text:
How Assam's Bihu Bird Count is becoming a national obsession
Birders from 11 states outside Assam participated in the Kati Bihu bird count from October 15- 18.
by Roopak Goswami December 9, 2021 Updated December 17, 2021
Red-breasted Parakeet: Photo courtesy Gaurab Talukdar
Guwahati: Praveen J, IIT Guwahati alumnus, was in Mauritius during the Kati Bihu festival this time. A keen birder, he was super excited to join the Kati Bihu Bird count.
The Assam Bird Monitoring Network, the organisers of the Bihu Bird Count – a bird monitoring program – say not only people from the state, but people from outside the state and country are participating.
"Having participated actively in Rongali Bihu count from the Karimganj district, I was super keen to join the Kati Bihu count as well. Thanks to the organisers for allowing me to join the count while I was on a vacation in the Maldives. We did Bihu count as a family watching the local subspecies of Striated Herons," Praveen told EastMojo.
Sikkim: New species of butterfly found in Dzongu
Birders have to send the species details, which they spotted in the country during the specified time of the event, in a checklist.
The Kati Bihu Bird Count this year saw 28 districts participating from Assam. Eleven states, including Maharashtra, Karnataka, Bihar, Chattisgarh, Arunachal Pradesh, West Bengal, Uttarakhand, Haryana, Gujrat, Tamil Nadu and Delhi (UT) also took part in the state-held Bihu Bird Count. Additionally, checklists were submitted from Kaafu and Maldives. However, the observation records shared by people from outside Assam were not considered for analysis.
The Kati Bihu Bird Count recorded 285 species through 300 checklists. Spotted Dove was the most frequently sighted species. It was followed by Common Myna and Red-vented Bulbul. Some rare sightings included Ashy-headed Green-Pigeon, Bay Woodpecker, Bengal Florican, Black-breasted Parrotbill, Black-faced Bunting, Black-tailed Crake, Cachar Bulbul etc.
Teas produced in Northeast India would be promoted as a brand
Jugal Borah from North Lakhimpur told EastMojo, "I used to do bird watching now and then, but this is my first time counting birds. During the time, we witnessed how some birds adjusted themselves to live alongside humans, how some birds helped the farmers; these are some invaluable lessons I learned in the event.
"I hope Assam Bird Monitoring Network will inspire and motivate us by arranging more programs like Kati Bihu Bird Count in the future," he said.
Parag Patgiri, pursuing a Master's Degree in Zoology from Bodoland University, has been interested in studying birds. However, he never got a chance to explore it scientifically.
"At Bodoland University, I met my professor Dr Hilloljyoti Singha, an expert ornithologist. After meeting him, I got the opportunity to expand the field of my study.
"He introduced me to Dr Jaydev Mandal, who is also an ornithologist. Mandal skilfully guided me on how to carry out the project of bird count with "eBird". And with their guidance, I started my birding passionately and on a deeper level" Patgiri told EastMojo.
"I noticed that the local people are very ignorant about the importance of a bird count. As a result of that ignorance, there is a lack of conventional documentation of birds. This is why we should organise different kinds of programs to make people aware of the importance of birds to form stable biodiversity," he added.
"This event is expected to deliver a better understanding of the seasonal and temporal distribution of birds in Assam among the common citizens across different terrains as well as imbibe the sense of attachment towards nature within people. The initiative will also try reaching out to people from other parts of the country as well as the globe, for people to participate and document birds during the Bihu Bird Count," Jaydev Mandal, the co-founder of Assam Bird Monitoring Network, told EastMojo.
Why are fewer Amur Falcons roosting at Nagaland's Pangti this year?
Kati Bihu 2022: Wishes and messages for your loved ones
Magh Bihu: Significance and wishes to share with loved ones
Assam: Kati Bihu celebrated with religious fervour in Golaghat
Budget: 50-year interest-free loans to states to continue another year February 1, 2023
Promotion of tourism to be taken up on mission mode: Sitharaman February 1, 2023
Tagged: Assam, Assam news, Bodoland University, Kati Bihu, Maldives, Rongali Bihu


This video shows how EXP board from iknowvations can be used to learn microcontrollers from TI using its low cost LaunchPad. When I got LaunchPad, I was very much eager to play with it but soon I found that there was no playing area as such.
I really admire TI people as their LaunchPad is the lowest costing microcontroller learning kit. They even provide you with extra controllers. Great ! But then I thought there should be something that will help everybody to start experimenting with LaunchPad. So we have come up with EXP board.
There are 4 touch pads also to experiment with touch switches.
This is part-5 of the series. This is another pattern. Will be publishing more parts of this series and in near future will try to publish the circuit & code also.


text:
Executive Channel Announces Restructure, Global Growth Plans
By Caroline Cottrill
Bruce Fink, executive chairman of Executive Channel International Pty Ltd (ECI) has announced a buy-out of its ECNlive UK and European operations and a significant restructure, with the now wholly owned ECI network poised for strong global expansion.
According to Mr Fink, ECNlive with operations in London, Paris and Amsterdam will operate alongside Executive Channel Australia as part of a cohesive global business in the office media arena. "Following completion of the purchase, the new global company is primed for an aggressive expansion of the ECNlive and Executive Channel networks," Mr Fink said. As part of the restructure, Charles Parry-Okeden, one of the company's three founding partners, has been appointed global CEO. ECNlive London has also appointed Stephen Gale as managing director.
Charles Parry-Okeden
Charles Parry-Okeden said, "This is an exciting time for Executive Channel. It's always been our intention that the UK, European and Australian networks operate as part of a global business." Executive Channel Australia has also undergone a significant restructure. Justin Kingston, with his extensive property management experience, has been promoted to chief operating officer and is responsible for all non sales related functions. Paul McBeth has been elevated to strategic sales and marketing director, and is responsible for driving national revenue. McBeth had been consulting to Executive Channel and has worked in senior marketing and strategy roles for media owners and media/advertising agencies. Tim Elder will head sales in New South Wales as well as Queensland as sales director and Guy Taffs appointed sales director of boutique markets. "ECI is now one of the biggest global players in office media. Our new cohesive global business focus, supported by our newly appointed talented teams here and in Europe, makes us a formidable force as we focus on our five year plan and growth," Mr Parry-Okeden said.
Guidelines McDonald's Michelin Man

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: Como converter 39 horas para o tipo 'TIME' Para realizar certa tarefa, contabilizo em meu sistema o tempo gasto da respectiva tarefa.
Hoje me deparei com o seguinte:

Data Inicial: 17/12/2018 16:49:14
Data Final: 19/12/2018 08:02:58

totalizando: 2362,29 minutos.

Converto para decimal: (2362,29 / 60) = 39,3716

Porém preciso converter esse valor (39,3716) para o formato TIME
Porém, segundo documentação hora (Transact-SQL)
O intervalo permitido é de: 00:00:00.0000000 a 23:59:59.9999999
Teria algum outro modo de representar esse valor com o tipo TIME?
Obs: Sei que, o valor 39,3716, representa 39 horas e 50 minutos, porém preciso que a coluna seja do time TIME, caso contrario, poderia definir a coluna como  VARCHAR e apresentar o valor 39:50.

A: Não, o tipo TIME foi criado para indicar um ponto específico na linha do tempo, não para indicar intervalos de tempo que é o que deseja. Esse requisito "preciso converter esse valor (39,3716) para o formato TIME" parte de uma premissa errada, é como querer medir quanto tem de leite em metros.
A forma correta para armazenar um intervalo de tempo é colocar a quantidade de unidade de tempo. Provavelmente o mais adequado é a quantidade de segundos, mas pode ter uma precisão maior ou menor, desde que você saiba qual é ela sempre, pode ser só minutos, por exemplo.
O .NET possui um tipo mais especializado que lida melhor com isso para você, é o Timespan. Infelizmente os bancos de dados não costumam ter um tipo assim, então precisa controlar isso na mão.
Armazenar é diferente de apresentar. Muitas pessoas não entendem isso. Você pode apresentar como quiser, pode até pegar essa unidade de tempo contida no intervalo e apresentar como se fosse um horário. Isso está errado. Dizer que "a duração é de 39 horas e 50 minutos" é correto, já dizer "a duração é 39:50" não é, essa notação é de horário e não tempo decorrido, mas se quiser pode fazê-lo, claro. Tem uma pergunta que fala sobre isso.
De qualquer forma como a duração cabe cálculos, o correto é armazená-la como um número e não como um texto. Eu iria de BIGINT ou equivalente no seu SGDB para guardar os segundos decorridos neste intervalo. Para a apresentação eu criaria funções que faz a conversão, algo parecido pode ser visto no link acima, seria só adaptar.
Eu sei, você provavelmente vai querer insistir que o jeito que está pensando está bom, mas não é o correto. As pessoas aprendem errado e trabalham com o erro por tanto tempo que fica até difícil compreender e assimilar o certo.
Ainda bem que o TIME não aceita o valor que deseja, se aceitasse provavelmente usaria errado sem perceber, caindo naquilo que eu falo sempre:

Fora isso eu não consigo imaginar porque 39,37 é o mesmo que 39 horas e 50 minutos. .37 é algo pouco acima de 1/3, portanto deveria ser pouco mais que 20 minutos quando falamos de uma hora.

redpajama_set_name: RedPajamaStackExchange


You can enter two background URLs to give your site a unique look. You can enter a background for the entire background of the template and also a second image that overlays on top of the main background image. As seen below the left image has two backgrounds a red repeating image and a second wine bottle image, while the image on the right only has 1 background image repeating through the back of the template.
All of this is done very easily in the template configuration located in the template manager area.


Vacuum Equipment Business | Advantech Inc.
Vacuum equipment business, our company's foundation business, manufactures and sells various parts such as vacuum parts, piping and processed products for various manufacturing equipment, pumps, valves and plants related manufacturers, mainly in the semiconductor, FPD and energy fields Doing.
Advantech manufactures in Ehime, Nagano, China.
ISO09001, ISO14001 certification acquired. We will aim at global expansion using our domestic and overseas networks.
Advantech manufacturing bases are manufactured in Ehime, Nagano, China. ISO09001, ISO14001 certification acquired. We will aim at global expansion using our domestic and overseas networks.
· Thorough quality control through consistent management.
· There are overseas manufacturing bases and overseas partner companies, and cost reduction is possible.
We will manufacture high quality products with high technical strength cultivated since our foundation.
High technical force promising reliable reliability and unique manufacturing of jigs and tools optimum for each production process in addition to high precision machine tools such as 5 axis NC manipulation center and NC lathe and experienced various welding. We will supply high quality products stably.
We also carry out overseas local procurement.
We will produce locally at a trusted customer who Advantech has provided technical guidance.
COPYRIGHT (c) ADVANTEC. ALL RIGHTS RESERVED.


