was the "invariance" problem in the terminology of his paper.)
Although there seems to be essentially no confusion about the
naming or proper description of finite ordinal levels, beyond that
there are serious difficulties that are classically known. Thus
mathematically there should be a non-enumerable totality of ordinal
numbers each of which has only an enumerable set of predecessor
ordinals. Then how many can we expect to properly name or describe
in the words of any precise mathematical language? And what can be
said about the description: "the least ordinal not thus properly
describable"? Relating to this is the mathematical property of the
ordinals that any subset of them has a least member. This shows
by illustration how difficult it is to form an ideal concept of
definable ordinals or of canonical names for ordinals.
Turing's hierarchy of levels in any one of his systems was based
on association of the levels with ordinals which were themselves
associated with natural integers through a recursive description of
a subset of all ordinals. Unfortunately this specification did not
have uniqueness, or "invariance" as he called it, a property that
he knew was most desirable.
In my own thinking, after a long period of study and after
encountering the problem that although one could talk about "ideal"
(or mathematical) ordinals one would need names for them in order to
be able to refer to them in formulae of a formal system. Ultimately
I came to the idea that instead of associating the levels of a system
with ordinal numbers they could instead be associated with definitions
of ordinals. This is not the same as indexing the levels by ordinals.
Two quite different definitions might easily define the same
mathematical ordinal yet also it could be very difficult to determine
the precise comparative relationship of two different definitions,
each of them stating the definition of a specific mathematical
ordinal.
And it is also easily observed that any specified language would
enable the formation of only an enumerable array of finite formulae
to serve as definitions for ordinals while mathematically the number
or cardinality of the ordinals is unlimited. (And as was mentioned
above, even the set of ordinals where each has only enumerably many
predecessors, this set should itself be of higher cardinality and
non-enumerable.)
An Hierarchy of Levels
So we evolved the idea that the formal system serving to extend a
given basic system (or "ground level") could be structured by making
use of special functions enabling references to levels. In effect,
if we used references indicating a higher level then it would become
permissible to "overview" the formulae and procedures of a lower
level.
And the basic idea, at least at the beginning of the construction
of an hierarchy of extension levels, is quite like Turing's concept
in that the first extension level, the first level above the ground
level, incorporates the possibility to prove as a theorem the Goedel
assertion for the logical system of the "ground level".
The first higher level, corresponding in our approach to any
acceptable definition of the first ordinal number, would be such that
the Goedel assertion from the ground level would now become provable
if that Goedel assertion had been simply added as an axiom usable on
the higher level. Or instead it would be effective to introduce as an
axiom an assertion of formal consistency of the ground level logic.
This type of assertion can be seen to imply the assertion of Goedel's
type. Formal consistency is the assertion to the effect that "on the
ground level nothing false can be proved."
The verbal argument for the truth of Goedel's formula depends
in effect on an overview of the specific formal system in which
it is stated and on the basis of the axioms, etc. of which it was
constructed. Also this argument depends on the assumption that that
formal system is actually perfectly consistent. Our idea of extension
and of "hierarchical introspective logics" is that there is to be
considered an hierarchy of levels of logic such that higher levels
have an "overview" of the proceedings and results obtainable on
lower levels. Thus the totality is "introspective" because it is looking
inward to study itself but this self-study is not unrestricted. And
effectively a higher level is supposed to be able to see the truth of
a Goedel assertion deriving from a lower level by a process analogous
to the verbal argument originally known for the truth of the Goedel
assertion in a formal system although there is no proof of that
assertion in that original formal system.
A logical system cannot effectively state its own consistency;
this relates to the incompleteness originally found by Goedel. But
one logical system CAN easily state the formal consistency of another
system. We use this idea in creating our hierarchy of levels so that
on a higher level it is possible to, in effect, assume the validity
and the consistency of proceedings possible on lower levels.
Axioms and Special Functions
(This portion of the talk should make use of transparencies
presenting the axioms or axiom schemata and the special functions
introduced.)
Our design for the extension hierarchy has the feature that
certain special logical functions are introduced as components of
formulae on all levels of the hierarchy except the ground level,
the ground level being simply the formal logic that we are extending.
The initial or ground level system should be adequate to be of the
sort studied by Goedel and to be able to deal with "number theory",
"functional calculus", and "set theory". And we prefer the viewpoint
of Zermelo, that sets can be based on ur-elements which are not
themselves sets. Also we like to think of ordinal numbers as being
basically a special type of mathematical objects such that they
satisfy appropriate axiomatic properties. Thus the ordinals, as
mathematical entities, are analogous to the quaternions. Of course
this doesn't really matter, it's a question of taste.
We assume, for convenience, that a specific "canonical"
"Goedel numbering" has been chosen, both for formulae (wff's) and
for "strings" of formulae that are presented as proofs. This is with
regard to the extended system, not just the "ground level". Then
each wff or string of wff's has its index which is simply one of the
natural integers. And the special functions that we introduce are
to be understood to use these index integers as arguments. We favor
the convention that the formula or string that is indexed may be
substituted for the indexing integer in an instance of any of these
special functions.
(Give oral verbal description with display of transparencies for
axioms and special functions. Explain that ORDDEF( "delta" ) is to
be affirmed or true ONLY when "delta" is a definition in a standard
form using only notation/symbols appropriate for the ground level.
And explain also that such a definition, to be approved, must be
recognizable as PROVABLY the def. of an unique ordinal by the logic
of the GROUND LEVEL. And remark that the function ORDDEF(~~) is
NOT recursive but is definable from a set of recursive functions relating
to provability and form.)
Regarding axioms, we are not sure that we have the final set of
main axioms and also we see some overlapping or redundancy of
axioms dealing with the KT(~~,~~) or "known true" function. It is also
a technical need that certain axioms should be present that simply
assert that the special functions are what they are supposed to be.
Inductive and General Incompleteness
The incompleteness found and exhibited in the paper of Paris and
Harrington has a relation to proofs by induction and the possibility
of achieving desired results by that means given favorable "axioms
of infinity". Parallelwise, for our system of extension, which builds
upon a given initial system or ground level, the extension process
naturally ends as the range of definable ordinals becomes exhausted.
So an extension system of our type is incomplete, which is fitting
in relation to general undecidability results, but the extension
process can be easily revived and renewed provided that new "axiomatic
ordinals" are introduced. That is, it is always possible to add an
axiom or axioms specifically directed to the theme of the existence
of ordinal numbers so that it becomes possible to define a "new"
ordinal level, with the validity of the "orddef" associated with this
level dependent on the new axiom or axioms, and such that the new
definable ordinal is larger than all of those that were previously
usable. And then the extension process is renewed and in the
re-extended system the proof of formal consistency of the system
as it was before becomes possible because the former range of the
system is all subject to overview from the perspective of levels
depending on the new axiom or axioms.
The interesting possibility is that it now becomes plausible or
conceivable that the incompleteness phenomenon will not cause us to
need to add anything like the Riemann Hypothesis or the Goldbach
Conjecture as an axiom (because of the impossibility of otherwise
establishing their truth) but rather that appropriate set theoretic
axioms relevant to the existence of ordinal numbers may be added
as needed.
If it had been possible to achieve logical completeness with a
system like Principia Mathematica then all of mathematical research
to the extent of its not depending on tasteful definitions and
inspired inventions of topics, and thus perhaps all "number theory",
in the sense of the study of questions of classical elementary form,
could be worked out straightforwardly by a robot operating in an
isolation chamber simply on the basis of the rules.
But the history of human progress in science and mathematics
reveals that observation of the phenomena of Nature has always
played a large ro^le. Mathematics itself can be viewed as having
evolved from the "language of precise quantitative communication"
and many precise logical concepts have become included with the
quantitative concepts. The Sumerian scribes wrote down records for
the quantities of grain received or sent from their central granaries.
So mathematical logic itself looks like a language that is
naturally capable of evolution like also mathematics as a language
and as an encyclopedia. But we feel that a "translatability" property
should hold true here. Thus, for example, a relatively modern proof
in geometry by Pascal should be translatable into a form, written in
Greek, that Euclid would have found acceptable. Or the proof of the
FLT by Wiles should be translatable into a version in French or Latin
such that, with its introductory material developing the theory of
elliptic curves and modular functions, it could have been studied and
accepted by Fermat, if he could devote enough time to the effort.
In this context of the theme of cultural evolution, as applied
specifically to mathematical logic, it seems possible that a system of
logic of the future could be translated into a form corresponding to a
system of the present time with the addition of a few axioms that give
what is needed to give the potentialities of the future system. Our
concept of an "hierarchical introspective logic" suggests that for
"number theory" at least that there might be no need to adopt any new
axioms other than essentially axioms of infinity that make it possible
to assert the existence of ordinals that otherwise could not be
thought of as definitely existing.
Already we have in set theory a variety of popular potential
axioms. It is not yet generally assumed, but the "Axiom of Choice"
is an extremely popular option in set theory. And if the adoption
of conceivable or popular axioms went further then the GCH or
"Generalized Continuum Hypothesis" could be adopted.
It seems plausible or at least conceivable that knowledge
actually gained from the study of Nature, plus cultural evolution,
would in time lead to decisions, positive or negative, about the
adoption of axioms relating to set theory that seem to us now as
quite optional in merits.


I make this lima bean and kale ragù from time to time, which is good on it's own, and adding clams to the dish works really well. It's healthful and tastes great.
Start by cooking 2 cups of butter beans. To do this, wash the beans then add them to a stock pot with enough water to cover the beans (about 2 inches above the level of the beans). Bring the beans to a boil, then reduce the heat and simmer for approximately 1 hr. The goal here is to slightly under cook the beans.
When the beans are slightly undercooked, sautée diced onion and minced garlic in a large flat pan, then add your beans to the pan. It helps to use a slotted spoon to transfer the beans to reduce the amount of liquid that you're adding to the pan. Add white wine and a bit of chicken stock to your beans, along with chopped kale, black pepper, cayenne pepper, lemon juice, and allow to cook on medium heat. You can also add a little salt here, but go easy on the salt, because your clams will add saltiness to the dish.
While your beans are cooking, wash your clams and cook them in a separate pan (they will release a lot of salt). Cook these covered on high heat for several minutes until the clams begin to open. As the clams open, remove them from the pan and place them on a plate (but leave the liquid, which is very salty).
Next, discard the liquid from the clams and rinse the salty liquid/residue from your pan. Sautée finely diced red bell pepper and minced garlic in your clean pan, then deglaze with white wine and finish with lemon juice.
Next, check your beans for doneness, and add more chicken stock as needed. Continue to cook and stir the beans until they are fully cooked. Some of the beans will also begin to break down as you stir, which is a good thing. The beans that break down help form the sauce for your dish.
When the beans are a few minutes from being done, reduce the heat and stir in a bit of olive oil. Then add your clams to the beans to finish the cooking, and add your sautéed red pepper and garlic to the clams.
When everything is cooked, check for seasoning at this point and add salt and pepper as needed. Remove the pan from the heat, drizzle with olive oil and lemon juice, add chopped parsley, and enjoy.


