text:
"New Oppo smartphone is very similar to the new OnePlus 10 Pro"
The specs of the Oppo Find X5 Pro have been leaked. It is striking that they are very similar to those of the new OnePlus 10 Pro, which was recently presented.
Read on after the ad.
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'These are the Oppo Find X5 Pro specs'
Oppo is working on a high-end smartphone, which will probably be called the Find X5 Pro. On the Chinese social network Weibo the specifications of the device have been published, writes, among other things, Android Police. What is striking is that the specs are almost identical to those of the OnePlus 10 Pro. This phone was unveiled in China this week and will be released in Europe later.
For example, the Oppo Find X5 Pro – just like the OnePlus 10 Pro – has a 6.7-inch screen with a high 2K resolution and a refresh rate of 120Hz. The devices also have the same Snapdragon 8 Gen 1 processor, a 5000 mAh battery and charge just as quickly. Charging with a cable goes with a maximum of 80 Watts, while this is 50 Watts wirelessly.
The OnePlus 10Pro.
The 32 megapixel selfie camera also seems to be the same. We do see some differences with the cameras on the back. The Find X5 Pro is rumored to have two cameras of 50 megapixels and one of 13 megapixels. The brand new OnePlus 10 Pro has a 48-megapixel main camera, 50-megapixel wide-angle lens and 8-megapixel zoom lens. Unfortunately, we do not know to what extent the photo quality of the phones differs.
It is also not illogical that the smartphones from Oppo and OnePlus are so similar. The brands have the same parent company: BBK Electronics. The software on Oppo and OnePlus smartphones is also growing more and more. The companies announced a while ago that the software pods will be merged, thus combining functions.
More about the Oppo Find X5 Pro
It is still unclear when Oppo will unveil the Find X5 Pro. Its predecessor, the Find X3 Pro, was released in March last year. The Chinese manufacturer may again opt for an announcement in that month. However, nothing has been announced about this yet.
Renders of the Oppo Find X5 Pro have been leaked. They show a separate camera design. The island on which the lenses are located is not square or rectangular, as is the case with most smartphones. The camera module slopes downwards at the bottom. This gives the smartphone a recognizable design.
Source: Android Planet by www.androidplanet.nl.
*The article has been translated based on the content of Android Planet by www.androidplanet.nl. If there is any problem regarding the content, copyright, please leave a report below the article. We will try to process as quickly as possible to protect the rights of the author. Thank you very much!


text:
Christie and Trump to lunch together in DC
Published Mon, Feb 13 20171:53 PM EST
Chris Christie, governor of New Jersey,
Daniel Acker | Bloomberg | Getty Images
New Jersey Governor Chris Christie was scheduled to have lunch in Washington, D.C., with President Donald Trump on Tuesday, according to NJ Advance Media.
The two were expected to discuss opioid addiction in the U.S., among other topics.
Christie said he declined several positions in the administration, but recent reports have emerged saying a possible shakeup in the White House staff may be coming.
Trump has not nominated anyone to be the director of the Office of National Drug Policy, according to NJ Advance Media.
CNBC reached out to the White House and Governor Christie for comment but did not receive an immediate reply.
Read more about this story at NJ Advance Media.
China confirms coronavirus can pass person-to-person; WHO calls emergency meeting

redpajama_set_name: RedPajamaCommonCrawl


text:
Articles tagged "Disney"
A Whirlwind Tour of Walt Disney World
by Paul Brady on Aug 3, 2011
I am not, as far as I can tell, in Walt Disney World's target demographic. I'm not four. I'm not a family man. I'm not Brazilian. I'm not even a fan of animated movies. But to drive through Central Florida after seeing a shuttle launch and pass up the parks? To miss out on a quintessentially American summertime diversion? To skip a chance to meet the one and only Mickey Mouse? I'm not nuts.
Traveling the American Road – A Whirlwind Tour of Disney
My plan was a whirlwind tour of all four of Disney's parks, trying my best to try what attractions had been added since my last visit, in 2007. Then, I was in town for the opening of Expedition Everest, a ride that challenged my poor tolerance for roller coasters and impressed me with its ability to make visitors feel like they were hiking the Himalayas, even in the heat of Central Florida.
This time, the big draw was the Wild Africa Trek, a new behind-the-scenes tour of the Animal Kingdom that takes visitors behind the fences, out to Disney's "savanna" and ends with a killer lunch on an African safari-inspired wildlife watching pavilion far from the crowds. To amp up the excitement, trekkers cross a crocodile enclosure on rope bridges, distressed to look rickety even if they were reinforced by steel cables. Anyone who's seen "Temple of Doom," though, can't get past their primal fear of a rope bridge collapse.
I survived, obviously, to see Epcot and its world pavilions. There's something hilarious about visiting "France," "Ireland," "Italy" and "Mexico" when you're a travel writer. In every one of the miniature countries, I was studying the architecture, comparing it to my memories, figuring out what it is we remember about the places we visit – and wondering why we forget the things we forget. Is that really what the Eiffel Tower looks like, I asked myself, cocking my head, as I couldn't clearly remember the original's shape.
At Disney's Hollywood Studios, I tested the limits of my stomach with a ride on the Star Tours simulator, a 3-D ride set in the Star Wars universe, in which C-3PO is an accidental tour guide and passengers fly through the galaxy. A visit to the American Idol Experience impressed not just for its slick production values – from where I was sitting, it could've been the real TV show – but also for the talent of the contestants on stage.
I ended my tour at the Magic Kingdom, the park that to me, a person visiting without my kids, seemed the least interesting. But the polish here was the most fine, the smiles on singers the most gleaming, the lawn edging the most precise, the background music the most bubbly. The good news, thanks perhaps to some friends inside Disney: I did manage to meet Mickey. I even put on a set of ears.
The Party's "Summer Vacation," a nostalgic music video
by Meg Nesterov on May 3, 2011
Summer vacation is just around the corner and it's time to enjoy some lighter forms of entertainment. If you're a twenty- or thirty-something, you may remember Disney's Mickey Mouse Club's attempt to become cool as the MMC. Before future stars Britney Spears, Christina Aguilera, and Justin Timberlake joined the cast, there was the pop ensemble The Party and their summer anthem "Summer Vacation." The song did respectably on the charts in 1990 and the video has all of the hallmarks of an early nineties music video: white dudes rapping, boardwalk skateboards, and lots of lycra. Enjoy this nostalgic trip to the beach: Tune in, groove on, and bust out!
Disney gets the keys to new resort
by Chris Owen on Apr 26, 2011
Construction is officially complete on Disney's Aulani Resort & Spa in Hawaii. The construction company working on the project handed over the keys recently and final preparation of the resort is underway. As with all things Disney, a celebration was in order and cast members paused for a little magical beach fun.
Disney Cast members will work on honing the guest service, installing the décor, landscape and furnishings that will make Aulani a memorable island vacation experience.
Built on 21 acres adjacent to a blue lagoon and a white-sand beach in Ko Olina, Disney's Aulani resort will have 359 hotel rooms and 481 two-bedroom equivalent Disney Vacation Club villas upon its completion. Rooms are housed in two 15-story towers with views of the ocean, mountains and grounds. On-site are two restaurants, an 18,000-square-foot spa, a wedding lawn, a conference center and extensive water activities, including a central swimming pool, 900-foot tube floating watercourse, wading pool, body slides, sunset-facing hot tubs, a quiet adult pool, a snorkel lagoon, and a conservation pool.
Aulani will also include a signature kids' club, Aunty's Beach House, where kids can explore Hawaiian culture through fun and games, arts and crafts, and other experiences. Disney also promises to "help guests connect with Hawaiian culture, traditions and hospitality in a profound way through local experts and activities that immerse you in the legends of the land" via Discovery Experiences that immerse guests into topics varying from the Flavors of Life to the Wonders of Nature.
Disney Breaks Ground on Shanghai Resort [VIDEO]
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Hawaii destinations close for repairs, new resorts prepare to open
by Chris Owen on Mar 24, 2011
Kona Village Resort, one of the iconic vacation destinations and resorts of Hawaii, announced this week that it will close for an extended period of time due to the significant structural and property damage it sustained during the recent earthquake-generated tsunami. But while some Hawaii destinations have closed for repairs, other new resorts prepare to open.
"We are very sad to close Kona Village Resort for an extended period of time, but the damage to the property from this natural disaster is severe enough to render it inoperable," said Patrick Fitzgerald, President and Chief Executive Officer, Hualalai Resort/Kona Village Resort. "We are fortunate that our guests and employees were evacuated safely. Our hearts are heavy with this decision's impact on employees and loyal guests who have visited us over the years. Our thoughts also are with the people of Japan during this difficult time."
The 45-year old Hawaii resort sustained significant structural damage from the storm surge including damaged buildings and attractions at the 82-acre property.
The Four Seasons Resort at Hualalai, the first and only AAA Five-Diamond and Forbes Five-Star resort on the Big Island also closed for repairs but reopened this week.
Meanwhile, construction proceeds full speed ahead at Disney's massive new Aulani Resort and Spa scheduled to open August 29, 2011.
Hawaiian Tourism Rocked by Japan Disaster
Hawaii Becomes First State With $4 Gas Average
Massive Tsunami Devastates Japan; Waves Hit Hawaii
Hawaii Earthquake Jolts Honolulu
Kilauea Volcano Erupts on Hawaii's Big Island [PHOTOS]
Flickr photo by Luckypines
Earthquake strands thousands at Tokyo Disneyland & DisneySea
by Joel Bullock on Mar 12, 2011
Thousands of guests were stranded in Tokyo Disneyland and DisneySea after the massive earthquake that rocked the island nation on Friday. Park employees have been giving out humanitarian aid including blankets, heaters, plastic raincoats, food, and water. While Disney has reported only a few minor injuries and minimal property damage, other reports state that the earthquake has liquefied the parking lot. Authorities suspect that shaking from the earthquake caused water from the coastline to mix with the soil. The Disney parks were built on a landfill making the parks susceptible to flooding.
According to Disney, Tokyo Disneyland and DisneySea will remained closed to allow for a thorough inspection of the attractions and facilities. Inside the Magic has pictures from social media sites of the stranded guests and damage to the park. Below is a video of park guests experiencing the earthquake while they were waiting for a parade.
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Shirdi is a small village in Kopargam taluk, in Ahmadnagar District of the Maharastra State. When Baba was physically present at the age of 20 in Shirdi, it was a small village of 80 thatched houses with mud walls. Today it is a big town with palatial modern buildings and shops. Thursday is the day Saibaba is specially worshipped. On this day devotees from all over India visit Shirdi for Darshan of Saibaba. The Temple opens for devotees at 5.15 a.m. (0515 hrs) with Kakad Aarti and remains open till the end of Shejarti.
For more on Puja timings. For devotees across the globe, there are arrangements to get Pujas conducted on their behalf and the prasad sent to them. Shirdi is well connected by road. There are also a few religious places of tourist interest in and around Shirdi. Shirdi Saibaba Temple is managed by Shri Sai Baba Sansthan since 1922. The trust ensures that a strict code of conduct is maintained in the temple premises.
Chandbhai, the headman of a village called Dhoopkhede (in Aurangabad, India), once lost his horse and was looking for it. Suddenly he heard a voice say "You look tired. come here and rest a while". He turned around and saw a young Fakir . The fakir smiled at him and said "What are you looking for in this jungle, Chandbhai". Chandbhai was surprised and he wondered how the fakir knew his name. Slowly, he said "I have lost my horse. I have looked for it everywhere, but cannot seem to find it". The fakir told him to look behind a clump of trees. Chandbhai was pleasantly surprised to find his horse grazing peacefully behind those trees. He thanked the fakir and asked his name. The fakir said "some people call me Sai Baba."
One day a rich millionaire named Booty came to Sai Baba and said he was going to construct a stone building for Shri Krishna. Baba helped him plan the building. Before the building was completed Baba fell very ill. On the 15th of October 1918, he breathed his last. His last wish was to be buried in Booty's building.
Booty's stone building came to be known as the Samadhi Mandir. Shri Sai Baba was buried here and a beautiful shrine was built over it. To this day, people flock to Shirdi to pay homage to Shri Sai Baba.


text:
\section{Introduction}

The quantization of a second-class constrained system can be achieved by the
reformulation of the original theory as a first-class one and then
quantizing the resulting first-class theory. This quantization procedure was
applied to various models \cit
{stuckn,vytgen,kor,stuck5,stuck3,biz2formx,biz2formPRD,stuck2,stuck4,vyt3,stuck6,stuck1,IJMPAnoi,EMC,io1,io2io3}
using a variety of methods to replace the original second-class model to an
equivalent model in which only first-class constraints appear. The
conversion of the original second-class system into equivalent theories
which have gauge invariance can be accomplished without enlarging the phase
space starting from the possibility of interpreting a second-class
constraints set as resulting from a gauge-fixing procedure of a first-class
constraints one and "undo" gauge-fixing \cite{jap,mitra,marc,vyt0,bizNPB}.
Another method to construct the equivalent first-class theory relies on an
appropriate extension of the original phase space through the introduction
of some new variables. The first-class constraint set and the first-class
Hamiltonian are constructed as power series in the new variables \cit
{FS,bfgen,battyut}.