Jeff is a fit and adventurist 60 years old.
A retired military that works full time at the Weber State University as the Director of Online Programs.
On top of that, Jeff owns a horse training facility where he boards horses, train horses and give a few lessons.
I rode a lot as a kid then had a 20 year break and have been riding now the last 15 years. I own 4 but board and train horses, so I'll have about 7-10 on my farm at any one time.
Motivation is simple, competing in a new country, great organization and one heck of an adventure. Super excited to be part of this. Last year I won the National championships for my weight and place 3rd in the National AHA championships. So last year was a big deal for me. Long term I just hope to compete at a fairly high level and develop nice horses along the way.
Biggest fear? Having my body hold up to the rigors of riding 50 miles every day.
Biggest strength? The excitement of being part of this. Getting to meet and associate with like minded folks is always fun and rewarding. Improvements needed is just get tougher. Being able to grind out each day to complete. I ride everyday since I start colts for the public and condition my own endurance horses.
People need to sign up for this race because in our day to day lives, there are less and less real adventures out there. This is a super adventure to see Mongolia and test our self's but have great support along the way.
Success on this ride is for me to finish each day with a horse that is "Fit to Continue". So looking forward to this. Please come to join us in this great adventure.


text:
Dealer Contact Info
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Special athletes 'brought tears to my eyes and joy to my heart'
By Candace Brand on May 23, 2018
SC associates with a Special Olympics Texas gold medalist.
On your mark, get set, give back!
It was in that spirit that three-dozen Santander Consumer USA (SC) associates volunteered recently at the Special Olympics Texas (SOTX) Area 11 Spring Games in Arlington, Texas.
"It was an emotional but rewarding experience for me, simply amazing!" said one SC associate who was assigned to the important job of cheering for the athletes as they arrived at the finish line for one of the events.
SC volunteers were from the Enterprise Fraud Management and Dealer Services team.
The SOTX organization was happy for the help, with 1,500 athletes competing in field and track events, this is SOTX Area 11's largest competition of the year. For eight weeks or more, athletes have trained in preparation for this grand event.
Volunteer 'Life Changers' provide help to hungry families
Calling all volunteers: Santander Consumer associates making a difference
"The mission of Special Olympics is to provide year-round sports training and athletic competition in a variety of Olympic-type sports for children and adults with intellectual disabilities," said Krista Dunham, SOTX program director.
The games give athletes "continuing opportunities to develop physical fitness, demonstrate courage, experience joy and participate in a sharing of gifts, skills and friendship with their families, other Special Olympics athletes and the community."
All volunteers arrived early in the morning to set up the events and rest areas for the athletes and their families. Afterward, the SC associates were divided into groups to assist with multiple events, which started at noon.
SC associates celebrate with an athlete at the finish line.
The impact on the volunteers was described by one of the cheering SC associates.
"I cannot get little Jaydan out on my mind," she said. "Just to see the joy on her face as I cheered her across the finish line brought tears to my eyes and joy to my heart."
It seems like a small task, to cheer, but to all the athletes it's the difference between finishing an event or even mustering up the courage to try.
One of Dunham's favorite stories is about an athlete competing in the tennis ball throw:
"This young lady stood at the line with her hand in the air and the ball all ready to throw. She stood there. People start cheering for her. She stood there. Volunteers remind her what she is supposed to do. She stands there …," according to Dunham.
"Finally, the volunteers decided that she wasn't going to throw the ball and helped her to her seat … No, she never threw the ball. But she had never stood for so long either."
SC associate helping an athlete to some water.
While a majority of the SC volunteers were cheering, others were assisting the athletes on and off the track. Some were supplying water and guiding the athletes to the medal podium. Others kept time, organized the athletes at the event, or kept score.
Another SC associate expressed her gratitude for being given the opportunity to volunteer thanks to the SC volunteer-paid time off (VPTO) program.
"After my experience, I felt very blessed to be able to participate by volunteering at such an awesome event. I feel that SC's VPTO is a beautiful thing, and it shows the community that as an organization we care by giving back. Thank You Santander."
To see more SC associates using their volunteer paid-time off, follow us on our social media pages, Facebook, Google+ and Twitter, as #SCGivesBack.
SC volunteers jump for joy at SOTX Area 11 Spring Games
Candace Brand
These are among the best used cars to buy right now, U.S. News rankings show
Used cars are the way to go for most American buyers. Around 40 million a year, in fact, considerably more than twice the number of new cars sold each year. But picking out the best used cars to buy right…
Longest-lasting vehicles for 2019: 8 SUVs, 5 American made – iSeeCars.com
If you're shopping for a vehicle with staying power, new or used, buy a sport utility vehicle. Even better, purchase an American-made SUV. The top seven vehicles and eight of the top 11 on the 2019 list of longest-lasting vehicles…
25 vehicles that hold value best over five years – iSeeCars.com
Not all cars, trucks and SUVs are created equal. Especially when it comes to new vehicles that hold their value best in the years after purchase. A study of more than 4.3 million new and used car sales by the…
Most-popular SUVs, trucks, cars in America right now
Popularity isn't the only measure of an SUV, car or truck. But looking at the most-popular SUVs and other vehicles isn't a bad place to start shopping. After all, there probably are reasons the most-popular are, well, the most popular….
Who makes the most reliable cars – Consumer Reports
They are the best of the best. The brands that pass the test of who makes the most reliable cars as ranked by Consumer Reports. Lexus and Toyota brands rated first and second, respectively, the reverse of last year but…
Discover Santander
RoadLoans
Make these New Year's resolutions to drive safer in 2015
Around this time of the year, many of us begin thinking about New Year's resolutions – those promises we make to ourselves with every intention of following through. Santander Consumer USA hopes that your resolutions include some safe-driving promises that…
DealerRater aims to help you find an auto dealership for your next purchase
So you're ready to buy a new car, truck, crossover or SUV. You've done all your homework and have pared your list to three models, say Toyota Avalon, Hyundai Sonata and Chrysler 200 (although it could be any of 57…
Drivers of these vehicles more lead-footed than others, reports insurance.com
The next time you see a Subaru WRX, Pontiac GTO or Scion FR-S speeding down the highway you should not be too surprised if a police cruiser – its red-and-blue lights flashing and siren blaring – goes zipping past you…
Dealer Payoff
© 2020 Santander Consumer USA Inc. All Rights Reserved. NMLS Consumer Access ID 4239.
Chrysler Capital is a registered trademark of FCA US LLC and licensed to Santander Consumer USA Inc.
Chrysler, Dodge, Jeep, Ram, Mopar and SRT are registered trademarks of FCA US LLC.
ALFA ROMEO and FIAT are registered trademarks of FCA Group Marketing S.p.A., used with permission.


Contemporary door styling. Exemplary, distinctive grille types for the look of individual windowpanes. Pella wood craftsmanship at a competitive price.
Elegantly crafted wood-grain fiberglass entry door systems. Premium fiberglass available in four wood grains. VividGrain? Pella quality at a competitive price.
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Consider wood if the following benefits for patio doors are most important to you: Unmatched . Review the 7 most common factors that affect your project cost.


text:
Kanton Mas-Cabardès (fr. Canton de Mas-Cabardès) je francouzský kanton v departementu Aude v regionu Languedoc-Roussillon. Skládá se z 15 obcí.

Obce kantonu
 Caudebronde
 Fournes-Cabardès
 Les Ilhes
 Labastide-Esparbairenque
 Lastours
 Les Martys
 Mas-Cabardès
 Miraval-Cabardes
 Pradelles-Cabardès
 Roquefère
 Salsigne
 La Tourette-Cabardès
 Trassanel
 Villanière
 Villardonnel

Externí odkazy

Mas-Cabardès

redpajama_set_name: RedPajamaWikipedia


Posted on November 5, 2018 by Maria J.
The future business KG A is the ideal partner for real estate investments in Dresden the future business KG A (FuBus) focuses mainly on the most exclusive districts of the city of Saxony. This includes, for example, Striesen. In this quarter, the average price per square meter is between 2,300 and 2,700 euros according to the research of the FuBus. Also the range of high-quality objects is still relatively scarce. The independent economic enterprise Empirica rose in a recent survey by the following values: Dresden can be expected until the year 2025 with an additional need for 17,400 units. The conclusion of future business KG A is, therefore: it is so definitely worth to invest in real estate or land in Dresden. According to the real estate experts the FuBus is also no danger of so-called overheating", since prices for Dresden real estate due to a strong demand to rise only gradually. The initial yields There are 7 percent.
About the future business KG A (FuBus) the future business KG A (FuBus) on 23 February 2000 in Dresden, Germany founded. The underwriter is the focus of a broad-based Group of companies. The future business KG acquires A (FuBus) participations, she cares about the purchase and recovery of pension and life insurance products and acquires selected real estate in prime locations. Investors have the opportunity to participate in A (FuBus) by purchasing public and market interest rate-independent investment products in the form of participatory rights or order notes the success of future business KG.


Kendra (Sublette) Lema '96 has long kept a list of 100 goals she'd like to accomplish in her lifetime. She recently crossed one of them off—becoming a member of the Mines Century Society, a distinguished group of loyal individuals whose lifetime giving to Mines exceeds $100,000. As a director of development engineering for ConocoPhillips, she was able to take advantage of the corporation's matching gifts program, which matches employee giving dollar-for-dollar.
"I set the goal relatively early in my career, and by contributing regularly and using the match, I was able to reach it earlier than maybe is traditionally seen," said Lema, who became the youngest alumna inducted into the society. While she didn't think of herself as an extraordinary student while at Mines, Lema believes the university prepared her to be extraordinary in her career. Because of this, she wanted to support Mines.
Lema's primary area of giving is the Lema Endowed Scholarship. The scholarship was created with her husband at the time, Raul Lema '98, and provides four years of scholarship support to a Mines student. Even though she was a civil engineering major, Lema liked the idea of not placing degree restrictions on the scholarship, which rotates between a female and male student every four years.
"What I consider a flexible and long-term commitment to individual students has helped me create relationships with them," Lema said. Lema visits with her scholars and stays in touch with them after they graduate and start their careers.
For more than 20 years, the Mines Century Society has recognized those individuals who, through a lifetime of giving, have shown an extraordinary commitment to the continuing excellence of the school. This society comprises more than 300 donors whose lifetime giving to the school exceeds $100,000. Their names are prominently displayed on the donor wall in the Starzer Welcome Center.
Giving levels within Mines Century Society are denoted by varying gemstones and precious metals. These resources symbolize Mines' commitment to earth, energy and environment.
To learn more about the Mines Century Society, visit giving.mines.edu.


text:
\section{Introduction}

In the usual approach to interacting fermion systems,
the starting point is the treatment of the ideal Fermi gas, followed
by the development of diagrammatic perturbation theory.   Finally,
contact with the phenomenological Landau Fermi-liquid theory is made,
with a discussion of Landau quasiparticles and collective modes.
An exception to this paradigm is found in one-dimensional
systems, where perturbation theory diverges, and the quasi-particle
structure is in general destroyed by  interactions.   In this case,
a different paradigm has developed from Tomonaga's observation
that the low-energy degrees of freedom of a 1D Fermi gas are completely
collective, and the development of the ``bosonization'' technique.
In these lectures, I will try to present a generalization of the
``bosonization'' description as a general treatment of Fermi surface
dynamics in {\it any} dimension.   This suggests some new interpretations
of the Fermi surface as an ``order parameter'' for metals, and of its
notional formation as $T \rightarrow 0 $ as a type of critical phenomenon.
One virtue of a treatment that starts with the collective description,
arriving at quasiparticles (for $ d > 1$) at the end, is that the special
features that distinguish $d=1$ and $d > 1 $ occur at the end of the treatment,
rather than right at the beginning.
It also makes clear that the existence of a Fermi surface is not necessarily
synonymous with the validity of the Landau quasiparticle description.
It seems in principle possible that systems with a Fermi surface but which
are non Landau Fermi-liquids may exist, and the bosonization methods
seem promising tools for investigating such possibilities.

I will start by developing a one-dimensional interpretation of
bosonization as Fermi-surface dynamics, then extend it to higher
dimensions, review spin-charge separation and fractionalization of
electrons into spin and charge degrees of freedom, and end with
some intriguing new results on persistence of special features
of the ideal gas in some solvable models with ``spinons''.

\section{Luttinger's Theorem}

I will take the conceptual starting point for the bosonization of
the Fermi surface to be the Luttinger theorem\cite{luttinger} expressing
the total particle number and momentum of the Fermi gas  purely
in terms of the
Fermi surface geometry.
I will initially describe the treatment of one-dimensional,
spinless fermions, and eventually extend it to three-dimensional
electrons with spin.

