Gauge Invariant Variational Methods for Hamiltonian Lattice Gauge

SLAC-PUB-2864 December

1981

(T/E)

GAUGE INVARIANT

VARIATIONAL METHODS FOR HAMILTONIAN LATTICE GAUGE THEORIES

0. Tel

Aviv

Horn

University,

Ramat

Aviv

Israel

M. Weinstein Stanford

Linear

Accelerator

Center,

submitted

* work

supported

in

part

by

to

the

Physical

Department

Stanford

Review

of

California

94305

D

Energy,

contract

DE-AC03-76SF00515. **

work

supported

in

part

by

the

US-Israel

(BSF)

-l-

Binational

Science

Foundation

ABSTRACT

This state

paper and

defined

develops

excited

in

state

of

allowing

mean-field,

which

methods,

into

methods

apply

that

physics

one

Hartree-Fock

approximations,

fully in

they of

The scheme

introduced

to

more

convert

space

by

their

very

invariant

same way to

at

least

periodic This

non-abelian

theories

and

shows

computing

the

partition

to

that

of

We discuss derives the

the for

partition

theory.

function

non-abelian function

is

abelian

theory

extremely

interesting

discussion glue-ball

of

level. of

the

the

system the

can

hadron

of

complicated of

gauge

extend spectrum.

-2-

both

out

which the

one

function

we are these of to

system. one

evaluation

function

Hat-tree-Fock

ideas

spin

evaluation

between

problem

of

a non-abelian

partition

the

the

and

Rayleigh-Ritz

to

theory,

theories,

abelian

problem

this

these

3+1 space-time

spin

the

that

describe

a classical

a discussion these

non-invariant

of

than

emerge with

for

carrying

because

as

non-abelian

equivalent

the

group

2+1 and

the

effective

but

such

correctly

in

reduce

gauge-projected

We close

has

tools

problem

ways

SU(2)

paper

this

We show

to

effective

differences

how one and

the

evaluation

pure

in

and

function

derived, more

by comparing

simple

of

no explicit

that

the

is

considerably

PQED with

this

form

theory

how to

the

theories

gauge

abelian

the

discuss

for

The explicit

However,

of

then

nature

ground

gauge

renormalization

enough

formulates

evaluation

PQED and

both

the

lattice

familiar

electrodynamics

dimensions.

calculating

techniques.

powerful

quantum paper

for

Hamiltonian

real

gauge

are

of

and are

the

methods

spectrum

A0 = 0 gauge.

advantage

and

variational

of

is

the

this

derived

in

the

presented.

wavefunction able

to

the

and

for

show

theories

fermions

allow

for

even

that at

a

computation

of

INTRODUCTION

1. Quantum have

for

chromodynamics a theory

of

satisfactory

treatment

confinement,

the

given

to

continuum

the

attempts

of to

techniques,

extract

successive

carrying state

of

the

promise

formalism

gauge

any

lattice of

strong weak

to

successfully

steps

must

this

paper

point

of

effects, which founder

coupling

one

within

of

fail

to

make

when

the

the

one

requirement

set

a formalism for

Aa=O gauge,

situation.

view

group

with

calculations in

of for

the

ground

which

holds

The virtues

of

out

this

are: it

provides

it

shares

applicable

a general

way of

converting

without

losing

invariant

one

with

perturbation

to

physical

interest

the

weak if

the

theory coupling

one

wants

any

the

to

ability

virtue

regime

(this

make

variational to of is

scheme compute;

being the

directly

regime

a correspondence

to

of the

continuum) 3.

been

regime.’

deal

we present

defined this

the

renormalization

keep

variational

upon

has

coupling,

able

gauge-theory

etc.,

calculations,

not

invariant

spectrum,

Lattice

real-space

In

e.g.

from

we

no theory,

instanton

the

candidate

the

including

as

improving

a gauge 2.

theory

such

states.

out

of

of

hadron

methods,

truncation

gauge-invariant

at

physics

aspects

the

even

clear

been

satisfactory nevertheless,

the

confinement.

the

only

basic

analyze

theory,

confinement

have

most

spectrum, to

of

be the

interactions;

the

ball

Attempts

Non-perturbative

1.

of

physics

physics

that

strong

perturbation

clarify the

the

glue

date.

may well

it

can

theories2

be demonstrated, and

periodic

at

least

lattice

-3-

for quantum

the

case

of

Z(2)-gauge

electrodynamics

(PQED),

to

that or

application

of

Hartree-Fock

and

better

this

method

analysis

results

of

which

to

these

improve models

either

the

allows

one

cannot

be obtained

using

generalized

from

case

mean-field to

obtain

these

new

methods

al one; 4.

the

method

theories 5.

the

is to

method

readlily

that in

of

principle

non-perturbative gauge with 6.

it

and

possible

continuum We begin

that

from

this

the

establishes non-zero of

of

with

to

2 begins

can

glue-ball in

out

a

spectrum

the

case

of

the

of the

the way of

orient with

formalism

of

in

a pure

theories

to

provide

schemes

in

We will

results

for

2+1 and

which

this

paper

the

reader

material

as not

and

is

how to

the obtainable works

show

confinement

gauge

gauge

show

how this

We conclude

non-abelian

this

are

3+1-dimensions

constant. to

then

which

exhibits

with

such

To demonstrate

2+1-dimensions

method

dealing

technique,

obtain

alone.

coupling

for

groups.

a variational

PQED in

PQED in of

be extended

computational

gauge

to

method

sections order

method

a general

continuous

physics

extension

different

carrying

theories;

formalism

that

description

Section

upon

values

the

In

gauge

approximation,

the

the

spectrum

non-perturbative

variational

discuss

a way of

for

hadron

the

by presenting

based

Hartree-Fock

abelian

fermions.

implementable

combine

provides

the

of

theories;

computation

theory.

is

theories

non-abelian

the

how one

for

with

we

all

a discussion

theories.