Various aspects of the equivalence \cite{DJ1} between self-dual model \cit
{TPN} and Maxwell-Chern-Simons (MCS) \cite{DJT1} have been studied using one
of the two methods mentioned in the above \cite{io2io3,BRR1,BR1}. In \cit
{DJ2,SN} one finds generalizations that involve both the Maxwell and
Chern-Simons terms \cite{DJ1,TPN,DJT1}. In the first part of this paper we
investigate the MCS-Proca model \cite{SN,BCSx,DT,Baz
\begin{equation}
S=\int d^{3}x\left( -\tfrac{a}{4}\partial _{\lbrack \mu }A_{\nu ]}\partial
^{\lbrack \mu }A^{\nu ]}-b\varepsilon _{\mu \nu \rho }A^{\mu }\partial ^{\nu
}A^{\rho }-\tfrac{m^{2}}{2}A_{\mu }A^{\mu }\right) ,  \label{actmcs}
\end{equation
from the point of view of the Hamiltonian quantization using the
gauge-unfixing (GU) approach. A generalization of the Proca action for a
massive vector field with derivative self-interactions in $D=4$ has been
constructed in \cite{LH}. In the second part we consider a higher order
extension of the MCS-Proca model (which contains higher-order derivatives
involving CS term) described by Lagrangian action
\begin{equation}
S=\int d^{3}x\left[ -\tfrac{1}{4}\partial _{\lbrack \mu }A_{\nu ]}\partial
^{\lbrack \mu }A^{\nu ]}+\tfrac{1}{2}\varepsilon _{\mu \nu \rho }\left(
\partial _{\lambda }\partial ^{\lambda }A^{\mu }\right) \partial ^{\nu
}A^{\rho }-\tfrac{1}{2}A_{\mu }A^{\mu }\right] .  \label{y1}
\end{equation
and we apply the same quantization procedure.

The paper is organized in four sections. In section \ref{sec2}, starting
from MCS-Proca model we construct an equivalent first-class model using
gauge-unfixing method and meanwhile we obtain the path integral
corresponding to the first-class system associated with this model. The
gauge-unfixing method relies on separating the second-class constraints into
two subsets, one of them being first-class and the other one providing some
canonical gauge conditions for the first-class subset. Starting from the
canonical Hamiltonian of the original second-class system, we construct a
first-class Hamiltonian with respect to the first-class subset through an
operator that projects any smooth function defined on the phase space into
an application that is in strong involution with the first-class subset. In
section \ref{sec3} we exemplify in detail the gauge-unfixing method on
extended MCS-Proca (MECS-Proca) model and construct the path integral of the
equivalent first-class systems associated with this second-class theory.
Section \ref{sec4} ends the paper with the main conclusions.

\section{The MCS-Proca model\label{sec2}}

The MCS-Proca model is described by the Lagrangian action (\ref{actmcs})
where $a$ and $b$ are some real constants. We work with the Minkowski metric
tensor of `mostly minus' signature $\sigma _{\mu \nu }=\sigma ^{\mu \nu
}=diag(+--)$. The canonical analysis \cite{D1,D2} of the model described by
the Lagrangian action (\ref{actmcs}) displays the second-class constraints
(scc)
\begin{eqnarray}
\chi ^{\left( 1\right) } &\equiv &p^{0}\approx 0,  \label{pgei} \\
\chi ^{\left( 2\right) } &\equiv &\partial ^{i}p_{i}-b\varepsilon
_{0ij}\partial ^{i}A^{j}-m^{2}A_{0}\approx 0,  \label{pcei}
\end{eqnarray
and the canonical\ Hamiltonian
\begin{eqnarray}
H_{c} &=&\int d^{2}x\left( -\tfrac{1}{2a}p_{i}p^{i}-A_{0}\partial _{i}\pi
^{i}+\tfrac{a}{4}\partial _{\lbrack i}A_{j]}\partial ^{\lbrack
i}A^{j]}\right.   \notag \\
&&\left. +b\varepsilon _{0ij}A^{0}\partial ^{i}A^{j}+\tfrac{b}{a}\varepsilon
_{0ij}A^{i}p^{j}-\tfrac{b^{2}}{2a}A_{i}A^{i}+\tfrac{m^{2}}{2}A_{\mu }A^{\mu
}\right) ,  \label{phamcan}
\end{eqnarray
where $p^{\mu }$ are the canonical momenta conjugated with the fields
A_{\mu }$. The number of physical degrees of freedom \cite{marc} of the
orginal system is equal t
\begin{eqnarray}
\mathcal{N}_{O} &=&\left( 6\ \text{canonical\ variables}-2\ \text{scc
\right) /2  \notag \\
&=&2.  \label{doforig}
\end{eqnarray}
The same result with respect to the number of degrees of freedom is obtained
in \cite{DT}. Moreover, in \cite{DT} it is shown that the MCS-Proca model
describes a topological mass mix with two massive degrees of freedom, with
masses $\sqrt{b^{2}+m^{2}}\pm \left\vert b\right\vert $.

According to the GU method we consider (\ref{pcei}) as the first-class
constraint (fcc) set and the remaining constraints (\ref{pgei}) as the
corresponding canonical gauge conditions. Further we redefine the
first-class constraint as
\begin{equation}
G\equiv -\tfrac{1}{m^{2}}\left( \partial ^{i}p_{i}-b\varepsilon
_{0ij}\partial ^{i}A^{j}-m^{2}A_{0}\right) \approx 0.  \label{pcein}
\end{equation
The other choice, considering (\ref{pgei}) as the\ first-class constraint
set and the remaining constraints (\ref{pcei}) as the corresponding
canonical gauge conditions, yields a path integral that cannot be written
(after integrating out auxiliary variables) in a manifestly covariant form
\cite{vyt3,IJMPAnoi}. The next step of the GU\ approach is represented by
the construction of a first-class Hamiltonian with respect to (\ref{pcein}).
The first-class Hamiltonian $H_{GU}$ is given by
\begin{eqnarray}
H_{GU} &=&H_{c}-\chi ^{\left( 1\right) }\left[ G,H_{c}\right] +\tfrac{1}{2
\chi ^{\left( 1\right) }\chi ^{\left( 1\right) }\left[ G,\left[ G,H_{c
\right] \right] -\cdots   \notag \\
&=&H_{c}+\int d^{2}x\left[ -p^{0}\partial ^{i}A_{i}+\tfrac{1}{m^{2}}\left(
\partial _{i}p_{0}\right) \left( \partial ^{i}p^{0}\right) \right] ,
\label{phamclsI}
\end{eqnarray
Performing the canonical transformatio
\begin{equation}
A_{0}\longrightarrow \tfrac{1}{m}p,\qquad p^{0}\longrightarrow -m\varphi ,
\label{ptransf1}
\end{equation
the constraint (\ref{pcein}) become
\begin{equation}
G\equiv -\tfrac{1}{m^{2}}\left( \partial ^{i}p_{i}-b\varepsilon
_{0ij}\partial ^{i}A^{j}-mp\right) \approx 0,  \label{pcei1n}
\end{equation
and the first-class Hamiltonian (\ref{phamclsI}) takes the form
\begin{eqnarray}
H_{GU} &=&\int d^{2}x\left[ -\tfrac{1}{2a}p_{i}p^{i}+\tfrac{a}{4}\partial
_{\lbrack i}A_{j]}\partial ^{\lbrack i}A^{j]}+\tfrac{b}{a}\varepsilon
_{0ij}A^{i}p^{j}\right.   \notag \\
&&-\tfrac{b^{2}}{2a}A_{i}A^{i}+\tfrac{1}{2}\left( \partial _{i}\varphi
-mA_{i}\right) \left( \partial ^{i}\varphi -mA^{i}\right)   \notag \\
&&\left. -\tfrac{1}{m}p\left( \partial _{i}p^{i}-b\varepsilon _{0ij}\partial
^{i}A^{j}-mp\right) -\tfrac{1}{2}p^{2}\right] .
\end{eqnarray
The number of physical degrees of freedom of the GU system is equal t
\begin{eqnarray}
\mathcal{N}_{GU} &=&\left( 6\ \text{canonical\ variables}-2\times 1\ \text
fcc}\right) /2  \notag \\
&=&2=\mathcal{N}_{O}.  \label{dofgu}
\end{eqnarray
The original second-class theory and respectively the gauge-unfixed system
are classically equivalent since they possess the same number of physical
and, moreover, the corresponding algebras of classical observables are
isomorphic. Consequently, the two systems become equivalent at the level of
the path integral quantization, which allows us to replace the Hamiltonian
path integral of the MCS-Proca model with that of the gauge-unfixed
first-class system.
\begin{eqnarray}
Z_{GU} &=&\int \mathcal{D}\left( A_{i},\varphi ,p^{i},p,\lambda \right) \mu
\left( \lbrack A_{i}],[\varphi ]\right) \exp \left\{ \mathrm{i}\int d^{3}
\left[ \left( \partial _{0}A_{i}\right) p^{i}+\left( \partial _{0}\varphi
\right) p\right. \right.   \notag \\
&&\left. \left. -\mathcal{H}_{GU}+\tfrac{1}{m^{2}}\lambda \left( \partial
^{i}p_{i}-b\varepsilon _{0ij}\partial ^{i}A^{j}-mp\right) \right] \right\} ,
\label{piGU}
\end{eqnarray
where the integration measure `$\mu \left( \lbrack A_{i}],[\varphi ]\right)
' associated with the model subject to the first-class constraint (\re
{pcei1n}) includes some suitable canonical gauge conditions and it is chosen
such that (\ref{piGU}) is convergent \cite{FHP}.