The particle and momentum density of a one-dimensional Fermi gas are given by
$$ {2\pi N /L} = \int dk \, n(k) $$
$$ {2 \pi P / \hbar L } = \int dk \, k n(k) $$
where for  free fermions at zero temperature
$$ n(k) = \theta ( E_F - \epsilon(k) ) . $$
We must now express this in terms of the Fermi surface geometry.
In this case, the Fermi surface is described by a set of Fermi points
$\{k_{Fi}\}$ at which there is a step discontinuity
$\Delta \nu_i = n(k_{Fi}+\delta)
- n(k_{Fi}-\delta)$ in $n(k)$, with $\Delta \nu_i $ = $\pm 1 $ and $\sum_i
\Delta \nu _i $ = 0.  We may then write
$$ {2 \pi N / L} = \sum_i \Delta \nu_i k_{Fi} , $$
$$ {2 \pi P / \hbar L } = \frac {1}{2} \sum_i \Delta \nu_i (k_{Fi})^2 . $$
For free electrons this is a ``trivial'' result.  However
the  deep result of Luttinger is that (with some reinterpretation) this
result remains valid (at least in perturbation theory) even when there are
interactions between the fermions. In this case $\Delta \nu $ is no longer
the value of a step discontinuity in $n(k)$: $k_{Fi}$ still marks a
singularity in $n(k)$, but (in one dimension) it is generally weaker
than a step singularity.  Instead, the absolute value of $\Delta \nu $
is an index characterizing the nature of the Fermi surface singularity,
and its sign characterizes the {\it orientation} of the surface, which
in one dimension has an outward normal pointing either to the
right or to the left.   The usual value $|\Delta \nu | = 1 $ indicates that
the singularity in $n(k)$ arises from the Pauli principle, but other
rational values such as $1/m$ can occur,
principally in connection with the fractional
quantum Hall effect, so I will develop the treatment for general
$\Delta \nu$.

Luttinger's theorem is proved using methods of diagrammatic perturbation
theory, which in any case fails to converge in one-dimensional systems.
It follows from the fact that particle number and momentum are additively
conserved quantities carried by the particles and conserved in total
at each interaction vertex of a diagram.  Since I am invoking Luttinger's
theorem outside the strict validity of its derivation from diagrammatic
perturbation theory, I am in essence taking it as an axiom that is in
principle justified by experimental fact.
The aim of this treatment will be
to show it can be taken as the starting point
for the discussion of Fermi surface dynamics.

One further comment is in order.
The Luttinger theorem for the total momentum assumes strict momentum
conservation; on a lattice, momentum is only conserved modulo
reciprocal lattice vectors.  However, unless the Fermi surface geometry
is commensurate with reciprocal lattice vectors, Umklapp processes are ``frozen
out'' at low temperature, and the non-conservation of momentum on a
lattice is technically an irrelevant perturbation to the low-energy
fixed point.

To proceed, I now formulate the Luttinger theorem in a differential,
{\it local} form. On lengthscales $ \xi $ where
$$ |k_{Fi} - k_{Fj}| \xi >> 1 \quad (i \ne j ) $$
we can {\it locally } define the Fermi surface $k_{Fi}(x)$.  Low-energy,
long-wavelength excited states will then be described purely in
terms of local Fermi surface fluctuations about the ``reference''
(ground state) Fermi surface $k_{Fi}^0$:
$$ k_{Fi}(x,t) = k_{Fi}^0 + \delta k_{Fi}(x,t) . $$
This is essentially a ``semiclassical'' treatment of the Fermi surface
where momentum and position are simultaneously specified on
a coarse-grained scale.
The local charge density $ \rho (x) $ (relative to the uniform
density ground state) is
then given by
$$ 2 \pi  \rho (x) = \sum_i \delta \nu_i \delta k_{Fi}. $$
Similarly, the  local momentum density $ \Pi (x) $ is
$$ 2 \pi \hbar^{-1} \Pi (x) = \sum_i \delta \nu_i \left (
k_{Fi}^0 \delta k_{Fi}(x) + \frac {1}{2} (\delta k_{Fi}(x))^2 \right ) .$$

Thus the generators of continuous symmetries (particle conservation,
or $U(1)$ gauge invariance, and translations) are expressed purely in terms
of the locations of the $T=0$ singularities of $n(k)$, now defined locally
on large lengthscales.  The quantities $\Delta \nu _i $ are ``adiabatic
invariants'' that remain unchanged as the  Hamiltonian is adiabatically
varied (and the $k_{Fi}^0$ in general change), provided the basic structure
of the Fermi surface does not change, and are defined by the differential
relation
$$ \delta \rho (x) / \delta k_{Fi}(x')
= (2 \pi )^{-1} \Delta \nu_i \delta (x-x') . $$
I again stress that the Pauli principle gives $\Delta \nu = \pm 1 $, and
while the unit step-discontinuity in the free-fermion $n(k)$ is
reduced to a step $Z < 1 $ in Landau Fermi-liquid theory, and a weaker
power-law discontinuity in the 1D Luttinger-liquid, the value of
$\Delta \nu $ which characterizes the Luttinger theorem remains fixed.

The case $\Delta \nu \ne \pm 1 $ occurs in the application to edge states
in the fractional quantum Hall effect (FQHE), which have been extensively
described by Wen\cite{wen}.   As a thought experiment, put  electrons
in a strong uniform magnetic field in the $z$-direction and confine
them to the $xy$-plane and the lowest Landau level.
Add a substrate potential
$V(y) $ that is translationally-invariant in the $x$-direction.  The
single-particle dispersion relation is
$$ \epsilon (k_x) \approx \frac {1}{2} \hbar \omega_c + V(k_x\ell^2) $$
where $\ell = (\hbar c /e B)^{1/2}$ is the ``magnetic length'', and
$V(y)$ is assumed to vary slowly on this scale.

In this geometry
the $N$-particle Laughlin state takes the form\cite{thouless}
$$ \Psi_m \propto \prod_{i<j}(z_i-z_j)^m \prod_i (z_i^Je^{-y_i^2/2\ell^2}) $$
where $z_j = \exp (i (x_j + iy_j)\ell) $.  This state has non-zero
$n(k)$ in the range
$$2\pi J/L = k_{F-} < k < k_{F+} =  2\pi (J + m (N-1)) /L ,$$
and the mean
occupation of states in this range is $1/m $. A caricature of this state
is given by occupying $N$ orbitals in this range so that $m-1$ empty
orbitals separate successive occupied orbitals, giving the mean occupation
$1/m$.   For $m=3$, the resulting occupation pattern is the binary string
$\ldots 1001001001001 \ldots $.

Such a state, interpreted as a Slater determinant, is the
Tao-Thouless state\cite{tt}, advanced as a rival model to the Laughlin state
in the early days following discovery of the FQHE.  Taken literally, this
state is not a good model for the FQHE, but the Tao-Thouless configuration
is in a real sense the ``root configuration'' of the Laughlin state:
if it is acted on by the projection operator on the Hilbert subspace of
wavefunctions that vanish as $(z_i-z_j)^m$ as any pair of particles
approach each other, the Laughlin state results.   The only
occupation number configurations contained in the Laughlin state are
those which can be obtained from the root Tao-Thouless state by a succession
of ``squeezing'' operations where a pair of occupied orbitals $k_1, k_2$
are replaced by $k_1', k_2'$ where $k_1+k_2 = k_1'+k_2'$ and
$k_1<k_1'<k_2'<k_2 $.

The bosonization treatment shows\cite{wen} that the occupation
number distribution $n(k)$  of the Laughlin state
vanishes as $|k-k_{Fi}|^{m-1}$ as the ``Fermi points'' are approached
from the interior of the occupied region.  The width of the
occupied region in $k$-space is $2\pi m \rho $ so the generalized
Luttinger theorem states that
$$  \delta \rho (x) / \delta k_{F\pm}(x')
= \mp (2 \pi m)^{-1} \delta (x-x') . $$
Since the {\it mean} value of $n(k)$ in the interior region is $1/m$, but
(for $m > 1$) $n(k)$ must be less than this near the edges, there
will be regions where $n(k)$ exceeds $1/m$.
(Numerical studies\cite{halrezayi} confirm that, going into the
interior of a wide strip of Laughlin state, $n(k)$ rises to
to a maximum, the oscillates as it relaxes to its uniform value.)
If $(k_{F+}-k_{F-})\ell >> 1 $
the edges are well separated, and the deviations from the mean
occupation $1/m$ will be localized in the edge regions.
In this case, if one edge moves as charge is added, it is easy to
see that the Luttinger theorem is satisfied, as the non-interacting
edges will preserve their shape as they move, and the ``extra occupation''
will go into the uniform interior region of the $n(k)$ distribution.  However,
the principle is still valid if the edges are spatially close and there is
{\it no} uniform density interior; in this case, the shape of the
$n(k)$ distribution will deform so as to satisfy the sum rule.
We thus see that the Luttinger theorem can have non-trivial non-Pauli-principle
extensions.

The total conserved quantities are thus written
$$ \Delta N = \sum_i \Delta N_i $$
where
$$ \Delta N_i = \int {dx \over 2 \pi } \Delta \nu_i \delta k_{Fi}(x) , $$
and
$$ \Delta P = \hbar \sum_i k_{Fi}^0\Delta N_i +
\frac{1}{2} \int {dx \over 2 \pi } \Delta \nu_i
(\delta k_{Fi})^2. $$
The crucial feature is that at the metallic $T = 0$ fixed point of the
system
the charges $\Delta N_i $ are {\it separately conserved}
as a consequence of momentum conservation at low energies.
In the microscopic Hamiltonian, of course, only the total charge
$\Delta N$ is conserved.  It is important to note that the Fermi
surface (in the sense of a singularity in the $n(k)$ distribution)
strictly only appears in the $T \rightarrow 0 $ limit, and it is only
in this limit that the Luttinger theorem becomes precise.  The existence
of the Fermi surface at $T = 0 $ {\it implies a separate charge conservation
law ($U(1)$ gauge symmetry) at each point on the Fermi surface}.  In some
sense, this is just a restatement of the Landau Fermi-liquid theory
principle that the lifetime of a Fermi-liquid quasiparticle becomes infinite
at the Fermi surface as $T \rightarrow 0 $.  From
this viewpoint, the formation of the Fermi surface as $T \rightarrow 0$
is a critical phenomenon, and it is not surprising that new symmetries
not present in the microscopic Hamiltonian appear at a critical point:
in a renormalization-group sense,
the symmetry-breaking terms present in the microscopic model will
correspond to irrelevant perturbations of the fixed-point effective
Hamiltonian.

We must now construct the most general effective Hamiltonian compatible
with separate charge conservation at each Fermi point.  For free
electrons, with $\Delta \nu = \pm 1 $, the Hamiltonian
$\Delta {\cal H}$ = $\Delta (H - E_F N)$ is given by
$$ \Delta {\cal H}_0 =  \frac{1}{2} \sum_i \int {dx \over 2\pi}
v_{Fi}( \delta k_{Fi}(x) )^2  $$
where the Fermi velocity is $v_{Fi}\Delta \nu_i $. (If charge is added
to change $k_{Fi}$ by $\delta k_{Fi}$ the mean momentum of the additional
particles is $k_{Fi}^0 + \frac {1}{2} \delta k_{Fi} $.)  The most general
possibility is essentially a Landau-type form
$$ \Delta {\cal H}^{eff} =
 \frac {1}{2} \sum_{ij} \int {dx \over 2 \pi } \int {dx' \over 2\pi }
\Gamma_{ij}(x-x') \delta k_{Fi}\delta k_{Fj} . $$
Here stability requires that
$$ \tilde{\Gamma}_{ij}(q) = \int dx \, \Gamma_{ij}(x) e^{iqx} $$
is a real positive-definite symmetric matrix.

It is useful to review the conservation laws and gauge symmetries of
(a)  of free electrons,
(b)  of the microscopic Hamiltonian,
(c) of this effective Hamiltonian, and
(d) of the Landau Fermi liquid ($D \ge 2$).
Free spinless fermions have a huge set of $U(1)$ gauge symmetries,
one for each
orbital with a conserved occupation number
in the $D$-dimensional reciprocal space ({\it i.e.},
there is a $D$-dimensional
manifold of gauge symmetries).

In contrast, only the global $U(1)$ symmetry is present in the microscopic
interacting model.  The degrees of freedom in the effective Hamiltonian
derived above correspond only to fermion orbitals close to the Fermi surface;
its explicit symmetries correspond to gauge changes where all orbitals directly
above and below
a given point on the Fermi surface have the same phase change, corresponding
to a $(D-1)$-dimensional manifold of gauge symmetries (one for each point
on the Fermi surface). (In 1D this is a discrete set, one per Fermi point.)

Finally, because the interactions in a Landau Fermi liquid leave a
low-energy spectrum of fermionic quasiparticles in one-to-one correspondence
with bare electron states ``near'' the Fermi surface, it has the
full $D$-dimensional manifold of gauge symmetries asymptotically close
to the Fermi surface, with conserved quasiparticle occupation numbers.