text:
UNFPA Executive Director to visit DRC
© UNFPA
KINSHASA, Democratic Republic of the Congo—Dr. Natalia Kanem, the Executive Director of UNFPA, the United Nations sexual and reproductive health agency, will be visiting the Democratic Republic of the Congo from 12 May.
This is the first mission of the UNFPA Executive Director and UN Under-Secretary General to the country. It will be an opportunity to meet with officials from the Government, the United Nations system and with members of civil society, development partners, communities, and women and young people.
The Executive Director's visit is a show of solidarity with the people of the DRC in their journey towards sustainable development. UNFPA stands for human rights and the dignity and safety of women and girls, and passionately believes that sustainable development and prosperity are only achievable when there is gender quality and women are empowered.
UNFPA supports the rights, safety and dignity of women and girls and equal opportunity for all, through a bold Call to Action with three goals in all 150 countries, including the DRC:
Ending preventable maternal deaths
Ending unmet need for family planning
Ending gender-based violence and harmful practices, including child marriage and female genital mutilation.
The Executive Director will visit the eastern part of the country, where she is to meet survivors of gender-based violence, and as champion of the Inter-Agency Standing Committee (IASC), she will advocate for the prevention of and response to GBV.
At a health centre in Kinshasa, the Executive Director will interact with clients and have an honest exchange with women leaders who play an important role in the fight for the rights of women and young girls.
During the trip, she will meet with the highest Congolese authorities and development partners. In addition, this will be an opportunity to support the Government's efforts in advancing social and economic development, including the Population and Housing Census (PHC2).
Siaka Traore – Communication Specialist, DRC
traore@unfpa.org
Tel: 00243 815 786 999
Daisy Leoncio – Regional Communications Adviser
Leoncio@unfpa.org
Video of UNFPA Transforms Women's Lives in Remote Itula, DRC
UNFPA Transforms Women's Lives in Remote Itula, DRC
The community of Itula, located in the lost valley of Shabunda, more than 3000 kilometres from the capital city Kinshasa, is no longer being left...
Video of TUZIKILIZANE FILM ENGLISH
Tusikilizane (Let's Listen to Each Other)
The marginalization of indigenous groups led to the outbreak of violence in the Democratic Republic of the Congo in 2016, displacing more than 300,...

redpajama_set_name: RedPajamaCommonCrawl


According to experts, is very important for children have spaces to play at home as they help to develop the kids' creativity. Experiences reveal that the most important thing is the space itself, the fact of having a spot at home so that children can play and explore their skills. For all these reasons, Kids Bedroom Ideas' editors have selected 25 cheerful kids playroom ideas that your children will love. Keep scrolling to get inspired and choose your favorite!
Remember, moms and dads: a good recommendation to follow is not overfilled the space with toys, instead you must consider having few things helps the children to appreciate and enjoy them. When space is saturated with things the main thing is the space so they can move, jump, jump, move, dance and be happy is lost. A child who enjoys what you have, learn to take care, to value and to enjoy it, but the most important thing for them is the time they spend with their mom and dad learning and having fun.


Please make sure that the parcel is marked with a clearly readable and correct consignee address.
In case of domestic and international parcels you are requested to add the consignee's telephone number to the consignee address.
Very important information is postal code. By the postal codes parcels are automatically sorted on sorting machine to which country and region parcels will be shipped.
Make sure twice that your written down postal code on label is correct!
As an obligatory is any additional important information regarding a shipment (for example, gate code) to secure fast and high quality delivery.If you do not want to give away such information, obligate is to write down on label receiver name, surname and phone number.


Medical insurance..it's one of those things students either avoid or ignore altogether when preparing to study abroad. I could tell you stories about the inconveniences of having to pay your medical costs up front or finding the closest hospital while you're traveling in a country where you don't speak the language. But instead I'll encourage you to just mull over this thought – you're in another country on the other side of the world, would you really want to break a leg, or worse, be close to death on a hospital bed without insurance?
Most states will require you to participate in a student health insurance program or in a health benefit plan while you're abroad. First check to see if your current plan or the plan you are under covers you abroad, and second, if it covers you beyond the basic trip to the ER. If not, here are a few steps for finding the right coverage.
Research indicates that most study abroad students don't have comprehensive insurance for travel outside the U.S. (even though they assume they do).
Parents' plans often offer limited coverage outside the U.S. or are restricted in the coverage length.
Students need to be aware of their limits of coverage (pre-existing conditions, deductibles, reimbursement, etc.).
Students should understand that medical care outside the U.S. may operate differently from those in the U.S. and aren't subject to the same rules and regulations.
Does the insurance plan meet all requirements of both the country you are traveling to and the program you are studying abroad through?
Are there specific health insurance requirements in order to get a visa to enter the country you are studying in?
Does the insurance provider limit or exclude coverage of any services or pre-existing conditions?
Are there limits on which doctors, hospitals or other medical facilities you can go to?
Will you have access to emergency assistance services or an emergency hotline that can direct you to the nearest medical facilities or help explain treatment options?
Does the insurance plan cover Emergency Medical Evacuation and Repatriation?
– Repatriation: As morbid as it sounds, should you die abroad, repatriation insurance covers the cost of transporting the body back to the U.S.
Do you have to pay for services up front and submit for reimbursement later?
What specific information is required from doctors or hospitals in order to be reimbursed?
Will you be required to submit all claims in English and in US dollars?
Look into whether or not your home college or university offers a student health insurance plan. This is pretty common among schools with well established and organized study abroad offices. More often than not, if your school does provide an insurance program, you'll actually have to physically sign a waiver if you don't want to participate in it.
Your ISIC provides basic sickness and accident coverage including emergency evacuation insurance, repatriation and accidental death or loss/ use of limb(s) coverage. A letter outlining the policy is included with the card.
And finally, make sure to take a copy of your health insurance card and important policy information with you both when you head to your abroad location as well as all other travels.
I can't stress enough how important having quality insurance is when studying abroad. I once had to make a trip to an international hospital in Beijing, and the cost for the visit plus my medication combined was over $200 – the equivalent of my round trip ticket to Shanghai. Luckily, my study abroad program had provided us with a comprehensive medical plan which covered the entire thing, but had they not, my travel fund would've taken a big hit. Give your wallet a break and put your mind at ease by selecting the right medical insurance when studying abroad.


You're just in time and I'm happy you've arrived. Let's keep each other company always, ok? That way we can always have each other's backs. And fronts and sides.
So what does living gracefully look like? Since officially becoming Grace last November (and before that really but even more so after November) I've asked myself that question multiple times a day – it was definitely the impetus for changing my name since I was not always able to be as graceful as I would've liked, heck, are any of us ever as graceful as we'd like to be? Well, perhaps not as much as we'd like to be but I imagine, in most cases, we are as graceful as much as we can be given the circumstance.
Graceful is defined as characterized by elegance or beauty of form, manner, movement, or speech; elegant. To live this way, regardless of your circumstance, is perhaps more idealistic than is reasonable for some, maybe many even, but I'm pretty sure it all just comes down to perspective. Also, striving for the ideal is within all of our abilities, regardless of your circumstance.
Of course when we look at elegance as part of gracefulness it's easy to see where there may be some divide in how what we see ourselves as being defined unfortunately affecting whether we can be indeed graceful. Leaving the superior, dignified, refined aspects of elegant aside, let's focus on the remarkably good part of elegance for just a moment – so much about us is that we are good, many times, remarkably so and we can hold on to that and work with that for the time being if that's all we've got.
Because then there's also beauty tucked in there and we all know about that being in the eye of the beholder so what if you don't feel beautiful, can you still be graceful, live life gracefully? Of course you can. We all can. We would ALL benefit from being graceful and over the coming days, weeks, etc. I'll be writing and encouraging you, giving examples and sharing my own experiences with you so that we can live gracefully, beautifully, and all the other good ways there are to live.


One of the best ways to learn about Carncot is to speak with someone who has experienced it for themselves. Click the email address below and we will put you in contact with one of our ambassadors – an existing parent – who will be able to talk to you about life at Carncot based on their direct experience.
Please download our application form and return it to the school. You should hear from us within two weeks. In the meantime, please do contact us if you have any questions. Of course, you are welcome to visit Carncot at any time.
Children learn in the same way as in high school, with specialist teachers in mathematics, art, PE, languages and music.


Knee arthroscopy is a minimally invasive technique used to treat knee conditions that historically were treated through larger open incisions. There are a variety of conditions that cause knee pain including tendon tears, meniscus tears, loose bodies within the knee area, and osteoarthritis. When non-surgical methods fail to solve the problem, surgical intervention may be the answer. Knee arthroscopy is an excellent technique in the general treatment option for knee injuries.
Knee arthroscopy is performed under general anesthesia. After adequate anesthesia has been administered, Dr. LaReau will use small incisions to gain access to your knee joint. The incisions are placed in specific locations to minimize the potential for injury to the surrounding nerves, blood vessels and tendons. Through one incisions, a camera is placed into the joint, through other incisions, small instruments can be used to address the problem. The length of the knee arthroscopic procedure varies depending on what Dr. LaReau identifies as he begins the procedure. Once the procedure has been completed, Dr. LaReau will indicate the rehabilitation plan for you and the repaired knee.
Dr. LaReau gave my life back to me when he performed the surgery to partially replace my knee. Before the surgery, it was difficult for me to walk any distance without severe pain. Now, thanks to Dr. LaReau, I am living my life normally once again. From my first visit to his office to my postoperative visits, he and his staff have been wonderful. They have been there to answer any questions and offer their support through the whole process and I can't thank them enough.


A resident takes a photograph of a back fire that burns near his home in Tujunga, Calif.
A Los Angeles City firefighter monitors a backfire in Tujunga, Calif.
A sign to thank firefighters is seen in an evacuated house in La Crescenta, near Los Angeles.
The wildfire is close to a house in La Crescenta.
A helicopter fights to save a house in La Crescenta.
People watch the wildfire in La Crescenta.
Wildlife Waystation founder Martine Colette is consoled by Los Angeles County fire captain Todd Emrick as she coordinates an evacuation of anilmals from Sylmar, Calif.
Firefighters control a fire next to houses in the suburb of Glendale on the outskirts of Los Angeles.
Firefighters control a fire burning behind houses in Glendale.
U.S. Forest Service hotshot crews stand on a hill monitoring a backfire in Sylmar, Calif.
Firefighters monitor a backfire in Sylmar, Calif.
A firefighter works to save a house in Glendale.
People watch the wildfire in Glendale.
Helicopters fight against the wildfire in Glendale.
The wildfire approaches houses in Glendale.
Firefighters arrive to protect houses in Glendale.
A firefighters' truck is decorated with a child's drawing in La Crescenta.
Firefighters observe the wildfire in Glendale.