Performing in\ the path integral the transformation
\begin{equation}
p\longrightarrow p,\qquad \lambda \longrightarrow \bar{\lambda}=\tfrac{1}
m^{2}}\lambda +\tfrac{1}{m}p  \label{ptransf2}
\end{equation
and partial integrations over all momenta $\left\{ p^{i},p\right\} $, the
argument of the exponential become
\begin{eqnarray}
S_{GU} &=&\int d^{3}x\left[ -\tfrac{a}{4}\partial _{\lbrack i}A_{j]}\partial
^{\lbrack i}A^{j]}-\tfrac{a}{2}\left( \partial _{0}A_{i}-\partial _{i}\bar
\lambda}\right) \left( \partial ^{0}A^{i}-\partial ^{i}\bar{\lambda}\right)
\right.  \notag \\
&&-b\varepsilon _{0ij}\bar{\lambda}\partial ^{i}A^{j}-b\varepsilon
_{i0j}A^{i}\partial ^{0}A^{j}-b\varepsilon _{ij0}A^{i}\partial ^{j}\bar
\lambda}  \notag \\
&&\left. -\tfrac{1}{2}\left( \partial _{i}\varphi -mA_{i}\right) \left(
\partial ^{i}\varphi -mA^{i}\right) -\tfrac{1}{2}\left( \partial _{0}\varphi
-m\bar{\lambda}\right) \left( \partial ^{0}\varphi -m\bar{\lambda}\right)
\right] .
\end{eqnarray
Making the notatio
\begin{equation}
\bar{\lambda}\equiv \bar{A}_{0}
\end{equation
the argument of the exponential takes the for
\begin{eqnarray}
S_{GU} &=&\int d^{3}x\left[ -\tfrac{a}{4}\partial _{\lbrack i}A_{j]}\partial
^{\lbrack i}A^{j]}-\tfrac{a}{2}\left( \partial _{0}A_{i}-\partial _{i}\bar{A
_{0}\right) \left( \partial ^{0}A^{i}-\partial ^{i}\bar{A}^{0}\right) \right.
\notag \\
&&-b\varepsilon _{0ij}\bar{A}^{0}\partial ^{i}A^{j}-b\varepsilon
_{i0j}A^{i}\partial ^{0}A^{j}-b\varepsilon _{ij0}A^{i}\partial ^{j}\bar{A
^{0}  \notag \\
&&\left. -\tfrac{1}{2}\left( \partial _{i}\varphi -mA_{i}\right) \left(
\partial ^{i}\varphi -mA^{i}\right) -\tfrac{1}{2}\left( \partial _{0}\varphi
-m\bar{A}^{0}\right) \left( \partial ^{0}\varphi -m\bar{A}^{0}\right) \right
.
\end{eqnarray
In terms of the notations $\bar{A}_{\mu }\equiv \left\{ \bar{A
_{0},A_{i}\right\} $, the last functional reads a
\begin{eqnarray}
S_{GU} &=&\int d^{3}x\left[ -\tfrac{a}{4}\partial _{\lbrack \mu }\bar{A
_{\nu ]}\partial ^{\lbrack \mu }\bar{A}^{\nu ]}-b\varepsilon _{\mu \nu \rho
\bar{A}^{\mu }\partial ^{\nu }\bar{A}^{\rho }\right.  \notag \\
&&\left. -\tfrac{1}{2}\left( \partial _{\mu }\varphi -m\bar{A}_{\mu }\right)
\left( \partial ^{\mu }\varphi -m\bar{A}^{\mu }\right) \right] .
\label{pacttot3.1}
\end{eqnarray
The functional (\ref{pacttot3.1}) associated with the first-class system
takes a manifestly Lorentz covariant form and describes the (Lagrangian) S
\"{u}ckelberg coupling between the scalar field $\varphi $ and the $1$-form
\bar{A}_{\mu }$ \cite{stueck}.

The MCS-Proca model can be correlated to another first-class theory whose
field spectrum comprise two types of $1$-form gauge fields. For this purpose
we consider the following fields/momenta combination
\begin{eqnarray}
\mathcal{P}_{i} &\equiv &p_{i}+b\varepsilon _{0ij}A^{j},  \label{x2.1} \\
\mathcal{F}_{i} &\equiv &A_{i}+\tfrac{1}{m^{2}}\partial _{i}p_{0},
\label{x2.2} \\
\mathcal{F}_{0} &\equiv &A_{0},  \label{x2.3}
\end{eqnarray
which are in (strong) involution with the first-class constraint (\ref{pcein
)
\begin{equation}
\left[ \mathcal{P}_{i},G\right] =\left[ \mathcal{F}_{i},G\right] =\left[
\mathcal{F}_{0},G\right] =0.  \label{x3}
\end{equation
We observe that the first-class Hamiltonian (\ref{phamclsI}) can be written
in terms of these gauge invariant quantitie
\begin{eqnarray}
H_{GU} &=&\int d^{2}x\left[ -\tfrac{1}{2a}\mathcal{P}_{i}\mathcal{P}^{i}
\tfrac{a}{4}\partial _{\lbrack i}\mathcal{F}_{j]}\partial ^{\lbrack i
\mathcal{F}^{j]}\right.   \notag \\
&&\left. +\tfrac{m^{2}}{2}\mathcal{F}_{i}\mathcal{F}^{i}-\tfrac{m^{2}}{2
\mathcal{F}_{0}\mathcal{F}^{0}+m^{2}\mathcal{F}_{0}G\right] .  \label{x1}
\end{eqnarray
By direct computation we find that $\mathcal{F}_{\mu }\equiv \left\{
\mathcal{F}_{0},\mathcal{F}_{i}\right\} $ is divergenceles
\begin{equation}
\partial ^{\mu }\mathcal{F}_{\mu }=0,  \label{x6}
\end{equation
and satisfy the equation
\begin{eqnarray}
\partial ^{\nu }\partial _{\lbrack \nu }\mathcal{F}_{0]} &=&\tfrac{1}{a
\partial ^{i}\mathcal{P}_{i}+\mathcal{O}\left( G\right) ,  \label{x4.1} \\
\partial ^{\nu }\partial _{\lbrack \nu }\mathcal{F}_{i]} &=&\tfrac{m^{2}}{a
\mathcal{F}_{i}+\tfrac{2b}{a^{2}}\varepsilon _{0ij}\mathcal{P}^{j}+\mathcal{
}\left( G\right) .  \label{x4.2}
\end{eqnarray
Enlarging the phase space by adding the bosonic fields/momenta $\{V^{\mu
},P_{\mu }\}$, we can write the solution to (\ref{x6}) as
\begin{equation}
\mathcal{F}_{\mu }=-\tfrac{1}{m}\varepsilon _{\mu \nu \rho }\partial ^{\nu
}V^{\rho }.  \label{x7}
\end{equation
When we replace (\ref{x7}) in (\ref{pcein}) the constraint takes the form
\begin{equation}
-\tfrac{1}{m^{2}}\left( \partial ^{i}p_{i}-b\varepsilon _{0ij}\partial
^{i}A^{j}+m\varepsilon _{0ij}\partial ^{i}V^{j}\right) \approx 0,  \label{x9}
\end{equation
and remains first-class. From the gauge transformation of the quantity
\partial _{i}p_{0}$ we obtain tha
\begin{equation}
\partial _{i}p_{0}=m\varepsilon _{0ij}P^{j}.  \label{x11}
\end{equation
In this moment we have a dynamical system with phase space locally
parametrized by $\{A_{i},p^{i},V^{\mu },P_{\mu }\}$, subject to the
first-class constraints (\ref{x9}) and too many degrees of freedo
\begin{eqnarray}
\mathcal{N}_{GU}^{\prime } &=&\left( 10\ \text{canonical\ variables}-2\times
1\ \text{fcc}\right) /2  \notag \\
&=&4\neq \mathcal{N}_{O}.
\end{eqnarray
In order to cut the two extra degrees of freedom, we add to the first-class
constraint set (\ref{x9}) two new first-class constraints
\begin{equation}
P^{0}\approx 0,\qquad -\partial _{i}P^{i}\approx 0,  \label{x10}
\end{equation
and we obtain a first-class system with a right number of physical degrees
of freedo
\begin{eqnarray}
\mathcal{N}_{GU}^{\prime } &=&\left( 10\ \text{canonical\ variables}-2\times
3\ \text{fcc}\right) /2  \notag \\
&=&2=\mathcal{N}_{O}.  \label{dofgu'}
\end{eqnarray
Using (\ref{x7}) and (\ref{x11}) in (\ref{phamclsI}), we obtain for the
first-class Hamiltonian the following form
\begin{eqnarray}
H_{GU}^{\prime } &=&\int d^{2}x\left[ -\tfrac{1}{2a}\left(
p_{i}+b\varepsilon _{0ij}A^{j}\right) \left( p^{i}+b\varepsilon
^{0ik}A_{k}\right) +\tfrac{a}{4}\partial _{\lbrack i}A_{j]}\partial
^{\lbrack i}A^{j]}\right.   \notag \\
&&+\tfrac{m^{2}}{2}\left( A_{i}+\tfrac{1}{m}\varepsilon _{0ij}P^{j}\right)
\left( A_{i}+\tfrac{1}{m}\varepsilon ^{0ik}P_{k}\right) -\tfrac{1}{4
\partial _{\lbrack i}V_{j]}\partial ^{\lbrack i}V^{j]}  \notag \\
&&\left. +\tfrac{1}{2m}\varepsilon _{0ij}\partial ^{\lbrack i}V^{j]}\left(
\partial _{k}p^{k}-b\varepsilon _{0kl}\partial ^{\lbrack
k}A^{l]}+m\varepsilon _{0kl}\partial ^{k}V^{l}\right) \right] .  \label{x12}
\end{eqnarray
For the first-class system endorsed with the phase space parameterized by
\{A_{\mu },p^{\mu }\}$ and subject to the first-class constraints (\re
{pcein}), the fundamental classical observables read as $\{A_{i}+\tfrac{1}
m^{2}}\partial _{i}p_{0},A_{0}$\ and\ $p_{i}+b\varepsilon _{0ij}A^{j}\}$,
and for that with the phase space parameterized by $\{A_{i},p^{i},V^{\mu
},P_{\mu }\}$ and subject to the first-class constraints (\ref{x9}) and (\re
{x10}) the fundamental classical observables are $\{A_{i}+\tfrac{1}{m
\varepsilon _{0ij}P^{j},-\tfrac{1}{m}\varepsilon _{0ij}\partial ^{i}V^{j}$\
and\ $p_{i}+b\varepsilon _{0ij}A^{j}\}$. As the number of physical degrees
of freedom is the same for both theories and we can identify a set of
fundamental classical observables (such that they are in one to one
correspondence and they possess the same Poisson brackets) the first-class
theories are equivalent. As a result, the GU and the first-class systems
remain equivalent also at the level of the Hamiltonian path integral
quantization. This further implies that the first-class system is completely
equivalent with the original second-class theory. Due to this equivalence we
can replace the Hamiltonian path integral of MCS-Proca model with that one
associated with the first-class syste
\begin{eqnarray}
Z^{\prime } &=&\int \mathcal{D}\left( A_{i},V^{\mu },p^{i},P_{\mu },\lambda
,\lambda ^{(1)},\lambda ^{(2)}\right) \mu ^{\prime }\left( [A_{i}],[V^{\mu
}]\right)   \notag \\
&&\exp \left\{ \mathrm{i}\int d^{3}x\left[ \left( \partial _{0}A_{i}\right)
p^{i}+\left( \partial _{0}V^{\mu }\right) P_{\mu }-\mathcal{H}_{GU}^{\prime
}-\lambda ^{(1)}P^{0}\right. \right.   \notag \\
&&\left. \left. +\lambda ^{(2)}\partial _{i}P^{i}+\tfrac{1}{m^{2}}\lambda
\left( \partial _{i}p^{i}-b\varepsilon _{0ij}\partial ^{\lbrack
i}A^{j]}+m\varepsilon _{0ij}\partial ^{i}V^{j}\right) \right] \right\} .
\label{x16}
\end{eqnarray
If we perform in path integral the transformation
\begin{equation}
\lambda \longrightarrow \bar{\lambda}=\tfrac{1}{m^{2}}\left( \lambda -\tfrac
m}{2}\varepsilon _{0ij}\partial ^{\lbrack i}V^{j]}\right)   \label{x14}
\end{equation
and partial integrations over $\{V^{0},p_{i},P_{0},P_{i},\lambda ^{(1)}\}$,
the argument of the exponential become
\begin{align}
S_{GU}^{\prime }=& \int d^{3}x\left[ -\tfrac{a}{4}\partial _{\lbrack
i}A_{j]}\partial ^{\lbrack i}A^{j]}-\tfrac{a}{2}\left( \partial
_{0}A_{i}-\partial _{i}\bar{\lambda}\right) \left( \partial
^{0}A^{i}-\partial ^{i}\bar{\lambda}\right) \right.   \notag \\
& -b\varepsilon _{0ij}\bar{\lambda}\partial ^{i}A^{j}-b\varepsilon
_{i0j}A^{i}\partial ^{0}A^{j}-b\varepsilon _{ij0}A^{i}\partial ^{j}\bar
\lambda}  \notag \\
& +\tfrac{1}{4}\partial _{\lbrack i}V_{j]}\partial ^{\lbrack i}V^{j]}+\tfrac
1}{2}\left( \partial _{0}V_{i}-\partial _{i}\lambda ^{\left( 2\right)
}\right) \left( \partial ^{0}V^{i}-\partial ^{i}\lambda ^{\left( 2\right)
}\right)   \notag \\
& \left. +m\varepsilon _{0ij}\bar{\lambda}\partial ^{i}V^{j}+m\varepsilon
_{i0j}A^{i}\partial ^{0}V^{j}+m\varepsilon _{ij0}A^{i}\partial ^{j}\lambda
^{\left( 2\right) }\right] .  \label{x17}
\end{align
In terms of the notation
\begin{equation}
\bar{\lambda}\equiv \bar{A}_{0},\qquad \lambda ^{\left( 2\right) }\equiv
\bar{V}_{0},  \label{x18}
\end{equation
the argument of the exponential takes a a manifestly Lorentz covariant for
\begin{eqnarray}
S_{GU}^{\prime } &=&\int d^{3}x\left( -\tfrac{a}{4}\partial _{\lbrack \mu
\bar{A}_{\nu ]}\partial ^{\lbrack \mu }\bar{A}^{\nu ]}-b\varepsilon _{\mu
\nu \rho }\bar{A}^{\mu }\partial ^{\nu }\bar{A}^{\rho }\right.   \notag \\
&&\left. +\tfrac{1}{4}\partial _{\lbrack \mu }\bar{V}_{\nu ]}\partial
^{\lbrack \mu }\bar{V}^{\nu ]}+m\varepsilon _{\mu \nu \rho }\bar{A}^{\mu
}\partial ^{\nu }\bar{V}^{\rho }\right) .  \label{x19}
\end{eqnarray
where $\bar{A}_{\mu }\equiv \left\{ \bar{A}_{0},A_{i}\right\} $ and $\bar{V
_{\mu }\equiv \left\{ \bar{V}_{0},V_{i}\right\} $. The functional (\ref{x19
) associated with the first-class system describes a Chern-Simons coupling
between the two $1$-forms $\bar{A}_{\mu }$ and $\bar{V}_{\mu }$ \cite{BCS}.