{From} this we conclude that the {\it existence} of the Fermi Surface
implies only the $(D-1)$-dimensional manifold of symmetries, not the
full $D$-dimensional manifold that reappears in the Landau theory.
This indicates that the Fermi surface can exist even when the Landau
quasiparticle picture is not applicable, and non-Fermi-liquid systems
can still have a Fermi surface obeying the Luttinger theorems.  This
is in fact the case in interacting 1D systems (which are not
Landau Fermi liquids), and leaves open the
possibility (discussed later) of such a possibility for $D > 1$.

The term ``Luttinger liquid''\cite{pwa} has been used by Anderson
to refer to a system with a Fermi surface that
obeys the Luttinger theorem, but which is not a Landau
Fermi-liquid.  As a historical note, when I coined the
term ``Luttinger liquid'' in the 1D context\cite{haldane81} I was referring
to  the exactly solvable Luttinger model\cite{lutmodel,liebmattis}
which in 1D  played the role of the ``zeroth order'' model to which residual
interactions are added, in analogy to Landau's use of the
free fermion model as the ``zeroth-order'' model for $D \ge 2$ Fermi liquids.
However,
it is serendipitous that Luttinger originated  both the theorem and the
model, and Anderson's interpretation of the term
``Luttinger liquid'' is particularly appropriate.

\section{Quantization of the Fermi-Surface variables.}
So far, I have derived expressions for $\Delta {\cal H}$, $\Delta P $, and
$\Delta N$ relative to the ground state as quadratic expressions in
terms of the local Fermi-surface displacements $\delta k_{Fi} (x) $.
To quantize these degrees of freedom, we need to find their
dynamical algebra.   This is found by the Tomonaga's {\it bosonization}
method\cite{tomonaga}.

The {\it total} electron density $\rho (x) $ has commuting Fourier
components $[\rho_q,\rho_{q'}] = 0 $, where
$$ \rho_q = \sum_{kk'} \delta_{k-k',q} (c^{\dagger}_kc_{k'} -
\langle 0 |c^{\dagger}_kc_{k'} |0 \rangle ) $$
The Tomonaga procedure is to decompose the total electron density into
components associated with each Fermi point:
$$ \rho_q \approx \sum_i \rho_{qi} .$$
This is done by defining small non-overlapping domains of width $\Lambda $ in
reciprocal space around each Fermi point:
$$ f_i(k) = \theta (\Lambda^2 - |k-k_{Fi}^0|^2 ) $$
Then
$$ \rho_{qi} = \sum_{kk'} f_i(k)f_i(k')\delta_{k-k',q}
(c^{\dagger}_kc_{k'} - \langle 0 |c^{\dagger}_kc_{k'} |0 \rangle ) .$$
The commutation relations are
$$[\rho_{qi},\rho_{q'j}] = \delta_{ij}\left
( \delta_{q+q',0} \sum_k f_i(k)f_i(k+q)
\langle 0 | n_{k+q} - n_{k} | 0 \rangle + X_i(q,q') \right ) ,$$
where
$$ X_i(q,q')  =  \sum_{kk'} f_i(k)f_i(k') [ f_i(k'+q)-f_i(k'+q') ]
(c^{\dagger}_kc_{k'} - \langle 0 |c^{\dagger}_kc_{k'} |0 \rangle ) .$$
For $|q| << \Lambda $, the factor $f_i(k)f_i(k+q) \approx 1 $ over
almost all the range where $\langle 0 | n_{k+q} - n_k | 0 \rangle $ is
non-negligible.  Similarly, the factor $f_i(k)f_i(k')[ f_i(k'+q)-f_i(k'+q') ]$
vanishes over most of the range of $k$ and $k'$.  Up to corrections of order
$|q|/\Lambda $, the commutator becomes
$$ [\rho_{qi},\rho_{q'j}] = \delta_{ij}\delta_{q+q',0} (qL/2\pi )
\Delta \nu_i ,$$
where $\Delta \nu_i  $  is the shift in $\langle 0 | n_k | 0 \rangle $ in
going from $k \approx k_{Fi} - \Lambda $ to $k \approx k_{Fi} + \Lambda $.
With the identification $2\pi \rho_i (x) = \Delta \nu_i \delta k_{Fi}(x) $,
we obtain the local form of the
dynamical algebra of the Fermi-surface dispacements: taking
$\Delta \nu _i $ to be the rational number  $\xi_i p_i /q_i $
with $p_i $ and  $q_i$ positive, and $\xi_i = \pm 1 $,
$$ [\delta k_{Fi}(x) , \delta k_{Fj} ] = (2 \pi i q \xi_i /p )\delta_{ij}
\delta '(x-x') . $$
(Here $\delta ' (x) $ is the derivative of the Dirac delta-function.)
Since the RHS of this commutation relation is a c-number, and the
Hamiltonian is quadratic, the effective Hamiltonian has been  reduced
to a harmonic oscillator problem.

The operator that creates an electron in a wave-packet of states
near the $i'th$ Fermi point has the form
$$ \Psi^{\dagger}_i(x) = A_i \exp \left ( i \int^x dx' (k_{Fi}^0 +
\delta k_{Fi}(x) \right ) .  $$
We may write this as
$$ \Psi^{\dagger}_i (x) = A_i e^{ i\varphi_i(x) } .$$
It changes the total charge at the Fermi point by one unit:
$$ [\Delta N_i , \Psi^{\dagger}_j (x) ] \ \delta_{ij} \Psi_i^{\dagger} (x) .$$
{From} the Tomonaga commutation relations,
$$ [\varphi_i(x), \rho_j(x') ] = i\delta (x-x') $$
so $\varphi_i(x)$ is the conjugate field to $\rho_i(x)$, and  is
subject to the {\it chiral constraint}
$$ \partial \varphi_i(x) / \partial x = k_{Fi}^0+  \delta k_{Fi}(x)
= k_{Fi}^0 + 2 \pi i (\xi_i q_i /p_i) \rho_i(x).$$
Thus $\rho_i (x) $ and $\varphi_i (x)$ are not independent canonical
fields, and $\rho_i(x)$ is proportional to the derivative of its own
conjugate field.
The explicit representation of $\varphi_i(x)$ is
\begin{equation}
\varphi_i(x) = \theta_i + \int_0^x dx' {\partial \over  \partial x' }
\varphi_i(x) .
\end{equation}
The integration constant $\theta_i$ is the conjugate phase to
the number operator $\Delta N_i$, and is {\it not}` constructed from the
hamonic fluctuation modes.

Integrating the commutation relations gives
$$ [ \varphi_i (x), \varphi_j(x') ] = (i \pi \xi_i q_i /p_i ) \delta_{ij}
{\rm sign }( x-x') . $$
{From} this we get (for $x \ne x'$)
$$ \Psi_i^{\dagger} (x) \Psi^{\dagger}_i(x')
= e^{i\pi q_i /p_i }
\Psi_i^{\dagger} (x') \Psi^{\dagger}_i(x). $$
For $\Psi_i^{\dagger}(x) $ to be a fermion creation operator, we
require that $p_i = 1 $ and that $q_i$ is {\it odd}.
Anticommutation of operators $\Psi_i^{\dagger}(x)$ and
$\Psi_j^{\dagger}(x')$ when $i$ and $j$ are different requires that
the {\it Klein factor} is
$$ A_i = \exp \left (  i (\pi /2)\sum_j {\rm sign} (i-j) \Delta N_j \right ) $$
where an arbitrary ordering of the Fermi points has been introduced.

\section{Diagonalizing the One-Dimensional Hamiltonian}

Let us first consider systems with no coupling between different
Fermi points, so $\Gamma_{ij}(x-x') = \gamma_i \delta_{ij} \delta (x-x')$,
with $\gamma_i > 0 $.  When $\Delta \nu = \pm 1 $, this
just describes free electrons, or Landau quasiparticles.  I will extend
the discussion to the Laughlin-state case $\Delta \nu = \pm 1/m$, following
WEN \cite{wen}, as this is relevant for fractional
quantum Hall effect edge states.

The Hamiltonian is a sum of decoupled terms associated with each
Fermi point, ${\cal H}$ = $\sum_i {\cal H}_i $,where
\begin{equation}
{\cal H}_i = \epsilon_F\Delta N_i + \frac{1}{2} \int dx \, \gamma_i
: (\rho_i(x))^2 :
\end{equation}
Using the commutation relation $[\rho_i(x),\rho_i(x')]$ =
$ i (2\pi m)^{-1}\xi\delta'(x-x')$, with $\xi = \pm 1$,
we get
\begin{equation}
[{\cal H},\rho_i(x) ] = i\xi_i v_{Fi} {\partial \over \partial x } \rho_i(x)
\end{equation}
where the Fermi velocity is given by $v_{Fi} = \gamma_i /(2 \pi m) $.
In terms of boson creation and destruction operators obtained by normalizing
the Fourier components of $\rho_i(x)$, the Hamiltonian is given by
\begin{equation}
{\cal H}_i = \epsilon_{Fi} \Delta N_i + v_{Fi}
\left ( \frac{1}{2} {2 \pi m \over L } (\Delta N_i)^2
+ \sum_q \theta (q \xi_i) q b^{\dagger}_qb_q  \right ).
\end{equation}
The $m$-dependence shows up only in the term involving the total charge
at the Fermi point.   It is straightforward to compute the Greens function
using the bosonic representation of the fermion creation operators:
\begin{equation}
\langle \psi_i(x,t)\psi_i{\dagger}(0,0) \rangle
\approx {Z_i e^{i(k_{Fi}x-\epsilon_{Fi}t)} \over (x-v_{Fi}\xi_it)^m }
\end{equation}
where $Z_i$ is an undetermined normalization.
The Fourier transform gives the singular part of the occupation factor
$n(k)$:
\begin{equation}
n(k) = n(k)_{reg} + Z_i |k-k_{Fi}|^{-1} (k-k_{Fi})^m .
\end{equation}
More generally, the electron creation operator will take the form
\begin{equation}
\psi^{\dagger}(x) = \sum_{\{n_i\}} A(\{n_i\}) e^{i \sum_i n_i \varphi_i(x)} ,
\end{equation}
where $A(\{n_i\})$ = 0 unless $\sum_i n_i $ = 1.   The occupation
factor $n(k)$ will in general have singularities at
$k$ = $\sum_i n_i k_{Fi} $.

I now turn to the full problem, when $\Gamma_{ij}(x)$
$\approx$ $\Gamma_{ij} \delta (x-x') $is not diagonal, and the
different Fermi points are coupled.   The problem is to diagonalize
\begin{equation}
{\cal H} = {2 \pi \over 2 L } \sum_{ij} \Gamma_{ij} \rho_{qi}\rho_{-qj}
\end{equation}
\begin{equation}
[\rho_{qi},\rho_{q'j} ] = (qL/2\pi) \delta_{q+q',0}(\delta_{ij}\xi_i /m_i ) .
\end{equation}
The normal modes $\rho^{\lambda}_q $
are obtained from a real non-symmetric matrix eigenproblem:
\begin{equation}
\rho^{\lambda}_q = \sum_i \chi^{\lambda}_i \rho_{qi} ;
\quad \rho_{qi} = \sum_{\lambda}\psi^{\lambda}_i \rho^{\lambda}_q .
\end{equation}
where
\begin{equation}
\sum_i \chi^{\lambda}_i\psi^{\lambda '}_i
 = \delta^{\lambda\lambda '} ;
\quad \sum_{\lambda} \chi^{\lambda}_i \psi^{\lambda}_j = \delta_{ij} .
\end{equation}
and
\begin{equation}
\psi^{\lambda}_i = (\Gamma^{-1})_{ij} \chi^{\lambda}_j ;\quad
(\xi_j/m_j) \chi^{\lambda}_j = v^{\lambda}\psi^{\lambda}_j .
\end{equation}
Here, the eigenvalues $v_j$ is real, since $\Gamma_{ij}$ is positive definite;
these are the set of renormalized normal-mode velocities.
To calculate the Greens' function, for example,  the expression for
$\rho_{qi}$ in terms of the  normal modes $\rho_q^{\lambda}$
must be substituted into $\varphi_i(x)$.  The result for the diagonal
Greens function at Fermi point $i$ is
\begin{equation}
G_i(x,t) = Z_i \prod_{\lambda} (x-v^{\lambda}t)^{-\alpha_{i\lambda}},
\end{equation}
where $\alpha_{i\lambda} $ = $(\chi^{\lambda}_i)^2/2\pi |v_{\lambda}| $.
Detailed examination shows that $\sum_{\lambda} \alpha_{i\lambda}  \ge m_i $,
so the Fermi surface singularity in $n(k)$ is always {\it weakened} by
coupling between different Fermi points.   General expressions for
correlation exponents when many different Fermi points interact
have also been developed by PENC and SOLYOM\cite{penc}.