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text:
David Bowie albums
Previous 1 2 3 4 5 6 7 8 9 10
The 1980 Floor Show
Lyrical themes
Diamond drugs
Exit Ken Scott
The Astronettes
The mixing
The gatefold
The release
Reissues, remixes, remasters
The precise release date of Diamond Dogs is somewhat unclear. Some sources claim it as 24 April 1974, with others having it as 31 May. However, in 2014 Bowie's own website put the UK release date at 24 May, citing a contemporary release sheet.
The album was given a $400,000 promotional campaign in America, which included huge billboards in Times Square and Sunset Boulevard, subway posters, adverts in the press, and a made-for-television promo clip.
Diamond Dogs topped the album charts in the United Kingdom and Canada, and peaked at number five in America – Bowie's highest placing there to date.
The album was first released on compact disc in 1985 by RCA, and featured censored artwork.
Different masters were used for the German pressing, which was issued in Europe, and Japan, which was sold in America.
Diamond Dogs was remastered ahead of a 1990 reissue by Rykodisc/EMI. This version contained two bonus tracks: 'Dodo' and the demo of 'Candidate', both of which were previously unreleased.
Looking back in 1990, Bowie said of the Rykodisc reissues:
I find the Ziggy Stardust record very thin. I don't like the sound on that, it's much thinner than I always thought it was. It sounded really powerful then; maybe systems have got better, it sounds kind of weedy. I thought Diamond Dogs sounds really good, I like that one, and I'll always love the Eno/Fripp/Belew stuff.
Q magazine, April 1990
A 1999 edition was remastered by Peter Mew at Abbey Road Studios as part of The David Bowie Series, and contained no bonus tracks.
2004 saw the third and last in a short series of 30th anniversary reissues, following The Rise And Fall Of Ziggy Stardust And The Spiders From Mars and Aladdin Sane.
This version had a second CD with eight bonus tracks: '1984'/'Dodo'; 'Rebel Rebel' (single version); 'Dodo'; 'Growin' Up'; 'Candidate'; 'Diamond Dogs' (K-tel edit); 'Candidate' (Intimacy mix, 2001); and 'Rebel Rebel' (2003 version, from the Charlie's Angels: Full Throttle soundtrack).
Diamond Dogs was remastered once again for 2016's Who Can I Be Now? (1974–1976) box set, and released separately on CD, vinyl, and as a digital download. The book which came with the box set also contained notes and recollections by Tony Visconti.
Diamond Dogs was probably David's most radical album to date. It was very different from anything he'd done, it was the first self-produced album too. He played almost all the guitar on the album except for the 'Rebel Rebel' riff. Between us we conjured up sounds no one had ever heard or used before. It was the beginning of many new recording adventures I shared with my old friend, lasting until the present day.
Tony Visconti, May 2016
Who Can I Be Now? book
Somebody Up There Likes Me
David Bowie and drugs
Welcome to the David Bowie Bible!
★ (Blackstar)
David Bowie (1967)
1.Outside
Contact | Privacy policy | About us
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The Bowie Bible is run for the love of anything and everything to do with David Bowie. If you've learned something new today and wish to show your appreciation, why not make a small donation via PayPal? It'll help with server costs, research material etc...
Love the Beatles? Of course you do. Don't miss our sister site, the Beatles Bible!
The Bowie Bible Copyright © 2021.
All rights reserved. All images © their respective owners. This unofficial site is in no way associated with David Bowie, Columbia, Sony, EMI, related and unrelated organisations or their representatives.


Everything You Need to Know About COBRA Health Insurance | WEX Inc.
Most people don't need to know about COBRA continuation health coverage until they've lost employer-based health insurance coverage—an often-unexpected occurrence. At that point, they need to know what it is, what it covers and for how long. Not to worry. We've got you covered with straightforward answers to all of your COBRA questions.
When you lose employer-based insurance because you've lost or quit your job, aged out of your parents' health insurance or gotten a divorce, you may be eligible for COBRA continuation coverage, which allows covered employees, their spouses, former spouses and dependent children to temporarily keep the same health plan, benefits and services you had before. Your employer should send you and your family a notice that you are eligible for COBRA, outlining your COBRA rights and how much it will cost.
In 1985, COBRA amended the Employee Retirement Income Security Act, the Internal Revenue Code and the Public Health Service Act to require group health plans to provide a temporary continuation of group health coverage that otherwise might be terminated. The law applies to all group health plans offered by private-sector employers with 20 or more employees as well as by state or local governments. The U.S. Department of Labor oversees COBRA compliance. The Departments of Labor and Treasury have jurisdiction over private-sector group health plans, while the Department of Health and Human Services administers the continuation coverage law as it applies to state and local governmental health plans.
Consolidated Omnibus Budget Reconciliation Act. Yep, there's a reason we just call it COBRA.
How long do I have to decide if I want COBRA coverage?
You will have an election period of at least 60 days to decide whether to continue your health plan with COBRA or to seek coverage through the Health Insurance Marketplace, through a spouse's employer or Medicaid.
What counts as a qualifying event for COBRA?
It costs the same as your previous monthly health plan, plus a 2 percent service charge.
However, because you will now be responsible for the premiums that were once subsidized by your employer, your parents' employer or your spouse's employer, the monthly cost shouldered by the consumer are much larger. On average, employers pay nearly 82 percent of the cost of their employees' health insurance, and nearly 70 percent of the total family premium, so this means COBRA insurance can be up to 82 percent costlier for a consumer than their previous monthly payment.
By law, COBRA continuation coverage can extend from between 18 to 36 months, depending on the type of qualifying event. Certain plans may provide longer periods of coverage.
How does WEX Health make COBRA benefits easier to access and use?
The WEX Health COBRA & Direct Billing program is the health benefit industry's leading enterprise-class central system of record for COBRA, member billing and state continuation administration.
In June 2018, we introduced new features and functionality to our WEX Health Cloud platform; this included rolling out the WEX Health Cloud COBRA & Direct Billing Mobile App, which provides consumers ease of use and quick accessibility to several convenient WEX Health COBRA & Direct Billing features, including: online registration and election, credit/debit card payments, recurring ACH and one-time checking/savings payments. The app, which uses PIN security, enables consumers to view communications as well as payment history.


text:
Trump Puts an End to Obama's Not So "Fair Housing" Rule Leaving Biden in Shock
It was recently announced that if Democratic presidential candidate Joe Biden wins the White House in November, he will seek to add on to Obama's Affirmatively Furthering Fair Housing rule that was added to the Fair Housing Actin in 2015.
In doing so, Biden would permanently be destroying the suburbs of America. However, I am here to tell you that as of Thursday morning, that is no longer possible, thanks to President Trump.
No, Trump has not figured out a way to ensure himself a win in the upcoming election. But he has put an end to the entire rule, barring Biden from expanding on it in any way, should he be elected.
The plan for such an end to the rule was given Wednesday night in an exclusive interview of Trump's Housing and Urban Development (HUD) with the Daily Caller. In the interview, the department's secretary, Ben Carson, describes the recently ended program and why the Trump administration was so against its continuation.
It was determined that Obama's administration added the AFFH rule in an effort to hold local HUD grantees accountable for how they were using the funds given to them by the federal government. This was mainly done by "producing and maintaining assessments of fair housing practices." However, having such assessments completed meant that 64 new employees had to be added to the HUD payroll, which, as the Daily Caller notes, "cost American taxpayers up to $15 million a year."
In addition, having such employees meant that those local communities were not really involved in their own governing. Instead, it was federally paid employees who were now assessing and determining where and who would receive housing funds.
Furthermore, with the high costs of receiving, producing, and delivering these specific assessments to HUD officials, as well as getting them through "zoning changes to maintain compliance," many local jurisdictions, and often the ones needing the funds the most, couldn't afford to even use the program. And so the funds were given to other areas, you know, wealthier regions just looking for some extra cash.
President Trump, with the help of those like Secretary Carson, decided this was unfair to low income communities and doing much more harm than good.
Secretary Carson told the Daily Caller on Wednesday, "In nearly every case, it is a fact that local governments are more adequately equipped to deal with their community's unique needs than any unelected bureaucrat in Washington."
And I have to agree. Who knows what the community needs, where low-income housing should be created, and to what extent, better than those who live in that community? Some "unelected bureaucrat in Washington," as Carson says, many who have never even traveled to the location they are doling out funds and assessments for, have no idea what the needs are. And therefore, they shouldn't be the ones calling the shots.
And then there is the threat that Biden posed to those communities, as well as the suburbs of America.
Biden, as a staunch believer that anything Obama did was holy and, therefore, should be continued, has made it clear he would not have only kept this program intact but would have expanded it to include most American suburbs. Suburban areas would then be under urban control. Those who live there would no longer have a say in communities. Instead, it would be federal government officials doing it for them.
Just another example of how the socialist left is pushing for more government and less "for the people" decisions.
As the man who is believed to have first learned about the recent change in policy, Stanley Kurtz, noted, Biden's extended influence on the program would essentially "abolish the suburbs." Kurtz said that making this a vital issue in Trump's campaign could be a game-changer for him, forcing "Democrats to either retreat from this attempt to put suburbs under urban control or lose the suburban voters that gave them their House majority."
In either case, it would be a win for Trump and the Republicans.
As Senator Mike Lee says of Trump's HUD decision, "Every American should be free to choose where to live, and every community should be free to compete for new residents as they see fit. What our communities do not need is another costly government mandate that makes it harder for local communities to govern. The reversal of President Obama's burdensome housing rule is a strong step towards more affordable housing."
Reliable.dev
Still unknown to many is a brilliant government sponsored mortgage program called the High LTV Refinance Option (HiRO) that could benefit millions of Americans and reduce their payments by as much as $3,000 per year! You can bet the banks aren't too thrilled about losing all that profit and might secretly hope homeowners don't find out before the.....
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People's driving habits are changing and insurance companies have been forced to drop their prices– don't miss out. If you have three years of clean driving record, you could be overpaying by up to 70% for car insurance. The best way to find out if you're paying too much is to get quotes from multiple insurers.....
Make Sure You're Not Overpaying for Car Insurance
Most people overpay for car insurance and don't realize it. Even if you shop around comparing rates from the 10 best companies, there's still ONE easier way to cut your rates way down! If you don't work far from home and simply spend less time on the road than the typical road warrior, then ONE.....
Set Your Car Insurance By‑the‑Mile
Seniors who live in qualified zip codes across the country are slashing their monthly electric bill by taking advantage of these special programs and going solar. With new tax credits and state incentive rebate programs available, you can upgrade to solar for $0 down and no out of pocket expenses. This makes going solar a.....
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Casey Neill has the kind of voice that makes you stop what you are doing. It's timeless, like something that might drift out of a jukebox in a roadside tavern somewhere in Middle America. But there are other qualities there that make it truly unmistakable: hints of Michael Stipe's vibrato tenor, some alt. country inflections of Jay Farrar and the occasional whiskey-soaked growl of Shane McGowan.
I caught a show by Casey and his band, The Norway Rats, at the Laurelthirst Public House one Wednesday as part of his month-long happy hour residency at Portland's favorite Americana pub. I've been listening to him since college, and have come to expect a mellow, easy-on-the-ears folk/country sound (with an occasional rocker thrown in); essentially a predictable mix best characterized by his 2007 album, Brooklyn Bridge. The 2007 album was recorded mostly in New York City, and documents his stint in Brooklyn, while also placing his songwriting and vocals at center stage.
As I should have expected, a lot has changed for Casey since those days in New York. His live show at the Laurelthirst was anything but mellow. Since coming back to Portland and re-establishing his roots a few years ago, Casey put together a super-group of the city's finest folk/Americana musicians—including Chris Funk and Jenny Conlee of the Decemberists, and Scott McCaughey of Minus 5. While not all the household names where there last Wednesday night, the band played a rousing set of rockers balanced by a few old favorites (see "Riffraff") and some folksy solos.
Neill embraced the talents of some of these same stellar musicians on his 2010 release "Goodbye to the Rank and File" with a nice blend of Celtic punk, rockabilly and folk. Neill's literate sensibilities ground each song in memorable images and turns of phrase. It's a nostalgic homage to a country life that may have never been, and puts you inside the cool silver waters of the quarry with a young lover on a hot summer night.
Look for his newest record "All You Pretty Vandals" to be released this year, produced by Chris Funk of the Decemberists.
In the meantime, catch one of his upcoming shows in March featuring his Pogues tribute band K.M.R.I.A. (And what does that stand for, you say? Kiss My Royal Irish A**, a phrase used in James Joyce's "Ulysses" and in the Pogues' song "Transmetropolitan").
Fri. March 16: K.M.R.I.A. – SEATTLE, WA @ The High Dive 9:30pm 513 N 36th St.
Sat. March 17: The Norway Rats – PORTLAND, OR @ McMenamins Kennedy School Gym. 4 p.m. Free.