If we set in (\ref{actmcs}) $a=1$ and $b=0$, we find Proca model \cite{vyt3}
and for $a=0$ and $b=1$ we obtain self-dual model \cite{TPN}. It has shown
\cite{vyt3,IJMPAnoi,io1} that from the point of view of Hamiltonian path
integral quantization the Proca model is equivalent with a gauge theory
described by
\begin{equation}
S=\int d^{3}x\left[ -\tfrac{1}{4}\partial _{\lbrack \mu }\bar{A}_{\nu
]}\partial ^{\lbrack \mu }\bar{A}^{\nu ]}-\tfrac{1}{2}\left( \partial _{\mu
}\varphi -m\bar{A}_{\mu }\right) \left( \partial ^{\mu }\varphi -m\bar{A
^{\mu }\right) \right] ,  \label{m1fs}
\end{equation
or, for a suitable extension of the phase space, with first-class theor
\begin{equation}
S=\int d^{3}x\left[ -\tfrac{1}{4}\partial _{\lbrack \mu }\bar{A}_{\nu
]}\partial ^{\lbrack \mu }\bar{A}^{\nu ]}+\tfrac{1}{4}\partial _{\lbrack \mu
}\bar{V}_{\nu ]}\partial ^{\lbrack \mu }\bar{V}^{\nu ]}+m\varepsilon _{\mu
\nu \rho }\bar{A}^{\mu }\partial ^{\nu }\bar{V}^{\rho }\right] .  \label{x21}
\end{equation
In \cite{io2io3} it has proved that the self-dual model is conected to a
first-class theory whose exponent of the exponential from Hamiltonian path
integral takes a manifestly Lorentz-covariant form
\begin{equation}
S=\int d^{3}x\left[ -\varepsilon _{\mu \nu \rho }\bar{A}^{\mu }\partial
^{\nu }\bar{A}^{\rho }-\tfrac{1}{2}\left( \partial _{\mu }\varphi -m\bar{A
_{\mu }\right) \left( \partial ^{\mu }\varphi -m\bar{A}^{\mu }\right) \right
,  \label{msd1f}
\end{equation
o
\begin{equation}
S=\int d^{3}x\left[ -\varepsilon _{\mu \nu \rho }\bar{A}^{\mu }\partial
^{\nu }\bar{A}^{\rho }+\tfrac{1}{4}\partial _{\lbrack \mu }\bar{V}_{\nu
]}\partial ^{\lbrack \mu }\bar{V}^{\nu ]}+m\varepsilon _{\mu \nu \rho }\bar{
}^{\mu }\partial ^{\nu }\bar{V}^{\rho }\right] .  \label{x22}
\end{equation
Setting $a=1$ and $b=0$ in the functionals (\ref{pacttot3.1}) and (\ref{x19
) we arrive at (\ref{m1fs}) and repectively (\ref{x21}). In principle, the
functional (\ref{msd1f}) can be obtained from (\ref{pacttot3.1}) setting
a=0 $ and $b=1$, but this is not a possible choice because the parameter $a$
should be allways nonzero. The same remark can be done regarding the
functionals (\ref{x19}) and (\ref{x22}).

\section{The MECS-Proca model\label{sec3}}

In order to construct an equivalent first-class system starting from the
MECS-Proca model in the framework of the GU approach we need to know the
structure of the constraints set of the model. As the second term in the
action (\ref{y1}) contains higher derivative terms $\left\{ \partial
_{\lambda }\partial ^{\lambda }A_{\mu }\right\} $, the canonical analysis
will be done by a variant of Ostrogradskii method \cite{O1,GT,N1,K1,R1}
developed in \cite{BMP} based on an equivalent first order formalism \cit
{P1} and applied to a number of particle and field theoretic models \cit
{BMP,MP,Paul,BMP1}. The method consists in converting the original higher
derivative theory to an equivalent first order theory by introducing new
fields to account for higher derivative terms. In the equivalent first order
formalism we define the variabiles $B_{\mu }$ a
\begin{equation}
B_{\mu }=\partial _{0}A_{\mu },  \label{y1.1}
\end{equation
and enforce the Lagrangian constraint
\begin{equation}
B_{\mu }-\partial _{0}A_{\mu }=0,
\end{equation
by Lagrange multiplier $\xi ^{\mu }
\begin{eqnarray}
\mathcal{L} &=&-\tfrac{1}{4}\partial _{\lbrack i}A_{j]}\partial ^{\lbrack
i}A^{j]}-\tfrac{1}{2}\left( B_{i}-\partial _{i}A_{0}\right) \left(
B^{i}-\partial ^{i}A^{0}\right)  \notag \\
&&+\tfrac{1}{2}\varepsilon _{0ij}\left( \partial _{0}B^{0}+\partial
_{k}\partial ^{k}A^{0}\right) \partial ^{i}A^{j}+\tfrac{1}{2}\varepsilon
_{i0j}\left( \partial _{0}B^{i}+\partial _{k}\partial ^{k}A^{i}\right) B^{j}
\notag \\
&&+\tfrac{1}{2}\varepsilon _{ij0}\left( \partial _{0}B^{i}+\partial
_{k}\partial ^{k}A^{i}\right) \partial ^{j}A^{0}-\tfrac{1}{2}A_{\mu }A^{\mu
}+\xi ^{\mu }\left( B_{\mu }-\partial _{0}A_{\mu }\right) .  \label{y1.2}
\end{eqnarray
The canonical analysis of the model described by the first order Lagrangian
\ref{y1.2}) displays the irreducible second-class constraints
\begin{eqnarray}
\Phi _{\mu }^{\left( \xi \right) } &\equiv &\Pi _{\mu }\approx 0,
\label{y2.1} \\
\Phi ^{\left( A\right) \mu } &\equiv &p^{\mu }+\xi ^{\mu }\approx 0,
\label{y2.2} \\
\Phi _{i}^{\left( B\right) } &\equiv &\pi _{i}+\tfrac{1}{2}\varepsilon
_{0ij}B^{j}-\tfrac{1}{2}\varepsilon _{0ij}\partial ^{j}A^{0}-\tfrac{1}{2
\varepsilon _{0ij}\partial ^{j}\Pi ^{0}\approx 0,  \label{y2.4} \\
\Phi ^{\left( B\right) } &\equiv &\pi _{0}-\tfrac{1}{2}\varepsilon
_{0ij}\partial ^{i}A^{j}-\tfrac{1}{2}\varepsilon _{0ij}\partial ^{i}\Pi
^{j}\approx 0,  \label{y2.3} \\
\Phi _{II}^{\left( B\right) } &\equiv &\xi _{0}-\tfrac{1}{2}\varepsilon
_{0ij}\partial ^{i}B^{j}\approx 0,  \label{y4} \\
\Phi _{III}^{\left( B\right) } &\equiv &\partial _{i}\xi ^{i}+A_{0}-\tfrac{
}{2}\varepsilon _{0ij}\partial _{k}\partial ^{k}\left( \partial
^{i}A^{j}\right) \approx 0,  \label{y5} \\
\Phi _{IV}^{\left( B\right) } &\equiv &\partial _{i}A^{i}+B_{0}\approx 0,
\label{y6}
\end{eqnarray
and the canonical Hamiltonia
\begin{eqnarray}
H_{c} &=&\int d^{2}x\left[ \tfrac{1}{4}\partial _{\lbrack i}A_{j]}\partial
^{\lbrack i}A^{j]}+\tfrac{1}{2}\left( B_{i}-\partial _{i}A_{0}\right) \left(
B^{i}-\partial ^{i}A^{0}\right) \right.  \notag \\
&&-\tfrac{1}{2}\varepsilon _{0ij}\left( \partial _{k}\partial
^{k}A^{0}\right) \partial ^{i}A^{j}-\tfrac{1}{2}\varepsilon _{i0j}\left(
\partial _{k}\partial ^{k}A^{i}\right) B^{j}  \notag \\
&&\left. -\tfrac{1}{2}\varepsilon _{ij0}\left( \partial _{k}\partial
^{k}A^{i}\right) \partial ^{j}A^{0}-\xi ^{\mu }B_{\mu }+\tfrac{1}{2}A_{\mu
}A^{\mu }\right] ,  \label{y3.1}
\end{eqnarray
where $\left\{ p^{\mu },\pi ^{\mu },\Pi _{\mu }\right\} $ are the canonical
momenta conjugated with the fields $\left\{ A_{\mu },B_{\mu },\xi ^{\mu
}\right\} $. The nonzero Poisson brackets among the constraints functions
read a
\begin{eqnarray}
\left[ \Phi _{\mu }^{\left( \xi \right) }\left( x\right) ,\Phi ^{\left(
A\right) \nu }\left( y\right) \right] _{x_{0}=y_{0}} &=&-\delta _{\mu }^{\nu
}\delta ^{2}\left( \mathbf{x}-\mathbf{y}\right) , \\
\left[ \Phi _{\mu }^{\left( \xi \right) }\left( x\right) ,\Phi _{II}^{\left(
B\right) }\left( y\right) \right] _{x_{0}=y_{0}} &=&-\delta _{\mu
}^{0}\delta ^{2}\left( \mathbf{x}-\mathbf{y}\right) , \\
\left[ \Phi _{\mu }^{\left( \xi \right) }\left( x\right) ,\Phi
_{III}^{\left( B\right) }\left( y\right) \right] _{x_{0}=y_{0}} &=&\delta
_{\mu }^{i}\partial _{i}\delta ^{2}\left( \mathbf{x}-\mathbf{y}\right) , \\
\left[ \Phi ^{\left( A\right) \mu }\left( x\right) ,\Phi _{III}^{\left(
B\right) }\left( y\right) \right] _{x_{0}=y_{0}} &=&\left( -\delta _{0}^{\mu
}+\tfrac{1}{2}\delta _{i}^{\mu }\varepsilon ^{0ij}\partial _{k}\partial
^{k}\partial _{j}\right) \delta ^{2}\left( \mathbf{x}-\mathbf{y}\right) , \\
\left[ \Phi ^{\left( A\right) \mu }\left( x\right) ,\Phi _{IV}^{\left(
B\right) }\left( y\right) \right] _{x_{0}=y_{0}} &=&\delta _{i}^{\mu
}\partial ^{i}\delta ^{2}\left( \mathbf{x}-\mathbf{y}\right) , \\
\left[ \Phi ^{\left( B\right) }\left( x\right) ,\Phi _{IV}^{\left( B\right)
}\left( y\right) \right] _{x_{0}=y_{0}} &=&-\delta ^{2}\left( \mathbf{x}
\mathbf{y}\right) , \\
\left[ \Phi _{i}^{\left( B\right) }\left( x\right) ,\Phi _{j}^{\left(
B\right) }\left( y\right) \right] _{x_{0}=y_{0}} &=&\varepsilon _{0ij}\delta
^{2}\left( \mathbf{x}-\mathbf{y}\right) , \\
\left[ \Phi _{i}^{\left( B\right) }\left( x\right) ,\Phi _{II}^{\left(
B\right) }\left( y\right) \right] _{x_{0}=y_{0}} &=&\varepsilon
_{0ij}\partial ^{j}\delta ^{2}\left( \mathbf{x}-\mathbf{y}\right) .
\end{eqnarray
The number of physical degrees of freedom of the orginal system is equal t
\begin{eqnarray}
\mathcal{\bar{N}}_{O} &=&\left( 18\ \text{canonical\ variables}-12\ \text{sc
}\right) /2  \notag \\
&=&3.  \label{dofoemcs}
\end{eqnarray
We notice that the number of physical degrees of freedom of the extended
model is higher than the number of physical degrees of freedom of the
MCS-Proca model
\begin{equation*}
\mathcal{\bar{N}}_{O}>\mathcal{N}_{O}.
\end{equation*
This result was expected due to the higher derivative nature of the
MECS-Proca model.