I now come to what is one of  the central ideas of the ``Luttinger
liquid theory''\cite{haldane81}.   This is that (unless crossover
to a non-Luttinger-liquid fixed point occurs), the Landau parameters
$\Gamma_{ij}$ can be determined by identifying the excitations
of a finite interacting system with periodic boundary conditions,
that are associated with changing the net charges at the different
Fermi points.   If we suppress the finite-wavelength harmonic
oscillator modes, the residual charge terms in the excitation
spectrum are:
\begin{equation}
\Delta P = \sum_i k_{Fi}^0\Delta N_i + {\pi \over L} \xi_im_i (\Delta N_i)^2 ,
\end{equation}
\begin{equation}
\Delta {\cal H}  = \sum_i \epsilon_{Fi}\Delta N_i + {\pi \over L}
\sum_{ij} \Gamma_{ij} \Delta N_i \Delta N_j .
\end{equation}
By fitting the low-energy excitations of a system studies by finite-size
numerical diagonalization or the Bethe Ansatz to this form, the low-energy
effective Hamiltonian is determined, and its asymptotic correlations,
{\it etc.} can be calculated from them.   The program was demonstrated
in detail on the spinless fermion system  equivalent to the XXZ spin chain
in a magnetic field\cite{haldane80}, where there are just two Fermi
points, R and L.  The correspondence between the parametrization
of \cite{haldane80} and that used here is that
$\Gamma_RR$ = $\Gamma_LL$ = $v_N+ v_J$, and $\Gamma_{RL}$ = $\Gamma_{LR}$
= $v_N-v_J$.

Finally, we must discuss what happens when we include electron spin.
If a magnetic field is present, this is just an application of the
``spinless'' treatment with double the number of Fermi points.
However, if the Fermi points have  spin degeneracy, the full
non-Abelian $SU(2)$ symmetry at each Fermi point must be considered.
In this case, coupling between the spin degrees of freedom at different
Fermi points is incompatible with the existence of independent
spin rotation symmetries at each Fermi point.  The renormalization
group treatment\cite{lutheremery} shows that either the couplings scale
to zero at the low-energy fixed point, or they scale to a strong coupling,
non-Luttinger liquid fixed point.

\section{Generalization to Higher Dimensions}

In the one-dimensional systems, the low-energy degrees of freedom
are described by an independent set of bosonic variables at each Fermi
point, representing harmonic fluctuations of the Fermi surface.
LUTHER\cite{luther} made a pioneering attempt to describe higher-dimensional
Fermi surface degrees of freedom by bosonization, but his ``tomographic''
construction restricts attention to particle-hole pairs carrying
a net momentum strictly normal to the local Fermi vector.
Recently, I found that the Tomonaga bosonization algebra
could be formulated
in a more explicitly higher-dimensional form; this formulation has been
reviewed by HOUGHTON and MARSTON\cite{houghton}
who use it to discuss corrections
to the $T$-linear specific heat of Fermi liquids.  [Since these lectures
were given, a very similar
treatment has also been independently been developed by
CASTRO-NIETO and FRADKIN\cite{fradkin}.]

The basic idea is again to remark that the Luttinger theorem expresses
the ground-state particle density and momentum density purely in terms of
the ($d$-1)-dimensional Fermi surface in the $d$-dimensional reciprocal
space defined by the singularity in the ground-state occupation number
distribution $n(k)$.  This of course is making the assumption that
the ground state $n(k)$ has such a feature, and that no BCS or density-wave
instability occurs at low temperatures.

It is generally believed that
in the absence of any other instability, a BCS instability in some channel
will always occur below some critical temperature,
and destroy the singularity in $n(k)$, so we must in principal
first exclude the BCS processes from the effective Hamiltonian, then
restore the (presumably relevant) perturbations.  The BCS terms can be
recognized as deriving from the special shape of the Fermi surface,
which in the presence of either time-reversal symmetry or spatial
inversion symmetry, has inversion symmetry in reciprocal space.
If, for the moment, we ignore or conceptually abolish this symmetry,
it should in principle be possible to have higher-dimensional interacting
systems with a stable Fermi-surface singularity in their ground state.
{From} this viewpoint the BCS instability, like density-wave
instabilities, is classified as a special feature associated with
a particular class of Fermi surface shapes.

In the Fermi-liquid theory, the Fermi surface singularity is a
step discontinuity across which $n(k)$ decreases by an amount $Z$,
but I will make no {\it a priori} assumption about the nature of the
singularity, and merely use the property that it defines a surface
satisfying the Luttinger theorem.   It will become clear that, in
dimensions greater than one, non-Fermi-liquid behavior (such as spin-charge
separation)
requires a sufficiently-strong singular forward-scattering term in
the phenomenological Landau parameters; the possible existence of such
singular terms, which ANDERSON\cite{pwa} has argued
are generically present in two-dimensional fermion systems, is
controversial, and
currently a subject of active
investigation, though to date, no  microscopic treatment has clearly
demonstrated
the existence of such terms.

In general dimensions, the Fermi surface is described by a function
$\vec{k}_F(s)$, where $s$ is a ($d$-1)-dimensional surface coordinate.
On large lengthscales, I again describe the system in terms of
local fluctuations of the Fermi surface geometry:
\begin{equation}
\vec{k}_F(x,s) = \vec{k}_{F0}(s) + \hat{n}(s)\kappa_{\parallel}(x,s)
+ \hat{t}_{\mu}(s)\kappa_{\perp}^{\mu}(x,s)
\end{equation}
where $x$ now represents a $d$-dimensional spatial coordinate,
$\hat{n}(s)$ is the local direction of the Fermi velocity (the outward
normal direction of the Fermi surface), and $\{\hat{t}^{\mu}(s)\}$ are
a basis of the $d-1$ unit vectors tangent to the Fermi surface.   A
treatment that is quadratic in the normal and tangential
fluctuations $\kappa_{\parallel}(x,s),\kappa_{\perp}^{\mu}(x,s)$
will be developed, with the recognition that the transverse
fluctuations are essentially gauge variables describing
infinitesimal {\it reparametrizations} of the (curvilinear)
surface coordinates $s \equiv \{s_1,\ldots, s_{d-1}\} $, without
change in the shape of the surface.  The physical
quantities such as the local fluctuation in total particle density
and momentum density can be completely expressed in terms of
the normal fluctuations $\kappa_{\parallel}(x,s)$.   Classically, the
gauge condition $\kappa_{\perp}^{\mu}(x,s) $ = 0 could be imposed;
however, since as a
quantum operator $\kappa_{\perp}^{\mu}(x,s)$ has non-trivial
commutation relations, the gauge condition is the action
$\kappa_{\perp}^{\mu}(x,s) |\Psi \rangle $ = 0 on physical
states $|\Psi \rangle $.

Thus
\begin{equation}
\Delta N = \int d^{d}x \, \rho (x)
\end{equation}
where the local change $\rho(x)$ in particle density relative to the
ground state is given by
\begin{equation}
\rho (x) = \int {\omega (s) d^{d-1}s \over (2 \pi ) ^d } \, \kappa_{\parallel}
(x,s) ,
\end{equation}
where $\omega (s)$ is the surface area measure.  Similarly,
\begin{equation}
\Delta \vec{P} = \hbar \int d^dx \, \vec{\Pi} (x) ,
\end{equation}
where
the local change in momentum density is
\begin{equation}
\vec{\Pi} (x) =
\int {\omega (s) d^{d-1}s \over (2 \pi ) ^d } \,
\left ( \vec{k}_{F0}(s) \kappa_{\parallel}(x,s)
+ \frac{1}{2} \hat{n}(s) :(\kappa_{\parallel}(x,s))^2 :\right ) .
\end{equation}
(Here the notation $:(\kappa_{\parallel}(x,s))^2 :$ anticipates
the normal-ordering needed in the quantized formulation.)  I note that
the first and second functional derivatives of $\vec{\Pi}(x)$
with respect to $\kappa_{\parallel}(x,s)$ define the two fundamental
geometric properties of the Fermi surface, its {\it shape}
$\vec{k}_{F0}(s)$ and its {\it orientation} $\hat{n}(s)$.

In the spirit of Landau theory, the effective Hamiltonian is
a quadratic form where $\Delta (H - \epsilon_F N)$ is given by
\begin{equation}
\frac{1}{2} \int d^{d}x \int d^{d}x'
\int {\omega (s) d^{d-1}s \over (2 \pi ) ^d }
\int {\omega (s') d^{d-1}s' \over (2 \pi ) ^d } \,
\Gamma(s,s';x-x') :\kappa_{\parallel}(x,s)\kappa_{\parallel}(x',s'): ,
\end{equation}
where
\begin{equation}
\Gamma(s,s';x) = {(2\pi)^d \over \omega (s) } v_F(s) \delta^{d-1}(s-s')
\delta^d(x-x') + f(\vec{k}_F(s),\vec{k}_F(s');x) .
\end{equation}
Note that the kinetic energy (``effective mass'') term appears as a
$(2d-1)$-dimensional delta-function term in $\Gamma$. The conventional
Landau $f$-function is given by
\begin{equation}
f(\vec{k}_F(s),\vec{k}_F(s')) = \int d^d x\,
f(\vec{k}_F(s),\vec{k}_F(s');x)
\end{equation}
and $\delta^{d-1}(s-s')$ =
$\omega (s) \delta^{d-1}(\vec{k}_F(s)-\vec{k}_F(s'))$.
The stability of the Fermi surface against spontaneous shape
deformations requires that
\begin{equation}
\tilde{\Gamma}(s,s';\vec{q} = \int d^dx \, e^{i\vec{q}\cdot \vec{x}}
\Gamma (s,s';x)
\end{equation}
is a positive-definite quadratic form in $s,s'$ for all
$\vec{q}$.

In metals, the Fermi surface will in general consist of a number of
distinct manifolds in the primitive
(or sometimes extended) Brillouin zone.  The formal
integral $\int d^{d-1}s $ can be considered to implicitly include sums over
such discrete band indices distinguishing distinct manifolds.
In the discussion here, I will also assume that the Fermi surface sheets
are smooth differentiable (orientable) manifolds with finite
curvature at all points.  If some microscopic parameter is
varied through a critical point at which the Fermi-surface topology
changes, at the critical point there will be a Van-Hove singularity
on the Fermi surface at which the curvature is infinite, and the
linearized treatment of fluctuations will fail.  Systems at or close to
such critical points may also be a place to look for non-Fermi-liquid
behavior.

Having quadratic expressions for the various conserved quantities
in terms of the fluctuation variables $\kappa_{\parallel}(x,s)$,
we now need the $d$-dimensional version of the quantum
algebra of the Fermi-surface displacements.   I will first give the
answer, then sketch its derivation using a generalization of
Tomonaga's method.   The commutation relations are
\begin{equation}
[\kappa_{\parallel}(x,s),\kappa_{\parallel}(x',s') ]
= (2\pi )^d i D[\delta^d(x-x')\delta^{d-1}(\vec{k}_F(s)-\vec{k}_F(s'))],
\end{equation}
where
\begin{equation}
D[f(x,s)] \equiv (\hat{n}(s)\cdot \vec{\nabla} ) f(x,s)
\end{equation}
is a ``covariant derivative''.  The fact that the RHS of the commutation
relation is a $c$-number means
that $\kappa_{\parallel}(x,s) $ can be expressed as a linear
combination of harmonic oscillator variables.