text:
\section{#1}}
\oddsidemargin 4mm
\evensidemargin -8mm
\begin{document}
\parskip=2mm
\topmargin -10mm
\pagestyle{plain}

\begin{tabbing}\hspace{12cm}\={\small preprint UTF-397}\\
\> gr-qc/9704077\\
\> April 1997
\end{tabbing}
\vspace{1cm}

\begin{center}
{\Large\bf Massive scalar field
near a cosmic string}
\vspace{5mm}

{\Large Devis Iellici\footnote{Electronic address:
iellici@science.unitn.it}} \\
\vspace{3mm}

{Dipartimento di Fisica, Universit\`a di Trento\\
and Istituto Nazionale di Fisica Nucleare,\\
Gruppo Collegato di Trento, Italia}
\end{center}
\vspace{1cm}
\begin{center}
\parbox{13cm}{\small{\bf Abstract.}
The $\zeta$ function of a massive scalar field near a cosmic string is
computed and then employed to find the vacuum fluctuation of the
field. The vacuum expectation value of the energy-momentum tensor is
also computed using a point-splitting approach. The obtained results
could be useful also for the case of self-interacting scalar fields
and for the finite-temperature  Rindler space theory.}
\end{center}

\hspace{10mm}{\small PACS number(s): 04.62.+v, 04.70.Dy }

\vspace{1cm}

\begin{center}
{\large\bf I. Introduction}
\end{center}

In the last few years many authors have considered quantum fields on
spacetimes containing conical singularities. These kind of manifolds
are relevant for studying fields in presence of real conical
singularities, namely existing in the Lorentzian section of the
manifold, as in the case of idealized cosmic strings and of
2+1-dimensional gravity. In other cases,  conical singularities appear
in the finite temperature theory, when adopting the Euclidean path
integral with periodic imaginary time formalism. It is the case of
spacetimes containing horizons, such as the Rindler and the
Schwarzschild spaces, which are much studied in the off-shell
approach to the one-loop quantum corrections to the Bekenstein-Hawking
entropy. It is worth noticing that, in these latter cases, the conical
singularity could be avoided adopting a canonical approach.

As a consequence of this interest in conical spaces, many techniques
have been developed to compute relevant quantities in presence of
conical singularities (see, among the others,
\cite{DO77,cheeger83,DO87a,DesJack88,AlOt90,GiRuVa90,KS91,CKV94,fursaev94,FUSOL95,ZCV}).
With few exceptions
\cite{CKV94,fursaev94,linet87,ShiraHire92,GuimLin93,GuimLin94,moreira95},
most authors considered massless fields only. Indeed, the
introduction of the mass complicates considerably the problem, so that
it becomes very difficult to obtain manageable forms for the physical
quantities. At this regard, it has to be noted that using the
heat-kernel approach \cite{CKV94,fursaev94} it seems that all the
problems vanish: in fact, for a scalar field of mass $m$ the
integrated heat-kernel  is simply related to the massless one as
\begin{eqnarray}
 K_t^{m}=e^{-tm^2}K_t^{m=0},
\label{heat}
\end{eqnarray}
and, at least for a simple cone $R^{D-2}\times C_\alpha$,
the integrated massless heat-kernel is well known
\cite{cheeger83,CKV94,fursaev94}:
\begin{eqnarray*}
K_t^{m=0}=\frac{\Sigma_{D-2}}{(4\pi t)^2}\left[\frac{V(C_\alpha)}
{4\pi t}+\frac{1}{12}\left(\frac{2\pi}{\alpha}-\frac{\alpha}{2\pi}
\right)\right],
\end{eqnarray*}
where $2\pi-\alpha$ is the deficit angle of the cone, $V(C_\alpha)$ is
the volume of the cone and $\Sigma_{D-2}$ is the volume of the
transverse dimensions. On the contrary of the massless case, the
Mellin transform of Eq. (\ref{heat}) exists in the massive case, and
so it is easy to compute the massive global $\zeta$ function and from
it quantities like the effective action, which have a simple closed
form. However, it is well known \cite{ZCV} that this procedure does
not yield the correct dependence on the deficit angle of the cone in
the massless limit and, moreover, these global quantities are finite,
apart for the volume divergences, while the local quantities show a
non-integrable singularity near the tip of the cone.\footnote{In the
cosmic string case, this problem could be overcome renormalizing the
action of the string as in \cite{fursaev94}.} Therefore, it is at
least dubious that this simple result for the massive case is correct.
Besides, in order to compute important quantities such as the vacuum
fluctuation of the field  and the stress tensor, local rather than
global quantities are needed.

The aim of this work is to give a more manageable tool to compute
local quantities for a massive scalar field around a cosmic string,
and we will accomplish this by computing the local $\zeta$ function,
which will be given as an expansion in powers of $mr$, where $r$ is
the distance form the string core. Therefore, the obtained expression
is useful near the string, where $mr\ll 1$.\footnote{Note that
$r=\hbar/mc$ is the Compton wavelength of the particle. When the
result are translated for the Rindler case, the approximation is valid
close to the event horizon.} However,
since the local quantities diverge at the string core, this is just the
interesting region if, for instance, one wants to consider the
back-reaction of the energy-momentum tensor on the background
metric \cite{Hiscock87,MEX97}. The expression we obtain is simple
enough to allow us to compute the renormalized vacuum expectation
value $\langle \phi^2(x)\rangle$ and the the energy-momentum tensor,
potentially up to arbitrary order in $(mr)^2$.
Notice that the most common tool used to compute local quantities is
the Green's function, which in the massive case is given in a
complicated integral  form \cite{linet87,ShiraHire92,GuimLin94,CKV94}
or as a sum of generalized hypergeometric functions
\cite{moreira95}.

The capability of treating the massive case is  useful also when
dealing with self-interacting theories, such as $\lambda \phi^4$. In
fact, consider an Euclidean path-integral approach to the theory of a
self-interacting scalar field: the generating functional of the theory
is the given by
\begin{eqnarray*}
Z&=&\int{\cal{D}}\phi\exp(-S[\phi]),\\
S[\phi]&=&\int d^Dx\sqrt{g}\left[\frac{1}{2}\partial_\mu\phi
\partial^\mu\phi+V(\phi)\right],
\end{eqnarray*}
where $V(\phi)$ is a potential. In the usual $\phi^4$-theory, it has
the form $V(\phi)=\frac{1}{2}m^2\phi^2+\frac{1}{4!}\lambda\phi^4$,
$\lambda$ being the coupling constant. In the one-loop perturbative
approach, the above action is expanded near the minimum $\hat\phi$
of the action up to second order terms
$$
S[\hat\phi+\varphi]=S[\hat\phi]+\frac{1}{2}\int d^Dx\sqrt{g}\,
\varphi[-\Delta+M^2]\varphi,
$$
where $\Delta$ is the Laplace-Beltrami operator and
$M^2=V''(\hat\phi)$. In the $\phi^4$-case, we have that
$M^2=m^2+\frac{1}{2}\lambda{\hat\phi}^2$.
Then, the standard step is to assume $\hat\phi$ is a constant
configuration, so that standard passages lead to effective
action
$$
W=-\ln Z=S[\hat\phi]+\frac{\hbar}{2}\ln\det[-\Delta+M^2]+
{\cal O}(\hbar^2),
$$
where, in general, $S[\hat\phi]$ is proportional to the volume of the
spacetime. The determinant in the second term is ultraviolet
divergent and it has to be defined by means of some regularization
procedure. Convenient ways to regularize the above determinant are,
for example, the heat-kernel plus Schwinger's proper-time
regularization or the $\zeta$-function regularization.

{}From the above outline, it is clear that the one-loop self-interacting
case is closely related to the massive case, and, in particular, the
regularization of the determinant is the same, provided that $m^2$
is replaced by $M^2$. Of course, the renormalization of the effective
action to remove the ultraviolet divergences is different. In this
paper we shall not discuss the self-interacting case and its
renormalization explicitly, but we observe that in the cosmic string
case, in which $\langle \phi^2(x)\rangle\propto r^{-2}$, it could be
more reliable a `Hartree-like' approximation as that employed in
\cite{ShiraHire92} and in which in the action $\lambda\phi^4$ is
replaced by $\lambda\langle \phi^2\rangle\phi^2$.