Imposing the constraints $\Phi _{\mu }^{\left( \xi \right) }\approx 0$ and\
\Phi ^{\left( A\right) \mu }\approx 0$ strongly zero and eliminating the
unphysical sector $\xi ^{\mu }$ and $\Pi _{\mu }$ (the reduced phase space
is locally parametrized by $\left\{ A_{\mu },B_{\mu },p^{\mu },\pi ^{\mu
}\right\} $) we arrive at a system subject to the second-class constraint
\begin{eqnarray}
\chi _{i}^{\left( 1\right) } &\equiv &\pi _{i}+\tfrac{1}{2}\varepsilon
_{0ij}B^{j}-\tfrac{1}{2}\varepsilon _{0ij}\partial ^{j}A^{0}\approx 0,
\label{y8.1} \\
\chi ^{\left( 1\right) } &\equiv &\pi _{0}-\tfrac{1}{2}\varepsilon
_{0ij}\partial ^{i}A^{j}\approx 0,  \label{y8.2} \\
\chi ^{\left( 2\right) } &\equiv &-p_{0}-\tfrac{1}{2}\varepsilon
_{0ij}\partial ^{i}B^{j}\approx 0,  \label{y8.3} \\
\chi ^{\left( 3\right) } &\equiv &-\partial _{i}p^{i}+A_{0}-\tfrac{1}{2
\varepsilon _{0ij}\partial _{k}\partial ^{k}\left( \partial ^{i}A^{j}\right)
\approx 0,  \label{y8.4} \\
\chi ^{\left( 4\right) } &\equiv &\partial _{i}A^{i}+B_{0}\approx 0.
\label{y8.5}
\end{eqnarray
The nontrivial Poisson brackets between the constraints functions are listed
bello
\begin{eqnarray}
\left[ \chi _{i}^{\left( 1\right) }\left( x\right) ,\chi _{j}^{\left(
1\right) }\left( y\right) \right] _{x_{0}=y_{0}} &=&\varepsilon _{0ij}\delta
^{2}\left( \mathbf{x}-\mathbf{y}\right) ,  \label{y10.1} \\
\left[ \chi _{i}^{\left( 1\right) }\left( x\right) ,\chi ^{\left( 2\right)
}\left( y\right) \right] _{x_{0}=y_{0}} &=&\varepsilon _{0ij}\partial
^{j}\delta ^{2}\left( \mathbf{x}-\mathbf{y}\right) , \\
\left[ \chi ^{\left( 1\right) }\left( x\right) ,\chi ^{\left( 4\right)
}\left( y\right) \right] _{x_{0}=y_{0}} &=&-\delta ^{2}\left( \mathbf{x}
\mathbf{y}\right) ,  \label{y10.2} \\
\left[ \chi ^{\left( 2\right) }\left( x\right) ,\chi ^{\left( 3\right)
}\left( y\right) \right] _{x_{0}=y_{0}} &=&\delta ^{2}\left( \mathbf{x}
\mathbf{y}\right) , \\
\left[ \chi ^{\left( 3\right) }\left( x\right) ,\chi ^{\left( 4\right)
}\left( y\right) \right] _{x_{0}=y_{0}} &=&-\partial _{k}\partial ^{k}\delta
^{2}\left( \mathbf{x}-\mathbf{y}\right) ,
\end{eqnarray
while the canonical Hamiltonian (\ref{y3.1}) takes the for
\begin{eqnarray}
H_{c} &=&\int d^{2}x\left[ \tfrac{1}{4}\partial _{\lbrack i}A_{j]}\partial
^{\lbrack i}A^{j]}+\tfrac{1}{2}\left( B_{i}-\partial _{i}A_{0}\right) \left(
B^{i}-\partial ^{i}A^{0}\right) \right.  \notag \\
&&-\tfrac{1}{2}\varepsilon _{0ij}\left( \partial _{k}\partial
^{k}A^{0}\right) \partial ^{i}A^{j}-\tfrac{1}{2}\varepsilon _{i0j}\left(
\partial _{k}\partial ^{k}A^{i}\right) B^{j}  \notag \\
&&\left. -\tfrac{1}{2}\varepsilon _{ij0}\left( \partial _{k}\partial
^{k}A^{i}\right) \partial ^{j}A^{0}+p^{\mu }B_{\mu }+\tfrac{1}{2}A_{\mu
}A^{\mu }\right] .  \label{y9}
\end{eqnarray
If we make the following linear combination of the constraints $\chi
^{\left( 2\right) }\approx 0$ and $\chi _{i}^{\left( 1\right) }\approx 0$
\begin{equation}
\chi ^{\left( 2\right) }\longrightarrow \chi ^{\left( 2\right) }+\partial
^{i}\chi _{i}^{\left( 1\right) },  \label{y11}
\end{equation
the matrix of the Poisson bracket among the constraint functions become
\begin{equation}
C_{\alpha _{0}\beta _{0}}=\left(
\begin{array}{ccccc}
\varepsilon _{0ij} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\
\mathbf{0} & 0 & 0 & 0 & -1 \\
\mathbf{0} & 0 & 0 & 1 & 0 \\
\mathbf{0} & 0 & -1 & 0 & -\partial _{k}\partial ^{k} \\
\mathbf{0} & 1 & 0 & \partial _{k}\partial ^{k} &
\end{array
\right) .  \label{y12}
\end{equation
Analyzing the above matrix we find that the constraints $\chi _{i}^{\left(
1\right) }\approx 0$ generate a submatrix (of the matrix of the Poisson
brackets among the constraint functions) of maximum rank, therefore they
form a subset of irreducible second-class constraints. Thus in the sequel we
examine from the point of view of the GU method only the constraints $\chi
_{A}\equiv \left\{ \chi ^{\left( 1\right) },\chi ^{\left( 2\right) },\chi
^{\left( 3\right) },\chi ^{\left( 4\right) }\right\} \approx 0$.

\subsection{St\"{u}ckelberg-like coupling}

The second-class constraints set $\chi _{A}\approx 0$ cannot be
straightforwardly separated in two subsets such that one of them being
first-class and the other providing some canonical gauge conditions for the
first-class subset. To make this possible we write the constraints set in an
equivalent for
\begin{equation}
\chi _{A}^{\prime }=E_{AB}\chi _{B},  \label{y13}
\end{equation
where $E_{AB}$ is an invertible matrix
\begin{equation}
E_{AB}=\left(
\begin{array}{cccc}
\partial _{k}\partial ^{k} & 0 & -1 & 0 \\
0 & 1 & 0 & 0 \\
-1 & 0 & 0 & 0 \\
0 & 0 & 0 &
\end{array
\right) ,  \label{y14}
\end{equation
The concrete form of the constraints $\chi _{A}^{\prime }\approx 0$ i
\begin{eqnarray}
\chi ^{\prime \left( 1\right) } &\equiv &\partial _{i}p^{i}-A_{0}+\partial
_{k}\partial ^{k}\pi _{0}\approx 0,  \label{y15.1} \\
\chi ^{\prime \left( 2\right) } &\equiv &-p_{0}+\partial _{i}\pi ^{i}\approx
0,  \label{y15.2} \\
\chi ^{\prime \left( 3\right) } &\equiv &-\pi _{0}+\tfrac{1}{2}\varepsilon
_{0ij}\partial ^{i}A^{j}\approx 0,  \label{y15.3} \\
\chi ^{\prime \left( 4\right) } &\equiv &\partial _{i}A^{i}+B_{0}\approx 0,
\label{y15.4}
\end{eqnarray
with the matrix of the Poisson brackets among the constraint functions
expressed by
\begin{equation}
C_{AB}=\left(
\begin{array}{cccc}
0 & 1 & 0 & 0 \\
-1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & -1 &
\end{array
\right) .  \label{y16}
\end{equation
According to the GU method we consider $G_{a}\equiv \left\{ \chi ^{\prime
\left( 1\right) }\text{, }\chi ^{\prime \left( 3\right) }\right\} \approx 0$
as the first-class constraints set and the remaining constraints
C_{a}\equiv \left\{ \chi ^{\prime \left( 2\right) }\text{, }\chi ^{\prime
\left( 4\right) }\right\} \approx 0$ as the corresponding canonical gauge
conditions. In \cite{mitra} it has proved that for a dynamical system
subject to the second-class constraints $\left\{ \chi _{\alpha _{0}}\approx
0\right\} _{\alpha _{0}=\overline{1,2M_{0}}}$, the subsets $\left\{ \chi
_{1},\chi _{2},\ldots ,\chi _{M_{0}}\right\} $ and $\left\{ \chi _{1},\chi
_{2},\ldots ,\chi _{M_{0}-1},\chi _{M_{0}+1}\right\} $ of the full set of
constraints are first-class sets on $\Sigma _{2M_{0}}$. Our choice
corresponds to the second first-class subset.