As a side comment, I note that if a static magnetic
field $\vec{B}(x)$ is present, the covariant derivative
becomes
\begin{equation}
D[f(x,s)] \equiv (\hat{n}(s)\cdot \vec{\nabla} ) f(x,s)
 + 2 \pi \Phi_0^{-1} \hat{n}(s) \times \vec{B}(x) \cdot
\hat{t}^{\mu} {\partial \over \partial s^{\mu} } f(x,s) .
\end{equation}
Here $\Phi_0$ is the London flux quantum  $2\pi \hbar /e$.
This derivative encodes the information that if a wave packet of states
centered at real-space position $x$ and Fermi-surface point $s$ is made,
the spatial coordinate evolves in the direction
$\hat {n}(s)$ and the Fermi-surface coordinate evolves in the direction
$\vec{B}(x)\times \hat{n}(s)$.   The {\it rate} at which this evolution
takes place is however encoded in the effective Hamiltonian, rather than
the Fermi-surface displacement algebra.   In what follows, I will
assume that no magnetic field is present.

We may also write $\kappa_{\parallel} (x,s) $ = $D[\varphi (x,s)]$, where
\begin{equation}
[\varphi (x,s) ,\kappa_{\parallel}(x',s') ]
= (2\pi )^d i \delta^d(x-x')\delta^{d-1}(\vec{k}_F(s)-\vec{k}_F(s')) ,
\end{equation}
so $\kappa_{\parallel}$ is the derivative of its own conjugate field.
The scalar phase field $\varphi (x,s) $ obeys the algebra
\begin{equation}
[\varphi (x,s), \varphi (x,s') ] =
i \pi \delta^{d-1}(\vec{k}_F(s) - \vec{k}_F(s'))
\delta^{d-1}(\hat{n}(s)\times (\vec{x} - \vec{x}'))
{\rm sgn}(\hat{n}(s) \cdot (\vec{x}-\vec{x}')) .
\end{equation}

This allows the commutation relations of both the normal and
tangential Fermi-surface fluctuations to be obtained from the identification
\begin{equation}
\vec{\kappa}(x,s) = \vec{\nabla} \varphi (x,s) .
\end{equation}

The microscopic derivation of the commutation relations follows Tomonaga's
approach to the 1D system.   First, all large momentum-transfer
scattering processes are in principle integrated out, in
a renormalization-group  sense, leaving an effective Hamiltonian
that keeps only electron states within a reciprocal space distance
$\Lambda $ from the Fermi surface.  The Fermi surface is then broken up into
``patches'' of area about $\lambda^{d-1}$, and reciprocal space
near the Fermi surface is broken up into little domains centered on
each patch.  It is convenient  to consider these domains as little spheres
of radius $\Lambda $ centered on a mesh of points representing a
triangulation of the Fermi surface, but since
we a seeking an effective long-wavelength theory,
the detailed cutoff structure
should not matter.  In this case, the ``patches'' would be circular
($d-1$)-spheres; since a space cannot be fully tiled with non-overlapping
spheres, the radius $\Lambda $ would have to be chosen so that areas
of Fermi surface that are double-counted because  circular patches
overlap are exactly compensated by omitted areas between the patches,
so the sum of all patch areas exactly equals the Fermi surface area.
Other tiling schemes could be used.  An important condition is that $\Lambda$
must be small enough so that within each domain, the Fermi-surface is
quasi-flat.  This means that there must be some finite upper bound
to the Fermi surface curvature, and excludes the possibility of a Van Hove
singularity on the Fermi surface.

Now let $\theta_{\alpha}(\vec{k})$ = 1 if $\vec{k}$ is inside the spherical
domain centered on the patch with label $\alpha$, and let it vanish otherwise.
Then we define
\begin{equation}
\kappa_{\alpha}(\vec{q})
=
\sum_{\vec{k}} \theta_{\alpha}(\vec{k} + \vec{q} ) \theta_{\alpha}(\vec{k})
\left ( c^{\dagger}_{\vec{k}+\vec{q}} c_{\vec{k}} - \delta_{\vec{q},0}
\langle n_{\vec{k}} \rangle _0 \right ) ,
\end{equation}
and
\begin{equation}
\Pi_{\alpha}(\vec{q})
=
\sum_{\vec{k}} \theta_{\alpha}(\vec{k} + \vec{q} ) \theta_{\alpha}(\vec{k})
\hat{n}_{\alpha}\cdot (\vec{k} - \vec{k}_{F\alpha} + \frac{1}{2} \vec{q} )
\left ( c^{\dagger}_{\vec{k}+\vec{q}} c_{\vec{k}} - \delta_{\vec{q},0}
\langle n_{\vec{k}} \rangle _0 \right ) .
\end{equation}
We must now approximately evaluate the commutation relations, and drop
``cutoff-dependent terms'' in the spirit of Tomonaga's treatment.
Then (ignoring any overlap between patches)
\begin{equation}
[\kappa_{\alpha}(\vec{q}),\kappa_{\alpha'}]
= \delta_{\alpha\alpha'} \left (
X_{\alpha}(\vec{q},\vec{q}') +
\delta_{\vec{q}+\vec{q}',0} g_{\alpha}(\vec{q} ) \right )
\end{equation}
where
\begin{equation}
X_{\alpha}(\vec{q},\vec{q}') =
\sum_{k} \theta_{\alpha} (\vec{k} + \vec{q} + \vec{q}' )
\left ( \theta_{\alpha} (\vec {k} +\vec{q} ) -
\theta_{\alpha}(\vec{k} + \vec{q}') \right ) \left
(c^{\dagger}_{\vec{k} + \vec{q}
+ \vec{q}' } c_{\vec{k}} - \delta_{\vec{q}+\vec{q}'}
\langle n_{\vec{k}}\rangle_0 \right  ).
\end{equation}
This operator-valued term can be neglected for
$|\vec{q}|, |\vec{q}'| \ll \Lambda $, as the factor
$(\theta_{\alpha}(\vec{k} + \vec{q}) - \theta_{\alpha}(\vec{k} +
\vec{q}') )$ vanishes except at the surface of the spherical domain,
and is a ``cutoff-dependent correction''.   Because (in contrast to
the original Tomonaga calculation in 1D) some of this ``correction''
involves states at the Fermi surface, this is perhaps not as
innocuous an approximation in higher dimensions, but appears
to be valid in the long-wavelength limit.   The residual term in the RHS of
commutation relation is the $c$-number term
\begin{equation}
g_{\alpha}(\vec{q}) = \sum_{k} \theta (\vec{k} + \vec{q} )
\theta (\vec{k} \langle (n_{\vec{k} + \vec{q} } - n_{\vec{k} } )\rangle _0 .
\end{equation}
In this case, for $|\vec{q} | \ll \Lambda $ the value of this is just the
number of allowed $k$-space points inside the volume of
reciprocal space swept out by displacing the patch of Fermi surface
by $\vec{q}$.    Note that it is independent of the detailed structure
of $\langle n_{\vec{k}} \rangle _0 $ near the Fermi surface and  only involves
the change in asymptotic values of
the occupation factor from deep inside to far outside the Fermi surface.  Thus
the commutation relation becomes
\begin{equation}
[\kappa_{\alpha}(\vec{q} ) ,\kappa_{\beta} (\vec{q}')]
= aV\delta_{\alpha\beta}\delta_{\vec{q}+\vec{q}'.0}
\hat{n}_{\alpha}\cdot \vec{q} ,
\end{equation}
where $a$ is the surface area of the patch on the Fermi surface.
This a generalization of an Abelian ($U(1)$) Kac-Moody algebra.
The other commutation relations
are similarly evaluated at long wavelengths
as
\begin{equation}
[\Pi_{\alpha}(\vec{q}),
\Pi_{\beta}(\vec{q}')] =
\hat{n}_{\alpha}\cdot \vec{q} \kappa_{\alpha}(\vec{q}+\vec{q}')\delta_{\alpha\beta}
\end{equation}
and
\begin{equation}
[\Pi_{\alpha}(\vec{q},\Pi_{,\beta}(\vec{q}')]
= \hat{n}_{\alpha}\cdot (\vec{q} - \vec{q}') \delta_{\alpha\beta}
\Pi_{\alpha} (\vec{q} + \vec{q}') +
\frac{1}{12} \delta_{\alpha\beta} aV \delta_{\vec{q} + \vec{q}' }
(\hat{n}_{\alpha} \cdot \vec{q} ) ^3
\end{equation}
The full structure is the generalization of the Kac-Moody and associated
Virasoro algebras to $d > 1$, where at each point on the Fermi surface,
the spatial coordinates separate into one special normal direction
$\hat{n}_{\alpha}$ along which derivatives are taken, and
$d - 1$ transverse directions.  This differs from the earlier
``tomographic'' bosonization
proposed by Luther\cite{luther} which only considers the case when
$\vec{q}$ is parallel to $\hat{n}_{\alpha}$, and does not allow any
natural coupling between Fermi surface points where the normals are
not parallel or antiparallel.  The thermodynamic limit may now be taken,
and Kronecker delta-functions on the discrete mesh of reciprocal
space points allowed by periodic boundary conditions become Dirac
delta-functions.

The fermion phase field $\varphi_{\alpha} (x) $ is formally
given by
\begin{equation}
  \varphi_{\alpha} (x) = \theta_{\alpha} (x_{\perp}) +
\int_0^{x} dx_{\parallel}'
(\hat{n}_{\alpha}\cdot \vec{\nabla}) \kappa_{\parallel\alpha}(x) ,
\end{equation}
where $\theta_{\alpha} (x_{\perp}) $ is an integration constant.
The operator $\exp ( i \theta_{\alpha} (x_{\perp}) )$ must be interpreted
as the operator that adds charge on the Fermi surface patch $\alpha$.
This will in a wave packet that is completely delocalized along
the direction in real space parallel to $\hat{n}(s)$, but is localized
in the transverse direction to within a distance $\Lambda^{-1}$ of
the transverse  spatial coodinate $x_{\perp}$.   The electron
creation operator will again be proportional
to $\exp (i \varphi_{\alpha}(x)) $;  as in the 1D case, a Klein factor can be
added to make electron operators defined in different Fermi-surface
patches anticommute.
Electron creation operators defined in the same patch, and sharing
(to within $\Lambda^{-1}$)
a common transverse spatial coordinate $x_{\perp}$ will also automatically
anticommute.       Anticommutation of creation operators in the same
patch, but at different transverse spatial coordinates must be imposed
through the integration constant $\theta_{\alpha} (x_{\perp})$:
\begin{equation}
e^{i\theta_{\alpha}(x_{\perp})}
e^{i\theta_{\alpha}(x_{\perp}')}
+
e^{i\theta_{\alpha}(x_{\perp})'}
e^{i\theta_{\alpha}(x_{\perp})}
 = 0
\quad (| x_{\perp} - x_{\perp}' | \gg \Lambda^{-1} ).
\end{equation}
If there are only one or two transverse dimensions, this can be represented
with Jordan-Wigner or ``anyon'' gauge fields.

It is straightforward to include spin degrees of freedom in the
preceding treatment, and define
\begin{equation}
\kappa_{\alpha\uparrow}(\vec{q}) +
\kappa_{\alpha\downarrow}(\vec{q}) =
2\kappa_{\alpha}(\vec{q})
; \quad
\kappa_{\alpha\uparrow}(\vec{q}) -
\kappa_{\alpha\downarrow}(\vec{q}) =
2\sigma^{z}_{\alpha}(\vec{q}) .
\end{equation}
Then
\begin{equation}
[\kappa_{\alpha}(\vec{q} ) ,\kappa_{\beta} (\vec{q}')]
= \frac{1}{2}aV\delta_{\alpha\beta}\delta_{\vec{a}+\vec{q}'.0}
\hat{n}_{\alpha}\cdot \vec{q} ,
\end{equation}
\begin{equation}
[\sigma^a_{\alpha}(\vec{q} ) ,\sigma^b_{\beta} (\vec{q}')]
= \delta^{ab}\left (
\frac{1}{2}aV\delta_{\alpha\beta}\delta_{\vec{a}+\vec{q}'.0}
\hat{n}_{\alpha}\cdot \vec{q}  + i \epsilon^{abc}\sigma^c{\alpha}
(\vec{q} + \vec{q}' ) \right ) .
\end{equation}
The spin degrees of freedom now obey a $d > 1$ version of the non-Abelian
$SU(2)$ Kac-Moody algebra.