As we said above, we are mainly concerned in the comic-string
background. We remind that the space-time around an infinitely long,
static and straight cosmic string \cite{vilenk85} has the topology
${\cal M}^4=R^2\times C_\alpha$, where $C_\alpha$ is the simple
two-dimensional cone with deficit angle $2\pi-\alpha$, and the metric
is
\begin{eqnarray}
ds^2=-dt^2+dz^2+dr^2+r^2 d\theta^2,\hspace{1cm} r\in(0,+\infty),\,
\,\theta\in[0,\alpha],\, \,t,z\in(-\infty,+\infty).
\label{metric}
\end{eqnarray}
The polar angle deficit $2\pi-\alpha$ is related to the mass per unit
length of the string $\mu$ by $2\pi-\alpha=8\pi G\mu$. Throughout this
paper we shall assume positive deficit angle, so that
$\nu\equiv2\pi/\alpha>1$. For GUT strings $\nu-1\sim 10^{-6}$
\cite{vilenk85}. In the rest of the paper we  adopt an Euclidean
approach and so we perform a Wick rotation $\tau=it$: the form of the
metric is the same as above with the replacement
$-dt^2\rightarrow+d\tau^2$. Moreover, for technical reasons we will
also consider the obvious generalization to the $D$-dimensional case,
with topology $R^{D-2}\times C_\alpha$.

We have assumed zero thickness for the string: this is clearly an
idealization, since an actual cosmic string should have a small  but
finite radius, of the order of the Compton wavelength of the Higgs
boson involved in the phase transition which gives rise to the cosmic
string \cite{vilenk85}. The internal structure of the string has
non-negligible effects even at large distances, as it has been shown
by Allen, Kay, and Ottewill \cite{AlKaOt96}. However, in the same work
it has been shown that for a minimally coupled scalar this dependence
on the internal structure is absent, and one can safely use the
idealized string. In order to avoid the complications discussed in
\cite{AlKaOt96}, which are related to the non-uniqueness of the
self-adjoint extension of the Laplace-Beltrami operator in the
idealized conical space-time, we shall consider the minimally coupled
case only. Actually, in Sec. IV we will compute the stress tensor
and, for sake of completeness, we will consider arbitrary coupling.
Therefore,  we have to remember that for $\xi\neq 0$ others effects
could be present in a realistic cosmic string.

Although the results we will obtain are easily translated to the
case of fields at finite temperature in the Rindler space
\footnote{The Euclidean metric of the cosmic string and
of the finite-temperature Rindler space are in fact the same.
In this latter case, $\theta$ corresponds to the periodic
Rindler time and the period $\alpha$ is the inverse temperature.
The absence of the conical singularity fixes the Unruh-Hawking
temperature $T=1/2\pi$ in suitable units \cite{birrel}.}
or near the horizon of a Schwarzschild black hole, we prefer
to consider the cosmic string background only. This because
the Euclidean Rindler space or Schwarzschild black hole
show a conical singularity only when the temperature is
different from the Unruh-Hawking one,  namely when one
considers the `off-shell' theories, but it has been shown that the
off-shell quantum states of a field are affected by several
pathologies on the event horizon \cite{HaNaSt84} and that these
theories do not have a consistent thermodynamics \cite{moiel}.

The rest of this work is organized as follows. In Sec. II we compute
the $\zeta$ function of a massive scalar field in the cosmic string
background. Then we use this result to compute, in the region $mr\ll
1$, the vacuum fluctuation of the field in Sec. III and the expectation
value of the stress tensor in Sec. IV. Section V contains the
conclusions. In the rest of the the paper we use units
for which $\hbar=c=G=1$.

\begin{center}
{\large\bf II. Computation on the $\zeta$ function}
\end{center}

We consider a quasi-free real scalar field in the background given by
the (Wick-rotated) metric (\ref{metric}). The action of the theory is
the given by
\begin{eqnarray*}
S[\phi]=\int d^Dx\sqrt{g}\frac{1}{2}\phi A\phi,
\end{eqnarray*}
where $A$ is known as the small fluctuation operator and in our
case reads
\begin{eqnarray}
A=-\Delta_D+m^2+\xi R.
\end{eqnarray}
Here $\xi$ is a parameter which fixes the coupling of the field to the
gravity by means of the scalar curvature $R$. We shall consider the
minimally coupled case, $\xi=0$. Then, by means standard passages
one sees that the generating functional of the theory can be expressed
in terms of the functional determinant of the small fluctuations
operator, which can be conveniently defined by means of the $\zeta$
function regularization \cite{hawking77}:
$$
\ln Z_\alpha=-\frac{1}{2}\ln\det\mu^{-2}A=\frac{1}{2}
\zeta'(0|A\mu^{-2})
$$
where $\zeta(s|A)$ is the global $\zeta$ function related to the
operator $A$, the prime indicates the derivative with respect to $s$
and $\mu$ is an arbitrary parameter with the dimensions of a mass
needed for dimensional reasons and not to be confused with the mass
per unit length of the string. The global $\zeta$ function can be
formally written as the integral over the manifold of a local $\zeta$
function $\zeta(s|A)(x)$:
\begin{eqnarray*}
\zeta(s|A)&=&\int_{{\cal M}^D_\alpha}\zeta(s|A)(x)\sqrt{g} d^Dx.
\end{eqnarray*}
Actually, when the manifold is non-compact only the local $\zeta$
function has a precise mathematical meaning, since the integration
requires the introduction of cutoffs or smearing functions to avoid
divergences.

In the massless case, the $\zeta$ function on the cone has been
explicitly computed in \cite{ZCV}. In order to compute it in the
massive case, the starting point is the well known relation between
the local $\zeta$ function and the heat kernel given by the Mellin
transform:
\begin{eqnarray*}
\zeta^{m}(s;x)=
\frac{1}{\Gamma(s)}\int_0^\infty dt\,t^{s-1}e^{-tm^2}K_t^{m=0}(x).
\end{eqnarray*}
The heat kernel of the massive field is related to the massless
one by means of the following obvious decomposition:
\begin{eqnarray*}
K_t^{m}(x)=e^{-tm^2}K_t^{m=0}(x).
\end{eqnarray*}
Now we take into account the following property of the Mellin
transform of the product of two functions \cite{report}
\begin{eqnarray*}
\int_0^\infty t^{s-1}f(t)g(t)dt=
\frac{1}{2\pi i}\int_{\sigma-i\infty}^{\sigma-i\infty}
F(z)G(s-z)dz,
\end{eqnarray*}
where $F$ and $G$ are the Mellin transforms of $f$ and $g$
respectively and $\sigma$ is a real number in the common strip of
convergence of the two  Mellin transforms. In this way we can write
the massive $\zeta$ function of  in terms of the massless one:
\begin{eqnarray}
\zeta^{m}(s;x)=\frac{1}{2\pi i \Gamma(s)}
\int_{\sigma-i\infty}^{\sigma-i\infty}
\Gamma(z)\zeta^{m=0}(z;x)m^{2z-2s}\Gamma(s-z) dz,
\label{general}
\end{eqnarray}
where we have used the Mellin transform
\begin{eqnarray*}
\int_0^\infty t^{s-1}e^{-tm^2}dt=m^{-2s}\Gamma(s),
\end{eqnarray*}
which converges for $\mbox{Re} s>0$.