Starting from the canonical Hamiltonian of the original second-class system
we construct a first-class Hamiltonian with respect to the first-class
subset in two steps \cite{vyt0}. First, we construct the first-class
Hamiltonian with respect to the constraint $G_{1}\approx 0
\begin{eqnarray}
H_{GU}^{1} &=&H_{c}-C_{1}\left[ G_{1},H_{c}\right] +\tfrac{1}{2}C_{1}C_{1
\left[ G_{1}\left[ G_{1},H_{c}\right] \right] -\cdots  \notag \\
&=&H_{c}+\int d^{2}x\left[ \tfrac{1}{2}\partial _{k}\left( p_{0}+\tfrac{1}{2
\varepsilon _{0ij}\partial ^{i}B^{j}\right) \partial ^{k}\left( p_{0}+\tfrac
1}{2}\varepsilon _{0lm}\partial ^{l}B^{m}\right) \right.  \notag \\
&&\left. -\left( p_{0}+\tfrac{1}{2}\varepsilon _{0ij}\partial
^{i}B^{j}\right) \left( \partial _{k}A^{k}+B_{0}\right) \right] ,
\label{y18}
\end{eqnarray
and then with this at hand we obtain the first-class Hamiltonian with
respect to the constraint $G_{2}\approx 0
\begin{eqnarray}
H_{GU} &=&H_{GU}^{1}-C_{2}\left[ G_{2},H_{GU}^{1}\right] +\tfrac{1}{2
C_{2}C_{2}\left[ G_{2}\left[ G_{2},H_{GU}^{1}\right] \right] -\cdots  \notag
\\
&=&\int d^{2}x\left\{ \tfrac{1}{4}\partial _{\lbrack i}A_{j]}\partial
^{\lbrack i}A^{j]}+\tfrac{1}{2}\left( B_{i}-\partial _{i}A_{0}\right) \left(
B^{i}-\partial ^{i}A^{0}\right) \right.  \notag \\
&&-\tfrac{1}{2}\varepsilon _{0ij}\left( \partial _{k}\partial
^{k}A^{0}\right) \partial ^{i}A^{j}-\tfrac{1}{2}\varepsilon _{i0j}\left(
\partial _{k}\partial ^{k}A^{i}\right) B^{j}  \notag \\
&&-\tfrac{1}{2}\varepsilon _{ij0}\left( \partial _{k}\partial
^{k}A^{i}\right) \partial ^{j}A^{0}+p^{\mu }B_{\mu }+\tfrac{1}{2}A_{0}A^{0}
\notag \\
&&+\tfrac{1}{2}\left[ A_{i}+\partial _{i}\left( p_{0}+\tfrac{1}{2
\varepsilon _{0jk}\partial ^{j}B^{k}\right) \right] \left[ A^{i}+\partial
^{i}\left( p_{0}+\tfrac{1}{2}\varepsilon _{0lm}\partial ^{l}B^{m}\right)
\right]  \notag \\
&&\left. -B_{0}\left( p_{0}+\tfrac{1}{2}\varepsilon _{0jk}\partial
^{j}B^{k}\right) \right\} .  \label{y20}
\end{eqnarray
The number of physical degrees of freedom of the dynamical system with the
phase space locally parametrized by $\left\{ A_{\mu },B_{\mu },p^{\mu },\pi
^{\mu }\right\} $ subject to the second-class constraints (\ref{y8.1})\ and
first-class constraints (\ref{y15.1}) and (\ref{y15.3}) and whose evolution
is governed by the first-class Hamiltonian (\ref{y20}) is equal t
\begin{eqnarray}
\mathcal{\bar{N}}_{GU} &=&\left( 12\ \text{canonical\ variables}-2\text{ scc
-2\times 2\ \text{fcc}\right) /2  \notag \\
&=&3=\mathcal{\bar{N}}_{O}.  \label{dofguemcs}
\end{eqnarray
Based on the equivalence between the first-class system and the original
second-class theory, we replace the Hamiltonian path integral of the
MECS-Proca model with that of the first-class system. The Hamiltonian path
integral for the first-class system constructed in the above reads a
\begin{eqnarray}
Z &=&\int \mathcal{D}\left( A_{\mu },B_{\mu },p^{\mu },\pi ^{\mu },\lambda
^{\left( 1\right) },\lambda ^{\left( 2\right) }\right) \mu \left( \lbrack
A_{\mu }],[B_{\mu }]\right) \mathrm{det}^{1/2}\left( \varepsilon
_{0ij}\delta \left( x-y\right) \right)  \notag \\
&&\times \delta \left( \pi _{i}+\tfrac{1}{2}\varepsilon _{0ij}B^{j}-\tfrac{
}{2}\varepsilon _{0ij}\partial ^{j}A^{0}\right) \exp \left\{ \mathrm{i}\int
d^{3}x\left[ \left( \partial _{0}A_{0}\right) p^{0}\right. \right.  \notag \\
&&+\left( \partial _{0}A_{i}\right) p^{i}+\left( \partial _{0}B_{0}\right)
\pi ^{0}+\left( \partial _{0}B_{i}\right) \pi ^{i}-\mathcal{H}_{GU}  \notag
\\
&&\left. \left. -\lambda ^{\left( 1\right) }\left( \partial
_{i}p^{i}-A_{0}+\partial _{k}\partial ^{k}\pi _{0}\right) -\lambda ^{\left(
2\right) }\left( -\pi _{0}+\tfrac{1}{2}\varepsilon _{0ij}\partial
^{i}A^{j}\right) \right] \right\} ,  \label{y21}
\end{eqnarray
where the integration measure `$\mu \left( \lbrack A_{\mu }],[B_{\mu
}]\right) $' includes some suitable canonical gauge conditions.

Performing partial integration over the momenta $\pi _{i}$ in the path
integral we get to the argument of the exponential in the for
\begin{eqnarray}
S &=&\int d^{3}x\left\{ \left( \partial _{0}A_{0}\right) p^{0}+\left(
\partial _{0}A_{i}\right) p^{i}+\left( \partial _{0}B_{0}\right) \pi
^{0}\right.  \notag \\
&&-\tfrac{1}{2}\left( \partial _{0}B_{i}\right) \varepsilon ^{0ij}\left(
B_{j}-\partial _{j}A_{0}\right) -\tfrac{1}{4}\partial _{\lbrack
i}A_{j]}\partial ^{\lbrack i}A^{j]}  \notag \\
&&-\tfrac{1}{2}\left( B_{i}-\partial _{i}A_{0}\right) \left( B^{i}-\partial
^{i}A^{0}\right) +\tfrac{1}{2}\varepsilon _{0ij}\left( \partial _{k}\partial
^{k}A^{0}\right) \partial ^{i}A^{j}  \notag \\
&&+\tfrac{1}{2}\varepsilon _{i0j}\left( \partial _{k}\partial
^{k}A^{i}\right) B^{j}+\tfrac{1}{2}\varepsilon _{ij0}\left( \partial
_{k}\partial ^{k}A^{i}\right) \partial ^{j}A^{0}-p^{i}B_{i}  \notag \\
&&-\tfrac{1}{2}A_{0}A^{0}-\tfrac{1}{2}\left[ A_{i}+\partial _{i}\left( p_{0}
\tfrac{1}{2}\varepsilon _{0jk}\partial ^{j}B^{k}\right) \right]  \notag \\
&&\times \left[ A^{i}+\partial ^{i}\left( p_{0}+\tfrac{1}{2}\varepsilon
_{0lm}\partial ^{l}B^{m}\right) \right] +\tfrac{1}{2}B^{0}\varepsilon
_{0jk}\partial ^{j}B^{k}  \notag \\
&&\left. -\lambda ^{\left( 1\right) }\left( \partial
_{i}p^{i}-A_{0}+\partial _{k}\partial ^{k}\pi _{0}\right) -\lambda ^{\left(
2\right) }\left( -\pi _{0}+\tfrac{1}{2}\varepsilon _{0ij}\partial
^{i}A^{j}\right) \right\} .  \label{y22}
\end{eqnarray
Integration over $p^{i}$ leads to a $\delta $ function of the form
\begin{equation}
\delta \left( \partial _{0}A_{i}-B_{i}+\partial _{i}\lambda ^{\left(
1\right) }\right)
\end{equation
which permits calculation of the integral over $B_{i}$. Performing partial
integration over Lagrange multiplier $\lambda ^{\left( 2\right) }$ and
momentum $\pi _{0}$, the argument of the exponential become
\begin{eqnarray}
S &=&\int d^{3}x\left\{ \left( \partial _{0}A_{0}\right) \left( p^{0}+\tfrac
1}{2}\varepsilon ^{0ij}\partial _{i}\partial _{0}A_{j}\right) \right.  \notag
\\
&&-\tfrac{1}{4}\partial _{\lbrack i}A_{j]}\partial ^{\lbrack i}A^{j]}-\tfrac
1}{2}\left[ \partial _{0}A_{i}-\partial _{i}\left( A_{0}-\lambda ^{\left(
1\right) }\right) \right] \left[ \partial ^{0}A^{i}-\partial ^{i}\left(
A^{0}-\lambda ^{\left( 1\right) }\right) \right]  \notag \\
&&+\tfrac{1}{2}\varepsilon _{0ij}\partial _{\lambda }\partial ^{\lambda
}\left( A^{0}-\lambda ^{\left( 1\right) }\right) \partial ^{i}A^{j}+\tfrac{
}{2}\varepsilon _{i0j}\left( \partial _{\lambda }\partial ^{\lambda
}A^{i}\right) \partial ^{0}A^{j}  \notag \\
&&+\tfrac{1}{2}\varepsilon _{ij0}\left( \partial _{\lambda }\partial
^{\lambda }A^{i}\right) \partial ^{j}\left( A^{0}-\lambda ^{\left( 1\right)
}\right) -\tfrac{1}{2}\left[ A_{i}+\partial _{i}\left( p_{0}+\tfrac{1}{2
\varepsilon _{0jk}\partial ^{j}\partial ^{0}A^{k}\right) \right]  \notag \\
&&\left. \times \left[ A^{i}+\partial ^{i}\left( p_{0}+\tfrac{1}{2
\varepsilon _{0lm}\partial ^{l}\partial ^{0}A^{m}\right) \right] -\tfrac{1}{
}A_{0}A^{0}+\lambda ^{\left( 1\right) }A_{0}\right\} .  \label{y23}
\end{eqnarray
Making the notatio
\begin{equation}
p^{0}+\tfrac{1}{2}\varepsilon ^{0ij}\partial _{i}\partial _{0}A_{j}\equiv
-\varphi ,\qquad A_{0}-\lambda ^{\left( 1\right) }\equiv \bar{A}_{0},
\label{y24}
\end{equation
and integrating over Lagrange multiplier $\lambda ^{\left( 1\right) }$, the
argument of the exponential from the Hamiltonian path integral reads a
\begin{eqnarray}
S &=&\int d^{3}x\left[ -\tfrac{1}{4}\partial _{\lbrack i}A_{j]}\partial
^{\lbrack i}A^{j]}-\tfrac{1}{2}\left( \partial _{0}A_{i}-\partial _{i}\bar{A
_{0}\right) \left( \partial ^{0}A^{i}-\partial ^{i}\bar{A}^{0}\right) \right.
\notag \\
&&+\tfrac{1}{2}\varepsilon _{0ij}\left( \partial _{\lambda }\partial
^{\lambda }\bar{A}^{0}\right) \partial ^{i}A^{j}+\tfrac{1}{2}\varepsilon
_{i0j}\left( \partial _{\lambda }\partial ^{\lambda }A^{i}\right) \partial
^{0}A^{j}  \notag \\
&&+\tfrac{1}{2}\varepsilon _{ij0}\left( \partial _{\lambda }\partial
^{\lambda }A^{i}\right) \partial ^{j}\bar{A}^{0}  \notag \\
&&\left. -\tfrac{1}{2}\left( \partial _{i}\varphi -A_{i}\right) \left(
\partial ^{i}\varphi -A^{i}\right) -\tfrac{1}{2}\left( \partial _{0}\varphi
\bar{A}_{0}\right) \left( \partial ^{0}\varphi -\bar{A}^{0}\right) \right] .
\label{y25}
\end{eqnarray
The functional (\ref{y25}) takes a manifestly Lorentz-covariant for
\begin{eqnarray}
S &=&\int d^{3}x\left[ -\tfrac{1}{4}\partial _{\lbrack \mu }\bar{A}_{\nu
]}\partial ^{\lbrack \mu }\bar{A}^{\nu ]}+\tfrac{1}{2}\varepsilon _{\mu \nu
\rho }\left( \partial _{\lambda }\partial ^{\lambda }\bar{A}^{\mu }\right)
\partial ^{\nu }\bar{A}^{\rho }\right.  \notag \\
&&\left. -\tfrac{1}{2}\left( \partial _{\mu }\varphi -\bar{A}_{\mu }\right)
\left( \partial ^{\mu }\varphi -\bar{A}^{\mu }\right) \right] .  \label{y26}
\end{eqnarray
where $\bar{A}_{\mu }=\left\{ \bar{A}_{0}\text{, }A_{i}\right\} $ and
describes a (Lagrangian) St\"{u}ckelberg-like coupling between the scalar
field $\varphi $ and the $1$-form $\bar{A}_{\mu }$.