\section{Diagonalization of the Harmonic Oscillators}

Within the approximation that keeps only the terms which are quadratic in
the Fermi surface fluctuations, the problem of the interacting Fermi system reduces
to a harmonic oscillator problem.  In fact this is (of course)
essentially just the zero-sound problem of Fermi liquid theory.   It is
convenient to first rescale the Fermi surface normal fluctuation
operators, and write
\begin{equation}
\tilde{\rho}_{q\alpha} = \left ( {v_{F\alpha} \over aV } \right ) ^{1/2}
\kappa_{\alpha}(\vec{q}) .
\end{equation}
Then the commutation relations become
\begin{equation}
[\tilde{\rho}_{q\alpha},\tilde{\rho}_{q'\beta}] =
\delta_{\alpha\beta} \delta_{\vec{q} + \vec{q}',0} \omega^0_{\alpha} (\vec{q}),
\end{equation}
where
$\omega^0_{\alpha}(\vec{q}) $ =
$ v_{F\alpha} (\hat{n}_{\alpha} \cdot \vec{q} )$.
The Hamiltonian is now
\begin{equation}
\frac{1}{2} \sum_{q} \sum_{\alpha\beta}\tilde{\Gamma}_{\alpha\beta} (\vec{q})
\tilde{\rho}_{q\alpha} \tilde{\rho}_{-q\beta} ,
\end{equation}
where $\tilde{\Gamma}_{\alpha\beta}$ is the positive
definite matrix
\begin{equation}
\tilde{\Gamma}_{\alpha\beta}(\vec{q}) = \delta_{\alpha\beta}
+ \Lambda^{d-1} { f(\vec{k}_{F\alpha} , \vec{k}_{F\beta} ; \vec{q})
\over (v_{F\alpha}v_{F\beta})^{1/2}  } ,
\end{equation}
which is an even function of $\vec{q}$.
We now see the reason why the 1D case (with no transverse degrees of
freedom) is special.   {\it In the scaling limit
$\lambda \rightarrow 0$, with  $(d -1) > 0 $, $\tilde {\Gamma}_{\alpha\beta}
\rightarrow \delta_{\alpha\beta}$ (free fermions) }, unless either
(a) $f(\vec{k}_{F\alpha},\vec{k}_{F\beta}) \rightarrow \infty$ or
(b) $v_{F\alpha}v_{F\beta} \rightarrow 0 $.  Put another way, for
$d > 1$
unless the effective  Landau parameters are singular, the coupling
between different
patches on the Fermi surface contains a factor (patch area)/(Fermi surface
area) = $1/N_{patch}$,
and the only modification of the collective excitation spectrum
is that a finite number of zero-sound collective modes are pushed up above
the continuum of modes with frequencies up to $v_F |q| $.

Formally, to diagonalize the harmonic oscillator problem, we must
express its normal modes $\rho^{\lambda}_q$ in terms of the local modes
$\tilde{\rho}_{q\alpha}$ defined on each patch:
\begin{equation}
\rho^{\lambda}_q = \sum_{\alpha} \chi^{\lambda}_{\alpha} (\vec{q})
\tilde{\rho}_{q\alpha}
\end{equation}
with the inverse relation
\begin{equation}
\tilde{\rho}_{q\alpha} =
\sum_{\lambda}
\psi^{\lambda}_{\alpha}(\vec{q})\rho^{\lambda}_{\alpha} .
\end{equation}
If this involved the 1D problem of the coupling of a single pair
of Fermi points with opposite-direction normals, this problem would
be simple to treat by expressing it in terms of canonically-normalized
boson creation and annihilation operators, and carrying out a
Bogoliubov transformation.
In the general case, this is not so convenient; instead it can be
recognized (of course) as the zero-sound problem, and regarded as a
real non-symmetric eigenproblem where all the eigenvalues $\omega^{\lambda}$
are real
because the matrix $\tilde{\Gamma}_{\alpha\beta}$ is positive definite.
Then
\begin{equation}
\sum_{\beta} \tilde{\Gamma}_{\alpha\beta}
 \omega^0_{\beta}\chi^{\lambda}_{\beta}
= \omega^{\lambda}\chi^{\lambda}_{\alpha}
\end{equation}
and
\begin{equation}
\sum_{\beta} \omega^0_{\alpha}
 \tilde{\Gamma}_{\alpha\beta}\psi^{\lambda}_{\beta}
= \omega^{\lambda}\psi^{\lambda}_{\alpha}
\end{equation}
with the orthogonality relation
\begin{equation}
\sum_{\alpha} \psi^{\lambda}_{\alpha}\chi^{\lambda '}_{\alpha} =
\delta^{\lambda\lambda'} .
\end{equation}
The eigenvalues $\omega^{\lambda}(\vec{q}) $ are real, with the
symmetry $\omega^{\lambda}(-\vec{q})$ = $-\omega^{\lambda}(\vec{q})$.
This formal solution is useful for carrying out calculations of
correlation functions, {\it etc.}

If the Fermi surface is regarded
as the analog of the ``order parameter''
of a metal, its shape fluctuations are its ``Goldstone modes''.
The cutoff $\Lambda$ means that only modes with $|q| < \Lambda $
should be counted as independent, and there is then one
linear-dispersion ``Goldstone mode'' per patch. These modes
have a spectrum of velocities that becomes continuous in the
in the limit $\Lambda \rightarrow 0 $.  For $\vec{q}$ along some direction
$\hat {\Omega}$, the density of mode velocities
remains finite at zero frequency
provided that some part of the Fermi surface is tangential to
$\hat{\Omega}$.  It is this feature that gives the universal $T$-linear
specific heat of fermion liquids in this formalism, in contrast to the
$T^d$ specific heat of systems where the Goldstone mode velocities
remain finite.  The $T$-linear heat capacity (or entropy) is extensive
in not only the real-space volume $V$, but also in the Fermi-surface
area, and derives from the (2$d$-1)-dimensional delta-function
term proportional to $\delta^d(x)\delta^{d-1}(\vec{k}_F(s) - \vec{k}_F(s'))$
in $\Gamma (s,s';x)$.  Furthermore, this term controls the upper limit
(the Fermi velocity) to the continuous spectrum of velocities of modes
traveling in a given direction.   {\it In the absence of a contribution
to this delta-function part of $\Gamma$ coming from singular terms in the
Landau parameters, there is no renormalization of the Fermi velocity or
the $T$-linear specific heat}.

The collective zero-sound modes give  $T^d$ corrections to the
low-temperature specific heat that depend on the Landau parameters.
In fact, in three dimensions, the leading corrections to the
$T$-linear specific heat is the $T^3\log T$ term elucidated in detail
by PETHICK and CARNEIRO\cite{pethick} within a standard Fermi-liquid approach:
HOUGHTON and MARSTON\cite{houghton} have recently reported that
such terms can also be recovered in the bosonized approach discussed here.

Because of this ``all-or-nothing'' character of the contribution of
the Landau $f(\vec{k}_F(s),\vec{k}_F(s'))$ couplings to a shift in the
Fermi velocity and the $T$-linear specific heat, it is instructive the
consider the case when $f(\vec{k}_F(s),\vec{k}_F(s'))$ is a smooth
function with a strong anomaly in a narrow cone around the
forward scattering direction, and allow some control parameter
to continuously evolve this anomaly into a true delta-function.
If the Fermi surface is spherical, as the cone of the
anomaly becomes narrower,  more terms in the spherical-harmonic
expansion of the Landau parameters are needed to adequately represent
it.   Roughly speaking, there will be one extra zero-sound collective mode
pushed out above the continuum of modes with $0 < \omega < v_F|q|$
for each additional spherical harmonic term that becomes significant.
These modes will proliferate and become dense in the range
$v_F |q| < \omega < (v_F + \delta v_F)|q|$ as the Landau parameters
develop a delta-function singularity in the forward scattering
direction.  Similarly, when the anomaly becomes pronounced, the
specific heat will develop a ``pseudo-$T$-linear'' regime characterized
by what will become the renormalized Fermi velocity, but which crosses over
to the true unrenormalized $T$-linear regime at lower temperatures; this
crossover temperature vanishes as the singularity in the Landau parameters
develops.

\section{Spin-Charge Separation}

{\it Spin-Charge separation} is seen quite generally in one-dimensional
systems.   It is associated with forward scattering of particles at
the {\it same} Fermi point, and is not directly related to the
other characteristic one-dimensional phenomenon where the long-wavelength
coupling of the low-energy degrees of freedom of {\it different}
Fermi points renormalizes the correlation function exponents
away from their free fermion values.   In fact, an exactly solvable model,
the ``supersymmetric $t-J$ model'' with inverse-square
interactions\cite{kuramoto} exists in which spin-charge separation
exists without correlation function exponent renormalizations.
In this case, the simple pole of the electronic Greens function splits
into a branch cut terminated by inverse square-root singularities:
\begin{equation}
{\rm Im.} G(k_F + \delta k,E_F + \delta E) \propto
\theta ((\delta E- v_s\delta k)(v_c\delta k - \delta E))
\left ( (\delta E- v_s\delta k)(v_c\delta k - \delta E)\right )^{-1/2} .
\end{equation}
An electron injected into the system in a wavepacket of states near
such a Fermi point, and localized in space will physically separate
into spatially separated charge and spin components, moving with velocities
$v_s$ and $v_c$ as the state evolves.

Could such a phenomenon occur in two dimensions, as proposed
by Anderson\cite{pwa}?
We have seen that in dimensions greater than one, the Fermi velocity is
defined by the upper limit of the continuum of velocities of the
``Goldstone modes'' (Fermi-surface shape fluctuation modes), and that
this cannot be renormalized by  non-singular Landau couplings.  For
free fermions, and Landau Fermi liquids, the spin and charge velocities
are strictly equal, which as we shall see can be interpreted in terms
of a ``gauge symmetry''.  (The equal spin and charge velocities, defined
by the dispersion relation associated with the low-energy pole of the
Landau Fermi-liquid single-particle Green's function should not
be confused with the propagation velocities of the various spin and
charge fluctuation collective zero-sound excitations that are present
in a Fermi liquid).    To get spin-charge separation in higher dimensions,
singular forward scattering terms that differ in the singlet and triplet
scattering channels would be required, as proposed by ANDERSON\cite{pwa}
in two dimensions.   However, it should again be emphasized that
his proposal remains controversial.

The phenomenological
description outlined here treats the Landau parameters as an input,
and cannot provide guidance about their microscopic origin or validity.
It may again be useful to consider what would occur if there was, for
example, a strong forward scattering in the triplet but not the singlet
channel, but not a true singularity.  In this case, at higher energy scales
the spin and charge  degrees of freedom would presumably separate over
shorter lengthscales,  but finally, at the longest lengthscales and
lowest energies, the spin and charge quantum numbers of the electron
would be confined together to form a Landau quasiparticle.  A deconfinement
transition would take place if the Landau parameters were ``tuned''
to become (sufficiently) singular.

While spin-charge separation in two or higher dimensions  remains obscure,
I will now examine it more closely in the one-dimensional context
from a symmetry viewpoint.

\section{Hidden Symmetries in Spin-Charge Separated Systems}

As noted earlier, the ideal Fermi
gas exhibits an infinite set of gauge symmetries,
as the occupation numbers
of each orbital are separately conserved.   When spin degrees of freedom
are included, there is a infinite set of non-Abelian $SU(2)$
symmmetries, one for each orbital.   This means that the spin of each
singly-occupied orbital can be independently rotated, the spin
degeneracy
of a state with $N$ singly-occupied orbitals is $2^N$, and it is a highly
reducible representation of the global $SU(2)$ group.   When
interactions are ``switched on'' this non-generic structure of the ideal
gas is lost, and the eigenstates will become irreducible
representations of the spin rotation group (assuming no spin-orbit coupling).
The essence of the Landau Fermi-liquid state is that, asymptotically at
the low-energy fixed point, the extra symmetries of the ideal gas are
restored, but in the form of the {\it quasiparticle} occupations.