Since we are interested in the $\zeta$ function of a massive scalar
field in the (Euclidean) space $ R^{D-2}\times C_\alpha$, we consider
the corresponding massless $\zeta$ function. It has been shown in
\cite{cheeger83} and \cite{ZCV} that it is the sum of two parts which
converge in separate strips of the complex $s$-plane and which are
summed only after their analytic continuation:
\begin{eqnarray*}
\zeta^{m=0}(z;x)=\zeta_<(s;x)+\zeta_>(s;x),
\end{eqnarray*}
where
\begin{eqnarray*}
\zeta_<(s;x)&=&\frac{r^{2s-D}}{(4\pi)^{\alpha\frac{D-2}{2}}
\Gamma(s)} \frac{\Gamma(s-\frac{D-1}{2})
\Gamma(\frac{D}{2}-s)}{2\sqrt{\pi}\Gamma(s-\frac{D-2}{2})},
\hspace{1cm}\frac{D-1}{2}<\mbox{Re} s<\frac{D}{2},\\
\zeta_>(s;x)&=&\frac{r^{2s-D}}{\alpha(4\pi)^{\frac{D-2}{2}}
\Gamma(s)} \frac{\Gamma(s-\frac{D-1}{2})}{\sqrt{\pi}}
G_\alpha(s-{\scriptstyle\frac{D-2}{2}}),
\hspace{1cm}
\frac{D}{2}<\mbox{Re} s<\frac{D}{2}+\nu,
\end{eqnarray*}
and the function $G_\alpha(s)$ is defined as
\begin{eqnarray*}
G_\alpha(s)=\sum_{n=1}^{\infty}
\frac{\Gamma(\nu_n-s+1)}{\Gamma(\nu_n+s)},
\hspace{1cm}\nu_n=\frac{2\pi}{\alpha}|n|,\hspace{0.5cm}
\nu\equiv \nu_1
\end{eqnarray*}
and can be analytically continued in the whole complex plane showing
simple poles in $s=1$, with residue $\alpha/4\pi$, and $s=\nu_n+k+1$,
$k=0,1,2,\dots,$ (if $\alpha\neq2\pi$) with obvious residue. Since
$\zeta_<$ and $\zeta_>$ do not have a common strip of convergence,
we must split Eq. (\ref{general}) in two parts: setting $\mbox{Re}
s>D/2$ we have
\begin{eqnarray*}
\zeta_>^{m}(s;x)&=&\frac{1}{2\pi i\Gamma(s)}
\frac{r^{-D}m^{-2s}}{(4\pi)^{\frac{D-2}{2}}\sqrt{\pi}\alpha}
\int_{\scriptsize \frac{D}{2}<{\mbox{\scriptsize Re}\,} z<
{\mbox{\scriptsize Re}\,} s}
\hspace{-8mm}dz\,\Gamma(z-{\scriptstyle\frac{D-1}{2}})
G_\alpha(z-{\scriptstyle\frac{D-2}{2}}) \Gamma(s-z)(mr)^{2z}
\nonumber\\
&=&\frac{r^{2s-D}}{(4\pi)^{\frac{D-2}{2}}\sqrt{\pi}\alpha\Gamma(s)}
\left\{\sum_{n=0}^\infty\frac{(-1)^n}{n!}(mr)^{2n}
\Gamma(s+n-{\scriptstyle\frac{D-1}{2}})
G_\alpha(s+n-{\scriptstyle\frac{D-2}{2}})\right.
\nonumber\\
&&+\left.(mr)^{D-2s}
\sum_{n=1}^\infty\sum_{k=0}^\infty\frac{(-1)^k}{k!}
\frac{\Gamma(\nu_n+k+1/2)}{\Gamma(2\nu_n+k+1)}
\Gamma(s-\nu_n-k-{\scriptstyle\frac{D}{2}})
(mr)^{2\nu_n+2k}\right\},
\end{eqnarray*}
where we have performed the integral shifting the integration path to
the right and picking up the residues of the poles of the integrand.
The same procedure can be applied to $\zeta_<^{m}(s;x)$, but in this
case we have to consider the poles in $z=s+n$ and $z=2+n$:
\begin{eqnarray*}
\zeta_<^{m}(s;x)&=&\frac{1}{2\pi
i\Gamma(s)}\frac{r^{-D}m^{-2s}}{(4\pi)^{\frac{D-2}{2}}\sqrt{\pi}\alpha}
\int_{\scriptsize\frac{D-1}{2}<{\mbox{\scriptsize Re}\,} z<
{\mbox{\scriptsize Re}\,} \frac{D}{2}}
\hspace{-5mm}dz\,\frac{\Gamma(z-\frac{D-1}{2})
\Gamma(\frac{D}{2}-z)}
{2\Gamma(z+1-D/2)}\Gamma(s-z)(mr)^{2z}\nonumber\\
&=&\frac{r^{2s-D}}{(4\pi)^{\frac{D-2}{2}}\sqrt{\pi}\alpha\Gamma(s)}
\left\{\sum_{n=0}^\infty\frac{(-1)^n}{n!}(mr)^{2n}
\frac{\Gamma(s+n-\frac{D-1}{2})\Gamma(\frac{D}{2}-s-n)}
{2\Gamma(s+n+1-D/2)}\right.
\nonumber\\
&&\left.+(mr)^{D-2s}\sum_{n=0}^\infty\frac{(-1)^n}{n!}(mr)^{2n}
\frac{\Gamma(n+1/2)\Gamma(s-n-D/2)}{2\Gamma(n+1)}\right\}.
\end{eqnarray*}
Now we analytically continue each term and sum to get the final
expression of the local $\zeta$ function of a massive scalar field:
\begin{eqnarray}
\zeta^{m}(s;x)&=&\frac{r^{2s-D}\mu^{2s}}{(4\pi)^{\frac{D-2}{2}}
\alpha\Gamma(s)}\left\{\sum_{n=0}^\infty\frac{(-1)^n}{n!}(mr)^{2n}
I_\alpha(s+n-{\scriptstyle\frac{D-2}{2}})
\right.\nonumber\\
&&\hspace{-11mm}+\frac{(mr)^{D-2s}}{\sqrt{\pi}}
\sum_{n=0}^\infty\frac{(-1)^n}{n!}(mr)^{2n}
\frac{\Gamma(n+1/2)\Gamma(s-n-D/2)}{2\Gamma(n+1)}\nonumber\\
&&\hspace{-11mm}\left.+\frac{(mr)^{D-2s}}{\sqrt{\pi}}
\sum_{n=1}^\infty\sum_{k=0}^\infty\frac{(-1)^k}{k!}
\frac{\Gamma(\nu_n+k+1/2)}{\Gamma(2\nu_n+k+1)}
\Gamma(s-\nu_n-k-D/2)
(mr)^{2\nu_n+2k}\right\},\nonumber\\
&&
\label{zetafun}
\end{eqnarray}
where the function $I_\alpha(s)$ is defined as \cite{ZCV}
\begin{eqnarray}
I_\alpha(s)&=&\frac{\Gamma(s-1/2)}{\sqrt{\pi}}
\left[G_\alpha(s)+\frac{\Gamma(1-s)}{2\Gamma(s)}\right],
\nonumber\\
I_\alpha(0)&=&\frac{1}{6\nu}\left(\nu^2-1\right),\nonumber\\
I_\alpha(-1)&=&\frac{1}{90\nu}\left(\nu^2-1\right)
\left(\nu^2+11\right),
\label{ibeta}
\end{eqnarray}
The function $I_\alpha(s)$ is analytic in the whole complex plane but
in $s=1$, where it has a simple pole. Near the pole we have
$$
I_\alpha(s)=\frac{1}{2(s-1)}\left(\nu^{-1}-1\right)+
\frac{1}{2}\left(\nu^{-1}-1\right)\left(\gamma-2\ln 2\right)
-\frac{\ln\nu}{\nu}+{\cal {O}}\left((s-1)^2\right),
$$
where $\gamma$ is the Euler's constant. In expression (\ref{zetafun})
we have also reintroduced the arbitrary mass $\mu$ using the formal
relation $\zeta(s|A\mu^{-2})=\mu^{2s} \zeta(s|A)$, and which has been
omitted in the derivation for simplicity. It must be noted that the
above $\zeta$ function can be obtained in a more direct way, albeit
much longer, performing the integrations in the Mellin transform of
the spectral representation of the massive heat kernel, and then
rearranging the sums and the Euler gamma functions in the generalized
hypergeometric functions obtained by the integrations. We prefer the
used method since it can be more easily applied to the computation of
others quantities, as we will see in the next sections.

Although expression (\ref{zetafun}) looks awful, it is very simple
when $mr\ll 1$: considering the physically interesting case
$D=4$ and terms up to\footnote{Note that we are interested in the
values of the $\zeta$ function at $s=0,1$, and we consider positive
deficit angles only, so that $\nu>1$.} $(mr)^4$
\begin{eqnarray*}
\zeta^{m}(s;x)&=&\frac{r^{2s-4}\mu^{2s}}
{4\pi\alpha\Gamma(s)}\left[I_\alpha(s-1)-
(mr)^2 I_\alpha(s)+\frac{1}{2}(mr)^4 I_\alpha(s+1)+
{\cal O}\left((mr)^6\right)\right]\nonumber\\
&&+\frac{m^4(m/\mu)^{-2s}}{8\pi\alpha\Gamma(s)}
\left[\Gamma(s-2)-\frac{(mr)^2}{2}\Gamma(s-3)\right.\\
&&\left.+2(mr)^{2\nu}
\frac{\Gamma(\nu+1/2)\Gamma(s-\nu-2)}
{\sqrt{\pi}\Gamma(2\nu+1)}+{\cal O}\left((mr)^4\right)\right].
\end{eqnarray*}
In the first term of the first row we recognize the massless $\zeta$
function on $R^2\times C_\alpha$ \cite{ZCV}, while the first term in
the second row becomes the $\zeta$ function of a massive scalar field
in the Minkowski space-time when $\alpha=2\pi$ \cite{AAA95}. The
others terms are clearly corrections due to the presence of the
conical singularity. When the singularity is absent, namely when
$\alpha=2\pi$, the first row vanishes, since $I_{2\pi}(s)=0$, while in
the second and third rows only the first term survives, since the
others cancel two by two.

At the same order in $mr$, the effective lagrangian density is
given by
\begin{eqnarray*}
{\cal L}(x)&=&\frac{1}{2}\frac{d}{ds}\zeta^m(s;x)|_{s=0}\nonumber\\
&=&\frac{1}{8\pi\alpha r^4}\left\{I_\alpha(-1)
-(mr)^2 I_\alpha(0)\right.\nonumber\\
&&-\frac{(mr)^4}{2}
\left[\frac{\nu-1}{\nu}\left(\gamma+\ln\frac{{r\mu}}{2}\right)+
\frac{\ln\nu}{\nu}+\ln\frac{m}{\mu}-\frac{3}{4}\right]\\
&&\left.-\frac{(mr)^6}{24}\left[2G_\alpha(2)+2\gamma+
2\ln\frac{mr}{2}-\frac{11}{6}\right]+2(mr)^{2\nu+4}
\frac{\Gamma(\nu+\frac{1}{2})\Gamma(-\nu-2)}
{\sqrt{\pi}\Gamma(2\nu+1)}\right\}.
\end{eqnarray*}
where  $G_\alpha(n)$, $n\geq 2$, are readily computed being in the
region of convergence of the series defining $G_\alpha$. For example
\begin{eqnarray*}
G_\alpha(2)&=&-\frac{1}{2\nu}\left[2\gamma+
\psi\left(\nu^{-1}\right)+
\psi\left(-\nu^{-1}\right)\right],
\nonumber\\
G_\alpha(3)&=&-\frac{1}{6\nu}\left[3\gamma-
\psi\left(\nu^{-1}\right)-
3\psi\left(-\nu^{-1}\right)
+\psi\left(-2\nu^{-1}\right)\right],
\end{eqnarray*}
where $\psi(x)=\Gamma'(x)/\Gamma(x)$. In the limit
$\alpha\rightarrow 2\pi$ the above effective lagrangian reduces to the
usual Coleman-Weinberg potential. It is important to note that the
computation of higher orders in the above expansion in powers of
$(mr)^2$ does not involve particular complications.

\begin{center}
{\large\bf III. Vacuum fluctuations}
\end{center}

Now we use the above $\zeta$ function to compute the (renormalized) value
of the vacuum fluctuation of the field \cite{AAA95}:
$$
\langle\phi^2(x)\rangle=\lim_{s\rightarrow 1}\zeta(s;x).
$$
Since even in the Minkowski space the vacuum expectation value
diverges, we renormalize the vacuum expectation value on the cone
subtracting the Minkowski value: in this way we have a remarkable
cancellation of the poles, yielding a finite result:
\begin{eqnarray}
\langle\phi^2(x)\rangle_\alpha-\langle\phi^2(x)\rangle_{2\pi}&=&
\frac{1}{4\pi\alpha r^2} \left\{I_\alpha(0)+\frac{(mr)^2}{\nu}
\left[(\nu-1)\left(\ln\frac{mr}{2}+\gamma-\frac{1}{2}\right)+
\ln\nu\right]\right.\nonumber\\
&&+\frac{(mr)^4}{8}\left[2G_\alpha(2)+2\gamma-1+
2\ln\frac{mr}{2}\right]\nonumber\\
&&\left.+(mr)^{2\nu+2}
\frac{\Gamma(\nu+1/2)\Gamma(-\nu-1)}{\sqrt{\pi}\Gamma(2\nu+1)}+
{\cal O}\left((mr)^6\right)\right\}.
\label{vacuum}
\end{eqnarray}
The computation of higher corrections is straightforward. One can
verify that Eq. (\ref{vacuum}) correctly vanish in the limit
$\alpha\rightarrow 2\pi$, and that up to order $(mr)^2$ it is
identical to the result obtained by Moreira Jnr \cite{moreira95}. In
particular, we notice the additional logarithmic divergences of the
vacuum fluctuations at the conical singularity due to the massive
corrections.

\begin{center}
{\large\bf IV. Energy-momentum tensor}
\end{center}

Another important vacuum average is that of the energy-momentum
tensor. While in the massless case it is quite easy to be computed,
since its form is fixed from symmetry arguments, in the massive case
the computation is much more difficult and, to our knowledge, the
massive corrections near the string have never  been shown explicitly.
Only in  \cite{ShiraHire92} and \cite{GuimLin94} the explicit form of
the energy-momentum tensor has been given for $mr\gg 1$, where they
found the expected exponential damping factor $\exp(-2mr)$.