\subsection{Chern-Simons-like coupling}

In the sequel we show that the MECS-Proca model may be related to another
first-class theory. Starting from the GU system constructed in the above,
subject to the second-class constraints (\ref{y8.1}) and the first-class
constraints (\ref{y15.1}) and (\ref{y15.3}) and whose evolution is governed
by the first-class Hamiltonian (\ref{y20}) we consider the following field
combination
\begin{eqnarray}
\mathcal{F}_{0} &\equiv &A_{0},\qquad \mathcal{F}_{i}\equiv A_{i}+\partial
_{i}\left( p_{0}+\tfrac{1}{2}\varepsilon _{0jk}\partial ^{j}B^{k}\right) ,
\label{y28.2} \\
\mathcal{P}_{i} &\equiv &p_{i}-\tfrac{1}{2}\varepsilon _{0ij}\partial
_{k}\partial ^{k}A^{j}-\tfrac{1}{2}\varepsilon _{0ij}\partial
^{j}B^{0},\qquad \mathcal{B}_{i}\equiv B_{i},  \label{y28.3}
\end{eqnarray
which are in (strong) involution with first-class constraints $G_{a}\approx
0
\begin{equation}
\left[ \mathcal{F}_{0},G_{a}\right] =\left[ \mathcal{F}_{i},G_{a}\right]
\left[ \mathcal{P}_{i},G_{a}\right] =\left[ \mathcal{B}_{i},G_{a}\right] =0,
\label{y29}
\end{equation
and moreover $\mathcal{F}_{\mu }\equiv \left\{ \mathcal{F}_{0},\mathcal{F
_{i}\right\} $ is divergenceles
\begin{equation}
\partial ^{\mu }\mathcal{F}_{\mu }=0.  \label{y30}
\end{equation
Similarly to the case of the MCS-Proca model, the first-class Hamiltonian
\ref{y20}) can be written in terms of these quantitie
\begin{eqnarray}
H_{GU} &=&\int d^{2}x\left[ \tfrac{1}{4}\partial _{\lbrack i}\mathcal{F
_{j]}\partial ^{\lbrack i}\mathcal{F}^{j]}+\tfrac{1}{2}\left( \mathcal{B
_{i}-\partial _{i}\mathcal{F}_{0}\right) \left( \mathcal{B}^{i}-\partial ^{i
\mathcal{F}^{0}\right) \right.  \notag \\
&&\left. -\varepsilon _{0ij}\left( \partial _{k}\partial ^{k}\mathcal{F
^{0}\right) \partial ^{i}\mathcal{F}^{j}+\tfrac{1}{2}\mathcal{F}_{i}\mathcal
F}^{i}+\tfrac{1}{2}\mathcal{F}_{0}\mathcal{F}^{0}+\mathcal{B}^{i}\mathcal{P
_{i}\right] .  \label{y27}
\end{eqnarray
In order to write the solution to equation (\ref{y30}) we enlarge the phase
space by adding the bosonic fields/momenta $\left\{ V^{\mu },P_{\mu
}\right\} $
\begin{equation}
\mathcal{F}_{\mu }=-\varepsilon _{\mu \nu \rho }\partial ^{\nu }V^{\rho }.
\label{y31}
\end{equation
When we replace (\ref{y31}) in (\ref{y15.1}) the constraint takes the form
\begin{equation}
\partial _{i}p^{i}+\varepsilon _{0ij}\partial ^{i}V^{j}+\partial
_{k}\partial ^{k}\pi _{0}\approx 0,  \label{y32}
\end{equation
and remains first-class. Computing the Poisson bracket among the quantity
\partial _{i}p_{0}$ and first-class constraint (\ref{y15.1}) and the Poisson
bracket between $P_{i}$ and (\ref{y32}) we obtain that these two quantities
are corelated through the relatio
\begin{equation}
\partial _{i}p_{0}=\varepsilon _{0ij}P^{j}.  \label{y34}
\end{equation
Using (\ref{y31}) and (\ref{y34}) we write the first-class Hamiltonian as
\begin{eqnarray}
H_{GU}^{\prime } &=&\int d^{2}x\left\{ \tfrac{1}{4}\partial _{\lbrack
i}A_{j]}\partial ^{\lbrack i}A^{j]}+\tfrac{1}{2}\left[ B_{i}+\partial
_{i}\left( \varepsilon _{0jk}\partial ^{j}V^{k}\right) \right] \right.
\notag \\
&&\times \left[ B^{i}+\partial ^{i}\left( \varepsilon ^{0lm}\partial
_{l}V_{m}\right) \right] +\tfrac{1}{2}\varepsilon _{0ij}\left[ \partial
_{k}\partial ^{k}\left( \varepsilon ^{0lm}\partial _{l}V_{m}\right) \right]
\partial ^{i}A^{j}  \notag \\
&&-\tfrac{1}{2}\varepsilon _{i0j}\left( \partial _{k}\partial
^{k}A^{i}\right) B^{j}+\tfrac{1}{2}\varepsilon _{ij0}\left( \partial
_{k}\partial ^{k}A^{i}\right) \partial ^{j}\left( \varepsilon ^{0lm}\partial
_{l}V_{m}\right) +p^{i}B_{i}  \notag \\
&&+\tfrac{1}{4}\partial ^{\lbrack i}V^{j]}\partial _{\lbrack i}V_{j]}-\tfrac
1}{2}\varepsilon _{0jk}B^{0}\partial ^{j}B^{k}+\tfrac{1}{2}\left(
A_{i}+\varepsilon _{0ij}P^{j}+\tfrac{1}{2}\varepsilon _{0jk}\partial
_{i}\partial ^{j}B^{k}\right)  \notag \\
&&\left. \times \left( A^{i}+\varepsilon ^{0il}P_{l}+\tfrac{1}{2}\varepsilon
_{0lm}\partial ^{i}\partial ^{l}B^{m}\right) \right\} .  \label{y35}
\end{eqnarray
If we count the number of physical degrees of freedom of the system with the
phase space locally parametrized by $\left\{ A_{i},B_{\mu },V^{\mu
},p^{i},\pi ^{\mu },P_{\mu }\right\} $ subject to the second-class
constraints (\ref{y8.1})\ and first-class constraints (\ref{y15.3}) and (\re
{y32}) and whose evolution is governed by the first-class Hamiltonian (\re
{y35}), we obtai
\begin{eqnarray}
\mathcal{\bar{N}}_{GU}^{\prime } &=&\left( 16\ \text{canonical\ variables}-
\text{ scc}-2\times 2\ \text{fcc}\right) /2  \notag \\
&=&5\neq \mathcal{\bar{N}}_{O}.  \label{dofguemcs'}
\end{eqnarray
Imposing the first-class constraints
\begin{equation}
P_{0}\approx 0,\qquad -\partial ^{i}P_{i}\approx 0,  \label{y33}
\end{equation
the number of physical degrees of freedom is conserved
\begin{eqnarray}
\mathcal{\bar{N}}_{GU}^{\prime } &=&\left( 16\ \text{canonical\ variables}-
\text{ scc}-2\times 4\ \text{fcc}\right) /2  \notag \\
&=&3=\mathcal{\bar{N}}_{O}.
\end{eqnarray
For each first-class theory, derived in the above, we are able to identify a
set of fundamental classical observables such that \ they are in one to one
correspondence and they possess the same Poisson brackets. The previously
exposed procedure preserves the equivalence between two first-class theories
since the number of physical degrees of freedom is the same for both
theories and the corresponding algebras of classical observables are
isomorphic. As a result, the GU and the first-class system remain equivalent
also at the level of the Hamiltonian path integral quantization. This
further implies that the first-class system is completely equivalent with
the MECS-Proca model. Due to this equivalence we can replace the Hamiltonian
path integral of the MECS-Proca model with that associated with the
first-class syste
\begin{eqnarray}
Z^{\prime } &=&\int \mathcal{D}\left( A_{i},B_{\mu },V^{\mu },p^{i},\pi
^{\mu },P_{\mu },\lambda ^{\left( 1\right) },\lambda ^{\left( 2\right)
},\lambda ^{\left( 3\right) },\lambda ^{\left( 4\right) }\right) \mu \left(
\lbrack A_{i}],[B_{\mu }],[V^{\mu }]\right)  \notag \\
&&\times \mathrm{det}^{1/2}\left( \varepsilon _{0ij}\delta \left( x-y\right)
\right) \delta \left[ \pi _{i}+\tfrac{1}{2}\varepsilon _{0ij}B^{j}+\tfrac{1}
2}\varepsilon _{0ij}\partial ^{j}\left( \varepsilon ^{0kl}\partial
_{k}V_{l}\right) \right]  \notag \\
&&\times \exp \left\{ \mathrm{i}\int d^{3}x\left[ \left( \partial
_{0}A_{i}\right) p^{i}+\left( \partial _{0}B_{0}\right) \pi ^{0}+\left(
\partial _{0}B_{i}\right) \pi ^{i}+\left( \partial _{0}V^{0}\right)
P_{0}\right. \right.  \notag \\
&&+\left( \partial _{0}V^{i}\right) P_{i}-\mathcal{H}_{GU}^{\prime }-\lambda
^{\left( 1\right) }\left( \partial _{i}p^{i}+\varepsilon _{0ij}\partial
^{i}V^{j}+\partial _{k}\partial ^{k}\pi _{0}\right)  \notag \\
&&\left. \left. -\lambda ^{\left( 2\right) }\left( -\pi _{0}+\tfrac{1}{2
\varepsilon _{0ij}\partial ^{i}A^{j}\right) -\lambda ^{\left( 3\right)
}P_{0}+\lambda ^{\left( 4\right) }\partial ^{i}P_{i}\right] \right\} .
\label{y36}
\end{eqnarray
After a partial integration over the momenta $\pi _{i}$ in the path integral
the argument of the exponential read a
\begin{eqnarray}
S_{GU}^{\prime } &=&\int d^{3}x\left\{ \left( \partial _{0}A_{i}\right)
p^{i}+\left( \partial _{0}B_{0}\right) \pi ^{0}+\left( \partial
_{0}V^{0}\right) P_{0}+\left( \partial _{0}V^{i}\right) P_{i}\right.  \notag
\\
&&+\tfrac{1}{2}\left( \partial _{0}B_{i}\right) \varepsilon ^{0ij}\left[
-B_{j}-\tfrac{1}{2}\partial _{j}\left( \varepsilon ^{0kl}\partial
_{k}V_{l}\right) \right] -\tfrac{1}{4}\partial _{\lbrack i}A_{j]}\partial
^{\lbrack i}A^{j]}  \notag \\
&&-\tfrac{1}{2}\left[ B_{i}+\partial _{i}\left( \varepsilon _{0jk}\partial
^{j}V^{k}\right) \right] \left[ B^{i}+\partial ^{i}\left( \varepsilon
^{0lm}\partial _{l}V_{m}\right) \right]  \notag \\
&&-\tfrac{1}{2}\varepsilon _{0ij}\left[ \partial _{k}\partial ^{k}\left(
\varepsilon ^{0lm}\partial _{l}V_{m}\right) \right] \partial ^{i}A^{j}
\tfrac{1}{2}\varepsilon _{i0j}\left( \partial _{k}\partial ^{k}A^{i}\right)
B^{j}  \notag \\
&&-\tfrac{1}{2}\varepsilon _{ij0}\left( \partial _{k}\partial
^{k}A^{i}\right) \partial ^{j}\left( \varepsilon ^{0lm}\partial
_{l}V_{m}\right) -\tfrac{1}{4}\partial ^{\lbrack i}V^{j]}\partial _{\lbrack
i}V_{j]}  \notag \\
&&+\tfrac{1}{2}\left[ A_{i}+\varepsilon _{0ij}P^{j}+\tfrac{1}{2}\varepsilon
_{0jk}\partial _{i}\partial ^{j}B^{k}\right] \left[ A^{i}+\varepsilon
^{0il}P_{l}+\tfrac{1}{2}\varepsilon _{0lm}\partial ^{i}\partial ^{l}B^{m
\right]  \notag \\
&&-p^{i}B_{i}+\tfrac{1}{2}\varepsilon _{0jk}B^{0}\partial ^{j}B^{k}-\lambda
^{\left( 1\right) }\left( \partial _{i}p^{i}+\varepsilon _{0ij}\partial
^{i}V^{j}+\partial _{k}\partial ^{k}\pi _{0}\right)  \notag \\
&&\left. -\lambda ^{\left( 2\right) }\left( -\pi _{0}+\tfrac{1}{2
\varepsilon _{0ij}\partial ^{i}A^{j}\right) -\lambda ^{\left( 3\right)
}P_{0}+\lambda ^{\left( 4\right) }\partial ^{i}P_{i}\right\} .  \label{y37}
\end{eqnarray
Integration over $p^{i}$ leads to a $\delta $ function of the form
\begin{equation}
\delta \left( \partial _{0}A_{i}-B_{i}+\partial _{i}\lambda ^{\left(
1\right) }\right)  \label{y38}
\end{equation
which permits calculation of the integral over $B_{i}$. Performing partial
integration over the field $V_{0}$, momenta $\{\pi _{0}$, $P_{0}$, $P_{i}\}$
and Lagrange multipliers $\{\lambda ^{\left( 2\right) }$, $\lambda ^{\left(
3\right) }\}$, the argument of the exponential from the Hamiltonian path
integral takes the for
\begin{eqnarray}
S_{GU}^{\prime } &=&\int d^{3}x\left\{ -\tfrac{1}{4}\partial _{\lbrack
i}A_{j]}\partial ^{\lbrack i}A^{j]}\right. -\tfrac{1}{2}\left[ \partial
_{0}A_{i}-\partial _{i}\left( -\lambda ^{\left( 1\right) }-\varepsilon
_{0jk}\partial ^{j}V^{k}\right) \right]  \notag \\
&&\times \left[ \partial ^{0}A^{i}-\partial ^{i}\left( -\lambda ^{\left(
1\right) }-\varepsilon ^{0lm}\partial _{l}V_{m}\right) \right]  \notag \\
&&+\tfrac{1}{2}\varepsilon _{0ij}\partial _{\lambda }\partial ^{\lambda
}\left( -\lambda ^{\left( 1\right) }-\varepsilon ^{0kl}\partial
_{k}V_{l}\right) \partial ^{i}A^{j}+\tfrac{1}{2}\varepsilon _{i0j}\left(
\partial _{\lambda }\partial ^{\lambda }A^{i}\right) \partial ^{0}A^{j}
\notag \\
&&+\tfrac{1}{2}\varepsilon _{ij0}\left( \partial _{\lambda }\partial
^{\lambda }A^{i}\right) \partial ^{j}\left( -\lambda ^{\left( 1\right)
}-\varepsilon ^{0kl}\partial _{k}V_{l}\right)  \notag \\
&&+\tfrac{1}{4}\partial _{\lbrack i}V_{j]}\partial ^{\lbrack i}V^{j]}+\tfrac
1}{2}\left( \partial _{0}V_{i}-\partial _{i}\lambda ^{\left( 4\right)
}\right) \left( \partial ^{0}V^{i}-\partial ^{i}\lambda ^{\left( 4\right)
}\right)  \notag \\
&&+\varepsilon _{0ij}\left( \partial ^{0}V^{i}-\partial ^{i}\lambda ^{\left(
4\right) }\right) A^{j}  \notag \\
&&\left. +\varepsilon _{ij0}\left( \partial ^{i}V^{j}\right) \left( -\lambda
^{\left( 1\right) }-\varepsilon ^{0kl}\partial _{k}V_{l}\right) \right\} .
\label{y39}
\end{eqnarray
Using the notatio
\begin{equation}
-\lambda ^{\left( 1\right) }-\varepsilon _{0jk}\partial ^{j}V^{k}\equiv \bar
A}_{0},\qquad \lambda ^{\left( 4\right) }\equiv \bar{V}_{0},  \label{y40}
\end{equation
the argument of the exponential from the Hamiltonian path integral reads a
\begin{eqnarray}
S_{GU}^{\prime } &=&\int d^{3}x\left[ -\tfrac{1}{4}\partial _{\lbrack
i}A_{j]}\partial ^{\lbrack i}A^{j]}-\tfrac{1}{2}\left( \partial
_{0}A_{i}-\partial _{i}\bar{A}_{0}\right) \left( \partial ^{0}A^{i}-\partial
^{i}\bar{A}^{0}\right) \right.  \notag \\
&&+\tfrac{1}{2}\varepsilon _{i0j}\left( \partial _{\lambda }\partial
^{\lambda }A^{i}\right) \partial ^{0}A^{j}+\tfrac{1}{2}\varepsilon
_{0ij}\left( \partial _{\lambda }\partial ^{\lambda }\bar{A}^{0}\right)
\partial ^{i}A^{j}  \notag \\
&&+\tfrac{1}{2}\varepsilon _{ij0}\left( \partial _{\lambda }\partial
^{\lambda }A^{i}\right) \partial ^{j}\bar{A}^{0}+\tfrac{1}{4}\partial
_{\lbrack i}V_{j]}\partial ^{\lbrack i}V^{j]}  \notag \\
&&+\tfrac{1}{2}\left( \partial _{0}V_{i}-\partial _{i}\bar{V}_{0}\right)
\left( \partial ^{0}V^{i}-\partial ^{i}\bar{V}^{0}\right)  \notag \\
&&\left. +\varepsilon _{0ij}\left( \partial ^{0}V^{i}-\partial ^{i}\bar{V
^{0}\right) A^{j}+\varepsilon _{ij0}\left( \partial ^{i}V^{j}\right) \bar{A
^{0}\right] .  \label{y41}
\end{eqnarray
The functional (\ref{y41}) takes a manifestly Lorentz-covariant for
\begin{eqnarray}
S_{GU}^{\prime } &=&\int d^{3}x\left[ -\tfrac{1}{4}\partial _{\lbrack \mu
\bar{A}_{\nu ]}\partial ^{\lbrack \mu }\bar{A}^{\nu ]}+\tfrac{1}{2
\varepsilon _{\mu \nu \rho }\left( \partial _{\lambda }\partial ^{\lambda
\bar{A}^{\mu }\right) \partial ^{\nu }\bar{A}^{\rho }\right.  \notag \\
&&\left. +\tfrac{1}{4}\partial _{\lbrack \mu }\bar{V}_{\nu ]}\partial
^{\lbrack \mu }\bar{V}^{\nu ]}+\varepsilon _{\mu \nu \rho }\left( \partial
^{\mu }\bar{V}^{\nu }\right) \bar{A}^{\rho }\right] ,
\end{eqnarray
where $\bar{A}_{\mu }=\left\{ \bar{A}_{0}\text{, }A_{i}\right\} $ and $\bar{
}_{\mu }=\left\{ \bar{V}_{0}\text{, }V_{i}\right\} $. The above functional
describes a (Lagrangian) Chern-Simons-like coupling between the $1$-form
\bar{A}_{\mu }$ and the $1$-form $\bar{V}_{\mu }$.