I now pose the question, if forward scattering processes at
a Fermi point are included, so as to induce spin-charge separation, but
the other interactions that couple different Fermi points are omitted,
is any remnant of the ``hidden'' quasiparticle gauge symmetries
retained?  There is some remarkable evidence from certain exactly
solvable one-dimensional models that this is indeed the case.
These models are perhaps the closest interacting models to
the ideal gas, and seem to be the simplest non-trivial interacting models.
They have scale-invariant inverse-square interactions, and ground state
wavefunctions which can be considered as (full)
Gutzwiller projections of free fermion states.

The simplest of these models is the $S=1/2$ spin chain which I and
Shastry introduced independently a few years ago\cite{haldane,shastry}:
\begin{equation}
H = J \sum_{i<j} d(i-j)^{-2} \vec{S}_i \cdot \vec{S}_j
\end{equation}
Here $d(j)$ = $j$, or $(N\pi ) \sin (\pi k /N ) $ if periodic
boundary conditions on a chain of $N$ sites is used.
This model only has spin degrees of freedom. but an extension of this
to the ``supersymmetric $t-J$ model'' was introduced by
KURAMOTO and YOKOYAMA\cite{kuramoto}:
\begin{equation}
H = J\sum_{i<j} d(i-j)^{-2}
\left (-P_G(\sum_{\sigma}c^{\dagger}_{i\sigma}c_{j\sigma} + h.c. )P_G
+ (\vec{S}_i\cdot \vec{S}_j - \frac{1}{4} n_in_j \right )
\end{equation}
where $P_G$ is the full Gutzwiller projection operator that prevents
multiple occupancy of any site.  This model has both spin and
charge degrees of freedom, and exhibits spin-charge
separation without coupling of low-energy degrees of freedom at different
Fermi points.  The periodic versions of these models exhibit
remarkable ``supermultiplet'' degeneracies meaning that their energy levels
form highly reducible representations of $SU(2)$.   This is analogous
to the free fermion gas degeneracies, but with a much less straightforward
structure, and is what I will interpret as the remnant of the orbital
occupation number symmetries that survives spin-charge separation.
Since these symmetries just involve the spin sector, it is convenient
to consider just the spin chain.

To put the results into context, it is first useful to consider
the conformal limit, where the low-energy spin degrees of freedom
are described by the $k=1$ Kac-Moody algebra (Wess-Zumino-Witten
conformal field theory\cite{affleck}).  In this language, one
writes a (say, right-moving) spin density field $\sigma^a_{q\alpha}$
associated with a single Fermi point as $J^a_m$, with $q = 2\pi m / L$,
and $m = 0, \pm 1, \pm 2 \ldots $.  Then the Kac-Moody algebra
takes its standard form
\begin{equation}
[J^a_m,J^b_{m'}] = km/2 \delta_{m+m',0} + i\epsilon^{abc}J^c_{m+m'}.
\end{equation}
and the Hamiltonian becomes ${\cal H}^{eff}$ = $v_sP$, where
the momentum $P$ is given by
\begin{equation}
P = {2\pi \over L } L_0 ,
\end{equation}
where
$L_0$ is the ``zero mode'' of the associated Virasoro algebra:
\begin{equation}
L_0 = {1 \over k + 2 }
\left ( J^a_0J^a_0 + 2 \sum_{m=1}^{\infty} J^a_{-m}J^a_m \right )
\end{equation}
where $[L_0,J^a_m] = -m J^a_m $, and $J^a_m |0 \rangle $ = 0 for
$ m > 0$.
The Hamiltonian is very degenerate, since $L_0$ takes only values
$n+h$, where $n = 0,1,2,\ldots $ and (for the $k=1$ SU(2) algebra)
$h = 0 $ for integer total spin, and $h = \frac{1}{4}$ for
half-integral total spin.  The standard descriptions of this spectrum
are through ``Abelian bosonization'' (which is essentially
what has been described
in this these lectures) or the ``Verma module'' basis
(see, {\it e.g.} the book by KAKU\cite{kaku} for an introduction),
but these do not describe the fractional-statistics
particle-like $S=\frac{1}{2}$ excitations
(``spinons'') that turn out to be the appropriate basis for describing the
inverse-square perturbation of the conformal limit.

It is useful to introduce a short-distance ``point-splitting'' cutoff
that regularizes the conformal field theory as follows:
\begin{equation}
{\cal H}^{eff} = \frac{1}{2} \int dx \int dx \,' j(x-x') \vec{\sigma}(x) \cdot
\vec{\sigma}(x') .
\end{equation}
The conformal field theory is recovered in the limit
\begin{equation}
j(x) \rightarrow {v_s \over 4 \pi } \delta (x-x') ,
\end{equation}
or $\tilde {j} (0) $ = $ v_s / 4 \pi $, where
\begin{equation}
\tilde{j} (q) =  \int dx \, j(x) e^{iqx} .
\end{equation}
Usually one takes some point-splitting function $j(x)$ that falls
off exponentially for large separations, but consider the case when
it has algebraic tails falling off as $x^{-(1+\alpha})$. Then as
$|q| \rightarrow 0 $,
\begin{equation}
\tilde{j} (q) = (v_s/4\pi )  + A |q|^{\alpha} + Bq^2 .
\end{equation}
For $0 < \alpha < 2 $, the non-analytic term is the leading correction to the
conformal limit; the new term in the Hamiltonian can be
written
\begin{equation}
H^{(2)} = \sum_{m=1}^{\infty} m^{\alpha} J^a_{-m}J^a_{m}.
\end{equation}
Since this commutes with $L_0$, this term can be studied numerically
by diagonalizing it with the finite-dimensional subspace of states
with a given value of $L_0$.  In the limit $\alpha \rightarrow 0 $,
$H^{(2)} \rightarrow \frac{3}{2} L_0 - \frac{1}{2} J_0(J_0+1) $, where
$J_0$ is the total spin quantum number.    In this limit, $H^{(2)}$ merely
splits the states at a given $L_0$ into groups with the same total spin.
However, for general $\alpha > 0 $, the spectrum of $H^{(2)}$ is completely
broken up into distinct energy levels, each of which
corresponds to an irreducible representation of $SU(2)$
with no unexpected additional degeneracies.   This represents the complete
destruction of all the higher symmetries of the conformal
field theory by the point-splitting cutoff.   A striking exception
to this is seen in the special case $\alpha = 1$, corresponding to the
inverse-square fall-off of $j(x)$; in this case the levels partially
regroup into ``supermultiplets'' which a highly reducible representations
of $SU(2)$.   No other ``special'' values of $\alpha$ are detected
by this calculation.

Clearly  a large residual part of the symmetry of the conformal
field theory survives in the presence of the inverse-square
corrections to the conformal limit.  This symmetry has recently
been identified as a ``quantum group'' symmetry\cite{yangian} called the
{\it Yangian}\cite{drinfeld,chari} $Y(sl_2)$, generated by $J^a_0$ and
\begin{equation}
{\cal J}^a = ih\epsilon^{abc}\sum_{m=1}^{\infty} J^b_{-m}J^c_{m}
\end{equation}
where $h$ is here the ``quantum deformation parameter'' defined
by the ``non-co-commutative co-product''
\begin{equation}
\Delta ({\cal J}^a)
= \openone \otimes {\cal J}^a + {\cal J}^a \otimes \openone
 + \frac{1}{2}ih \epsilon^{abc} J^b_0 \otimes J^c_0 .
\end{equation}
It is perhaps out of place to describe the technical
aspects of ``quantum groups'' (which are in fact algebras,
not groups) in any detail here; suffice it to say that
quantum groups are infinite-dimensional algebras that are
``quantum deformations'' of Lie algebras, with the feature that they have
a tensor-product operation (the ``co-product'') where (unlike Lie algebras)
the result of a sequence of tensor products depends on the {\it order}
in which they are made (analogous to the action of a sequence of operators
in quantum mechanics).  ``Quantum groups'' are intimately related to
braiding and fractional statistics.  A physical explanation of the
appearance of quantum groups in connection with spin-charge separation
is that if a spin-1/2 fermion is factorized into independent
spin and charge factors, the two components are each {\it semions},
fractional-statistics entities half-way between fermions and
bosons\cite{laughlin}.     The Yangian $Y(sl_2)$ is the quantum
group which has $sl_2$ (the Lie Algebra of $SU(2)$ generators) as
a subalgebra.

The discrete spin chain also has this ``quantum group'' symmetry,
$[H,{\cal J}^a]$, with
\begin{equation}
{\cal J}^a = { h \over 2} \sum_{i <j }  \cot (\pi (i-j)/N)
\epsilon^{abc} S^b_i S^c_j .
\end{equation}
The energy levels are given by\cite{yangian} the construction
\begin{equation}
E = 2J(\pi / N)^2 \sum_{i=1}^M  m_i(m_i - N) ;\quad  e^{iK} = \prod_i \exp
 (2 \pi i m_i /N ) ,
\end{equation}
where $\{m_i\}$ are a set of distinct integers in the range
$0 < m_i < m_{i+1} < N $, subject to the ``generalized Pauli
principle'' that not only are they {\it distinct}, but also that
{\it they cannot be consecutive}.  These quantum numbers
can be represented by a binary sequence of
length $N-1$, where a ``1'' represents a value in the
set $\{m_i\}$. This means that the ground state
sequence is $1010101\ldots 1010101$, and has no consecutive pairs of
zeroes, which represent spinon excitations.
Removing a ``1'' from the sequence thus creates  a {\it two}-spinon
state $\ldots 101000010101 \ldots$, which can
be rearranged to give states such as $\ldots 1010100101010010101\dots $
A sequence such as this represents a fourfold-degenerate
state with $SU(2)$ representation content $ ( \frac {1}{2} ) \otimes
(\frac{1}{2})$ = $0 \oplus 1$.

A sequence
such as ``\dots 1000001 \ldots'' where there are four {\it successive}
``00'' combinations represents a ``four-string'' in CHARI and PRESSLEY's
\cite{chari} representation theory of $Y(sl_2)$, and hence
contributes a $S=2$ factor
in the tensor product of $SU(2)$ representations that makes up the
representation of $Y(sl_2)$.  Physically, this was previously
interpreted\cite{fdmh2}
as ``four spinons in the same orbital'' with a selection rule
that spinons ``in the same orbital'' could only be in a symmetric
spin state\cite{stat}.   This empirically-observed rule, discovered
by detailed examination of the results from numerical
diagonalization\cite{fdmh2}, now is seen to precisely correspond
to the $Y(sl_2)$ representation theory\cite{chari}.

This  example suggests that ``quantum-group'' techniques may turn out to
have important applications in connection with fractional statistics,
as a more algebraic formulation that makes contact with ``occupation
number'' descriptions and the Pauli principle.

\section{Conclusion}

In these lectures, I sketched out the logic of an
approach to Fermi fluids based on the idea that the
Fermi surface is an analog of an order parameter, and that
the low-energy degrees of freedom can be fully treated in terms
of  ``bosonized'' variables describing local fluctuations
of the shape of the Fermi surface.  The Luttinger theorem
relating the volume of the Fermi surface to the particle
density is seen to be the key principle.
While bosonization has been a key tool in treating the one
dimensional systems, it clearly shows promise in higher
dimensions too.  Much remains to be done to make this method
a real working tool for  higher dimensions.
On the one hand, it will be interesting to see how much of
the standard Fermi liquid results can be reproduced
using such methods.  On the other hand, they seem to have potential
for the study of possible non-Fermi-liquid states, since they
are not based on a perturbative expansion about the
non-interacting Fermi gas.
I also considered spin-charge separation, primarily in one dimension,
and described some recent hints that ``quantum group'' methods
may be important in cases where fermion variables fractionalize
into fractional-statistics objects.

This work was supported in part by NSF grant DMR 91-96212.

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