We use a point-splitting approach, in which the vacuum expectation
value of the energy momentum tensor is given by the coincidence
limit of a non-local differential operator applied to the propagator
of the field \cite{birrel}:
\begin{eqnarray*}
\langle T_{\mu\nu}(x)\rangle=i\lim_{x'\rightarrow x}D_{\mu\nu}(x,x')
G_{\cal F}(x,x'),
\end{eqnarray*}
where
\begin{eqnarray}
D_{\mu\nu}(x,x')\equiv(1-2\xi)\nabla_\mu\nabla_{\nu'}+
-2\xi \nabla_\mu\nabla_\nu
+(2\xi-1/2)g_{\mu\nu}\left[\nabla_\alpha \nabla^{\alpha'}
-m^2\right],
\label{dimunu}
\end{eqnarray}
and the prime indicates that the derivative has to be taken with
respect to $x'$.   Since the propagator can be obtained from the
off-diagonal $\zeta$ function \cite{birrel} as
\begin{eqnarray*}
G_{\cal F}(x,x')=i\lim_{s\rightarrow 1}\zeta(s;x,x'),
\end{eqnarray*}
we can compute the energy momentum tensor from the
$\zeta$ function as (see also \cite{hawking77,CVZ90})
\begin{eqnarray*}
\langle T_{\mu\nu}(x)\rangle=-\lim_{s\rightarrow 1}
\lim_{x'\rightarrow x}
D_{\mu\nu}(x,x')\zeta(s;x,x').
\end{eqnarray*}

The partial derivatives of the off-diagonal $\zeta$ function which
appear in the above expression can be computed from the spectral
representation in $D$ dimensions (${\bf x}\equiv(\tau,{\vec{z}})$,
$k=|{\bf k}|$)
\begin{eqnarray*}
\zeta_D(s;x,x')&=&\frac{2 (4\pi)^{-\frac{D-2}{2}}}
{\alpha\Gamma(\frac{D-2}{2})}
\int_0^\infty dk\, k^{D-3}\sum_{n=-\infty}^\infty\int_0^\infty
d\lambda\,\lambda [\lambda^2+k^2+m^2]^{-s}\nonumber\\
&&\times J_{\nu_n}(\lambda r')
J_{\nu_n}(\lambda r)e^{i{\bf k}\cdot({\bf x}-{\bf x}')+
i\frac{2\pi}{\alpha}n(\theta-\theta')},
\end{eqnarray*}
and then taking the coincidence limit. In this way one can easily show
that
\begin{eqnarray*}
\partial_\theta\partial_{\theta'}\zeta_D(s;x,x')|_{x=x'}&=&
-\partial_\theta^2\zeta_D(s;x,x')|_{x=x'}
\nonumber\\
\partial_{z_i}\partial_{{z_i}'}\zeta_D(s;x,x')|_{x=x'}&=&
-\partial_{z_i}^2\zeta_D(s;x,x')|_{x=x'}=
2\pi\zeta_{D+2}(s;x)\nonumber\\
\partial_\tau\partial_{\tau'}\zeta_D(s;x,x')|_{x=x'}&=&
-\partial_\tau^2\zeta_D(s;x,x')|_{x=x'}=2\pi\zeta_{D+2}(s;x)
\nonumber\\
\partial_{r'}\zeta_D(s;x,x')|_{x=x'}&=&\frac{1}{2}\partial_r
\zeta_D(s;x).
\end{eqnarray*}
As far as $\partial_\theta^2\zeta_D(s;x,x')|_{x=x'}$ is concerned,
in the $m=0$ case it is easy to see that we have
\begin{eqnarray*}
\partial_\theta^2\zeta_D^{m=0}(s;x,x')|_{x=x'}=
-\frac{r^{2s-D}\Gamma(s-\frac{D-1}{2})}
{(4\pi)^{\frac{D-2}{2}}\sqrt{\pi}\alpha\Gamma(s)}
H_\alpha(s-{\scriptstyle\frac{D-2}{2}}),
\end{eqnarray*}
where the function $H_\alpha(s)$ is defined and studied in the
Appendix {\bf A}. The massive case can then be treated using the
off-diagonal version of Eq. (\ref{general}) with the partial
coincidence limit $r=r'$, $z=z'$ and $t=t'$:
\begin{eqnarray*}
\partial_\theta^2\zeta_4(s;\theta,\theta')|_{\theta=\theta'}&=&
\frac{1}{2\pi i\Gamma(s)}\int \Gamma(z)
\partial_\theta^2\zeta_4^{m=0}(z;\theta,\theta')|_{\theta=\theta'}
\Gamma(s-z) m^{2z-2s}dz\nonumber\\
&=&\frac{-1}{2\pi i\Gamma(s)} \int_{\mbox{\scriptsize Re } z>3}
\frac{r^{2z-4}}{4\pi\sqrt{\pi}\alpha}
\Gamma(z-3/2)H_\alpha(z-1)\Gamma(s-z) m^{2z-2s}dz.\\
&=&-\frac{r^{2s-4}\mu^{2s}}{4\pi\sqrt\pi\alpha\Gamma(s)}\left\{
\sum_{n=0}^\infty\frac{(-1)^n}{n!}(mr)^{2n}\Gamma(s+n-3/2)
H_\alpha(s+n-1) \right.\nonumber\\
&&\left.\hspace{-2cm}+(mr)^{4-2s}
\sum_{n=1}^\infty\sum_{k=0}^\infty\frac{(-1)^k}{k!}
(mr)^{2\nu_n+2k}\nu_n^2\frac{\Gamma(s-\nu_n-k-2)
\Gamma(\nu_n+k+1/2)}
{\Gamma(2\nu_n+k+1)}\right\}.
\end{eqnarray*}
where, as usual, we have performed the integration shifting the
integration contour to the right and picking up the residues at
$z=s+n$ and $z=\nu_n+k+2$ to get an expansion in powers of $mr$
similar to Eq. (\ref{zetafun}).

As far as the second derivatives with respect to $r$ and $r'$ are
concerned, using the following identity, which can be proved using
some recursion formulas for the Bessel functions \cite{GR},
$$
[\partial_r J_\nu(\lambda r)]^2=-\frac{\nu^2}{r^2}J_\nu^2(\lambda r)
+\frac{1}{2r}\partial_r r\partial_r J_\nu^2(\lambda r)+\lambda^2
J_\nu^2(\lambda r),
$$
one can see that
\begin{eqnarray*}
\partial_r\partial_{r'}\zeta_D(s;x,x')|_{x=x'}&=&
\frac{1}{2r}\partial_r r\partial_r\zeta_D(s;x)+\frac{1}{r^2}
\partial_\theta^2
\zeta_D(s;x,x')|_{x=x'}+\chi_D(s;x),\\
\partial^2_{r'}\zeta_D(s;x,x')|_{x=x'}&=&
-\frac{1}{2r}\partial_r\zeta_D(s;x)-\frac{1}{r^2}\partial_\theta^2
\zeta_D(s;x,x')|_{x=x'}-\chi_D(s;x),\\
\end{eqnarray*}
where the function $\chi_D(s;x)$ is defined in Appendix {\bf B}.
If $D=4$ and $m=0$ we simply have
\begin{eqnarray}
\chi_{D=4}^{m=0}(s;x)=4\pi(s-2)\zeta_{D=6}^{m=0}(s;x),
\label{chizero}
\end{eqnarray}
while the massive case is more complicate and is studied in the
Appendix.

Now we have all the pieces needed to compute the massive correction to
the energy-momentum tensor. For the actual calculation it is
convenient to follow \cite{BROT86} and \cite{ALO92}. We define the
renormalized stress tensor as
$$
\langle T_{\mu\nu}(x)\rangle_\alpha^R\equiv
\langle T_{\mu\nu}(x)\rangle_\alpha-
\langle T_{\mu\nu}(x)\rangle_{2\pi}
$$
which, considering only the first massive correction, turns out to be
\begin{eqnarray*}
\langle T_{\theta\theta}(x)\rangle_\alpha^R&=&
\frac{1}{4\pi\alpha r^2}\left[2H_\alpha(0)-(6\xi-1)I_\alpha(0)+
(mr)^2\left(H_\alpha(1)+\xi\frac{\nu-1}{\nu}\right)\right],\\
\langle T_{rr}(x)\rangle_\alpha^R&=&
\frac{-1}{4\pi\alpha r^4}
\left[2H_\alpha(0)-I_\alpha(-1)-(2\xi-1)I_\alpha(0)+
(mr)^2\left(H_\alpha(1)+\xi\frac{\nu-1}{\nu}\right)\right],\\
\langle T_{tt}(x)\rangle_\alpha^R&=& \frac{-1}{8\pi\alpha
r^4}\left[I_\alpha(-1)+2(4\xi-1)I_\alpha(0)+
(mr)^2I_\alpha(0)\right],\\
\langle T_{zz}(x)\rangle_\alpha^R
&=&\langle T_{tt}(x)\rangle_\alpha^R,
\end{eqnarray*}
The tensor can be written in the more familiar form
\begin{eqnarray}
\langle {T_\mu}^\nu(x)\rangle_\alpha^R&=&
\frac{-1}{1440\pi^2 r^4}\left [\left(\nu^4-1\right)
\mbox{diag}(1,1,-3,1)+10(6\xi-1)\left(\nu^2-1\right)
\mbox{diag}(2,-1,3,2)\right.\nonumber\\
&&+15(mr)^2(\nu-1)
(12\xi-1-\nu) \mbox{diag}(0,1,-1,0)\nonumber\\
&&\left.-15(mr)^2\left(\nu^2-1\right)
\mbox{diag}(1,0,0,1)\right].
\label{stress}
\end{eqnarray}
In the limit $m\rightarrow 0$ the result is in agreement with that
obtained by others authors \cite{DO87a,FRSE87}.
The components $\langle T_{rr}(x)\rangle_\alpha^R$ and
$\langle T_{\theta\theta}(x)\rangle_\alpha^R$ satisfy the
equation
$$
\frac{d}{dr}(r {T^r}_r)={T^\theta}_\theta,
$$
which follows from the conservation law $\nabla_\mu {T^\mu}_\nu=0$.
It is interesting to note that in the corrections to the stress tensor
of order $(mr)^2$  are not present the logarithmic divergences which
are instead present in the vacuum fluctuations at the same order (see
Eq. (\ref{vacuum})). The logarithmic terms appear in higher order
corrections, which can also be computed with the method we have
developed. Actually only the logarithms in the term of order $(mr)^4$
give rise to divergences, since at higher orders they are multiplied
by positive powers of $r$. Indeed, the correction of order $(mr)^4$ to
$\langle T_{\theta\theta}(x)\rangle_\alpha^R$ is
\begin{eqnarray*}
\frac{1}{4\pi\alpha r^2}\left\{-\frac{(mr)^4}{8}
\left[2\frac{\ln\nu+1}{\nu}+(2\xi-1)A(r)+4\xi-1\right]\right.\\
\left.+(mr)^{2+2\nu}\nu[2\xi+(4\xi-1)\nu]\frac{\Gamma(-\nu-1)
\Gamma(\nu+1/2)}{\sqrt{\pi}\Gamma(2\nu+1)}\right\},
\end{eqnarray*}
from which one can also compute the
correction to $\langle T_{rr}(x)\rangle_\alpha^R$
by means of the conservation law. The correction
to $\langle T_{tt}(x)\rangle_\alpha^R=
\langle T_{zz}(x)\rangle_\alpha^R$ is
\begin{eqnarray*}
-\frac{1}{8\pi\alpha r^4}\left\{\frac{(mr)^4}{4}
\left[2B(r)-(4\xi-1)(A(r)+1)-\frac{1}{\nu}\right]\right. \\
\left.+(mr)^{2+2\nu}2(4\xi-1)\nu^2\frac{\Gamma(-\nu-1)
\Gamma(\nu+1/2)}
{\sqrt{\pi}\Gamma(2\nu+1)}\right\}.
\end{eqnarray*}
In the above equations we have set
\begin{eqnarray*}
A(r)&=&2G_\alpha(2)+2\gamma+2\ln\frac{mr}{2}.\\
B(r)&=&\frac{\nu-1}{\nu}\left(\ln\frac{mr}{2}+
\gamma-\frac{1}{2}\right)+\frac{\ln\nu}{\nu}.
\end{eqnarray*}