\section{Conclusion\label{sec4}}

In this paper the MCS-Proca model and a higher order derivative extension of
it have been analyzed from the point of view of the Hamiltonian path
integral quantization, in the framework of gauge-unfixing approach. The
first step of this approach is represented by the construction of an
equivalent first-class system. In order to construct the equivalent
first-class system with MECS-Proca model we realize a partial gauge-unfixing
(we maintain the second-class constraints (\ref{y8.1})), meanwhile in the
case of the MCS-Proca model we accomplish a total gauge-unfixing. The second
step involves the construction of the Hamiltonian path integral
corresponding to the equivalent first-class system for each model. The
construction of the equivalent first-class system does not require an
extension of the original phase space. The Hamiltonian path integral of the
first-class systems takes a manifestly Lorentz-covariant form after
integrating out the auxiliary fields and performing some field
redefinitions. In order to obtain a manifestly covariant path integral it is
not mandatory to enlarge the phase space adding extra fields. When we work
in the original phase space, we obtain the Lagrangian path integral
corresponding to St\"{u}ckelberg-like coupling between a scalar field and $1
- form, while the extension of the phase space leads to the Lagrangian path
integral for two kinds of $1$-forms with Chern-Simons-like coupling.

\section*{Acknowledgement}

The author wishes to thank E. M. Cioroianu for useful discussions and
comments. Also, I am very grateful to Prof. S. Deser for calling my attention to the ref. \cite{DT}

redpajama_set_name: RedPajamaArXiv


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Animal Documents

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redpajama_set_name: RedPajamaGithub


Japan has now frozen the assets of 103 companies and organizations and 108 individuals.
TOKYO – Japan froze the assets of 19 companies on Friday to step up pressure on North Korea to return Japanese citizens that it abducted in the 1970s and 1980s and to halt its nuclear weapons and missile development.
The companies, which have already been sanctioned by the United States, deal in finance, coal and minerals, transport including shipping and the sending of North Korean workers overseas.
A Foreign Ministry official said the unilateral move shows Japan's commitment to sanctions ahead of a U.N. Security Council meeting in New York later Friday to discuss the North Korea situation.
Japan has now frozen the assets of 103 companies and organizations and 108 individuals under either its own sanctions or Security Council resolutions. Of those, 56 groups and 62 individuals were unilateral.
Chief Cabinet Secretary Yoshihide Suga said Japan will continue to put pressure on North Korea to seek a resolution to both the abductee and nuclear and missile issues.
North Korean agents abducted Japanese citizens to train spies to pass as Japanese.
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North Korean leader Kim Jong-Un and Cuba's President Miguel Diaz-Canel review the honor guards while Kim's wife Ri Sol Ju and Diaz-Canel's wife Lis Cuesta Peraza walk behind during the Cuban leader's departure at Pyongyang airport, Nov. 6, 2018 North Korean leader Kim Jong Un and Cuba's President Miguel Diaz-Canel wave from a motorcade in Pyongyang, Nov. 4, 2018.


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