An interesting point to be discussed is the dependence on the
parameter $\xi$ which fixes the coupling of the scalar field with the
gravity. In the introduction we said that we would consider the
minimally coupled case only, $\xi=0$. This because it has been shown
in \cite{AlKaOt96} that the idealized conical space is a good model of
the space time of a cosmic string  only for the minimally coupled
case. For  nonminimally coupled  fields quantities like $\langle
\phi^2(x)\rangle$ will depend on the details of the metric in the core
of the string even very far from the string. However, one could be
interested, e.g., in  finite-temperature fields in the Rindler space,
where there is a true conical singularity and  not just an
idealization of a non-singular metric. In these cases it is
interesting to consider also nonminimally coupled fields.

{}From the mathematical point of view, it is not clear which is the
meaning of the field equation $[-\Delta+m^2+\xi R]\phi=0$ when the
curvature $R$ has Dirac's delta singularities, and our choice of
$\xi=0$ allowed us to avoid the problem. A possible way to  define the
problem is to smooth out the singularity, as done in
\cite{AlOt90,ALO92,FUSOL95}: as a result, one finds that when the
regularization of the singularity is removed to recover the conical
space also the dependence of the Green's function on the parameter
$\xi$ vanishes \cite{ALO92}. Then we can argue that also the $\zeta$
function is independent of $\xi$, since the modes used to construct
the $\zeta$ function are essentially the same as those for the Green's
function.

The dependence on the parameter $\xi$ comes back into play when
considering the stress tensor. It is worth while noticing that even
when the manifold if flat everywhere (in our case there is a Dirac's
delta singularity in the curvature at the core of the string) the
parameter $\xi$ remains in the theory as a relic of the fact that
$T_{\mu\nu}$ is obtained by varying the metric $g_{\mu\nu}$ in the
field Lagrangian \cite{birrel}. One can then see that, if $R=0$,  the
global conserved quantities as total energy should not depend on the
value of $\xi$. This is  because the contributions to those quantities
due to $\xi$ are discarded into boundary surface integrals which
generally vanish. However, this is not the case dealing with the
cosmic string because such integrals diverge at the origin. Notice
that similar problems appear working in subregions of the Minkowski
space in presence of boundary conditions \cite{birrel}. In the
point-splitting approach, the stress tensor is obtained applying the
$\xi$-dependent operator $D_{\mu\nu}(x,x')$ (see Eq. (\ref{dimunu}))
to the Green's function or to the $\zeta$ function. Therefore, if the
above argument that the Green's function  (or the $\zeta$ function) is
independent of $\xi$ holds, it is not contradictory to compute the
energy-momentum tensor applying $D_{\mu\nu}(x,x')$ with $\xi\neq 0$ to
the the Green's function computed setting $\xi=0$, as we have done
above and as done by most authors.

\begin{center}
{\large\bf V. Conclusions}
\end{center}

In this paper we have studied the $\zeta$ function of a massive scalar
field in a cosmic string background, and we have obtained an
expression, Eq. (\ref{zetafun}), which is useful in the region near the
core of the string, $mr\ll 1$. By means of this  expression
we have  computed the massive corrections to the vacuum
fluctuations, Eq. (\ref{vacuum}), and to the energy-momentum tensor,
Eq. (\ref{stress}), up to order $(mr)^4$, going beyond the known
results. Higher corrections are also computable.

Possible extensions of this paper are the inclusion of a magnetic
flux carried by the string, which gives Aharonov-Bohm effects, and
the case of spin-$1/2$ fields.  Also the limit $mr\gg 1$ is worth
studying, for example rewriting the $\zeta$ function (\ref{zetafun})
in terms of the hypergeometric function ${}_1F_2[a;b,c;z]$ and
then using its asymptotic behaviour for large $z$. However,
the vacuum fluctuations and the energy-momentum tensor
in this limit have been already obtained by others authors
with different methods \cite{ShiraHire92,GuimLin94}.

As a possible application of the results of this paper we point out the
study of the backreaction of the energy momentum tensor of a
massive scalar field on the background metric of the cosmic string,
as done by Hiscock \cite{Hiscock87} and Guimar{\~a}es \cite{MEX97}.
The computation seems feasible since the first massive correction to
the energy momentum tensor does not involve logarithmic terms.
We hope to cover this topic in a future paper.
\\

\noindent {\large\bf Acknowledgments}

\noindent I would like to thank Valter Moretti and, in particular,
Sergio Zerbini for useful discussions and suggestions.

\vspace{1cm}

\begin{center}
{\large\bf Appendix A}
\end{center}

In this Appendix we study the function $H_\alpha(s)$, which is
defined as the analytic continuation of following series:
\begin{eqnarray*}
H_\alpha(s)=\sum_{n=1}^\infty\nu_n^2
\frac{\Gamma(\nu_n-s+1)}{\Gamma(\nu_n+s)},
\hspace{1cm}\nu_n=\frac{2\pi}{\alpha}|n|.
\end{eqnarray*}
It can be studied and analytically continued proceeding exactly
as for the function $G_\alpha(s)$ studied in the Appendix of \cite{ZCV}.
Then one sees that the series converges for $\mbox{Re} s>2$ and that
the analytic continuation for $[n/2]<\mbox{Re} s<2$ (here $[n/2]$
represents the integer part of $n/2$) is given by
\begin{eqnarray*}
\sum_{j=0}^{[n/2]+1}c_j(s)\nu^{3-2s-2j}\zeta_R(2s+2j-3)
+\sum_{k=1}^\infty \nu_n^2 f_n(\nu_k,s),
\end{eqnarray*}
where $\zeta_R$ is the Riemann $\zeta$ function and the function
$f_n(\nu,s)$ is generally unknown, but vanishes for $s=1/2,0,-1/2,
-1,\dots$. The coefficients $c_j(s)$ vanish for $s=-n/2$,
($n=-1,0,1,2,\dots$) for all $j>(n+1)/2$, and the first ones
have been given in \cite{ZCV}.

The function $H_\alpha(s)$ has then a simple pole in $s=2$ and
near this pole we have
\begin{eqnarray*}
H_\alpha(s)=\frac{1}{2\nu(s-2)}+\frac{1}{\nu}(\gamma-\ln\nu)+
{\cal O}\left((\nu-2)^2\right).
\end{eqnarray*}
Moreover, it has simple poles at $s=1+\nu_n+k$, $k=1,2,\dots$, due to
the gamma function in the numerator of the terms of the series, with
obvious residue. Finally, it is possible to compute the value of the
function $H_\alpha(s)$ for some useful value of $s$:
\begin{eqnarray*}
H_\alpha(0)&=&\frac{1}{120\nu}(\nu^4-1),\nonumber\\
H_\alpha(1)&=&-\frac{1}{12\nu}(\nu^2-1).
\end{eqnarray*}

\begin{center}
{\large\bf Appendix B}
\end{center}

In this Appendix we study the function $\chi_D(s;x)$, which is defined
by the analytic continuation of the following spectral representation
\begin{eqnarray}
\chi_D(s;x)=\frac{2 (4\pi)^{-\frac{D-2}{2}}}{\alpha\Gamma((D-2) /2)}
\int_0^\infty dk\, k^{D-3}\sum_{n=-\infty}^\infty\int_0^\infty
d\lambda\,\lambda^3 [\lambda^2+k^2+m^2]^{-s}J_{\nu_n}^2(\lambda r),
\label{chi}
\end{eqnarray}
which is the same as $\zeta_D(s;x)$ with $d\lambda\,\lambda
\rightarrow d\lambda\,\lambda^3$. Note that the massless case
is trivial, since in that case the function $\chi$ is simply related
to the $\zeta$ function, see Eq. (\ref{chizero}). In the massive case,
we can proceed in analogy to the massive $\zeta$ function, namely
employing Eq. (\ref{general}): considering $D=4$ and using Eq.
(\ref{chizero})
\begin{eqnarray*}
\chi^{m}_{D=4}(s;x)&=&\frac{1}{2\pi i \Gamma(s)}
\int_{\sigma-i\infty}^{\sigma-i\infty}
\Gamma(z)\chi^{m=0}_{D=4}(z;x)m^{2z-2s}\Gamma(s-z) dz
\nonumber\\
&=&\frac{4\pi}{2\pi i \Gamma(s)}
\int_{\sigma-i\infty}^{\sigma-i\infty}
\Gamma(z)(z-s)\zeta^{m=0}_{D=6}(z;x)m^{2z-2s}\Gamma(s-z) dz.
\end{eqnarray*}
Then we go on as usual splitting $\zeta^{m=0}_{D=6}(z;x)$
as $\zeta=\zeta_<+\zeta_>$ and performing the integrals
shifting the integration contour to the left and picking up
the residues of the poles. The final result is an expansion
in powers of $mr$:
\begin{eqnarray}
\chi^{m}_{D=4}(s;x)=\frac{r^{2s-6}}{4\pi\alpha\Gamma(s)}
\left\{\sum_{n=0}^\infty\frac{(-1)^n}{n!}(s+n-2)(mr)^{2n}
I_\alpha(s+n-2)
\right.\nonumber\\
+\sum_{n=0}^\infty\frac{(-1)^n}{n!}(n+1)(mr)^{6+2n-2s}
\frac{\Gamma(n+1/2)\Gamma(s-n-3)}{2\sqrt{\pi}\Gamma(n+1)}
\nonumber\\
\left.+(mr)^{6-2s}
\sum_{n=1}^\infty\sum_{k=0}^\infty
\frac{(-1)^k}{k!}(\nu_n+k+1)(mr)^{2\nu_n+2k}
\frac{\Gamma(\nu_n+k+1/2)\Gamma(s-\nu_k-k-3)}{2\sqrt{\pi}
\Gamma(2\nu_n+k+1)}\right\}.
\nonumber\\
&&
\label{chifinal}
\end{eqnarray}

redpajama_set_name: RedPajamaArXiv


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