A

divided

among

the

lattice

gauge

theories

follows. unfamiliar

a discussion

of

-4-

the

with general

problem

and

then

explains third

the

idea

section

Here

behind

we turn

we establish

technique

our

to

the

context

of goes

and

explain

how one

confines

or

doesn’t

confine.

groundstate

the

problem

energy

to

d-dimensional general the

including

show

the

view,

field

analysis

presented

to

in

to

this

out

show

section and

however,

the

simple

example

explained

in

detail.

Since

more

al3

complicated

the

really

sort

of this

problem,

of of

physics

example knowledge

-5-

reasons

for

two

we wish

to

with

those

comparison

tension. for

will

Al though

dealing

the

with

the for

pedagogical

is,

analysis.

to

calculations

problem

to

point

a large

The arguments established

will

not a single

contains of

less

improved

from

of

is

a

are

simplifying

results

discussion the

of

for

out

This

Second,

use

the

of

the

carries

analysis

string

the

how one

important,

required

promise

methods

so

the

is of

four

be systematically

of

this

section

function

There

our

theory

estimate

of

approach. can

not

make

et

the

field

that

analytic

question

Hartree-Fock

considerably.

Drell,

of

by

holds

system

amenable

is

this

Section

and most

computation analysis

we wish

the

general mean

it

theory,

non-abelian

degree,

work

of

system.

implements

the

partition

functions.

simpler

mean

technique

abel ian

of

the

the

a straightforward

this

the

of

the

PQED.

variational

whether of

lattice

one

a variational of

First,

of

which

In the

scheme.

Hartree-Fock

result

computation

partition

in

arguing

computing

discussion.

the

about

mechanical

such

physics

way

a general

addresses

results

from

that

allow

of

this

obtained

the

specifically

evaluation

compare

of

variational the

the

The main

statistical

and

of

discuss

scheme,

of

invariant

a discussion

notation,

within

reduction

gauge

the

most

in

earlier

be self-contained; plaquette of

previous

the work

will

be

features will

not

be required fifth

in

section

gauge

presented

and

in

are

abelian

detail

is

summarize which

results

obtained

can

in

for

by

spatial

Hami 1 tonian,

H, and

gauge

in

remains

between

date

and

of

the

discuss

theory

mean-field

an SUCP)-gauge

discussion

for of

of

the

the

and

section

case

of

of case

The

the

formulation

specific

the

of

the

formalism

for

a non-abelian the

paper

the

directions

we in

Ao=O gauge

fact

invariant

that

those

abelian

gauge

theory

with

every

Such site

generators

‘i’

respect

of

to

has

the the

observabl

es.

Their

degrees

of

the possible,

arbitrary

one

with

Ao=O restricts quantization

which

a theory

commute

al 1 physical

setting

make canonical with

with

a lattice.

with

Hamiltonian

coincide

to

These

by assumption,

to

to

a continuum

associated

the

STATES

gauge-invariant

chosen

lattice.

sufficiently

theory

to

made explicit,

locally are

G(j),

d-dimensional

freedom

the

concluding

transcribing

reflects

method

string-tension

to

discussion.

THE PROBLEM OF GAUGE INVARIANT

quantized

existence

the

formulating

generators,

this

the

be developed.

d+l-dimensions

obtain

canonically

the

of

generalization

is

of

the

theories

gauge

theories

In

rules

the to

forth.

The

of

The correspondence

a computation

idea

bulk

Hamiltonian

then,

approximations

2.

would

The

set

this

the

extension

and

non-abelian

out

theory

the

explained.

and

carrying

understand

theories.

Hartree-Fock

theory

to

explains

non-abelian is

order

time

but

the

independent

transformations.

Since

all

al 1 physical

gauge

transformations

observables,

the

commute Hilbert

-6-

space

with of

both the

the theory

Hamiltonian divides

and into

an infinite

of

condition

number

that

the

representation get by

the

into

understand

theory

all

one

The missing O-S(i),

to

space.

It the

local

notes

the

Maxwell

matter

and

in

that

however,

Hamiltonian.

The

independent

Hamiltonian

can

background

charge

distribution.

this

requires

that

gauge-invariant G(i).

For

sector

of

equations

states, the gauge

the

of

gauge

invariant

weak

coupling

perturbative

all

operator

are

nothing

Thus,

ie.

those theory, states

the

Ao=O gauge

in

the

motion.

field,

the

equation

Hilbert commutes

they

of

and

The presence existence

of

the

annihilated

by

as for

the

theory,

Abelian the

a classical of

sector

states

source

QED in

of

the

free

the of

quantization to

yields

generators

thus

usual

which

of

GCj,>=9.l(h>-~(j.>

attention

for

the

a Heisenberg

but

the

To

of

electric

not

and

measures

case

in

states

the

we restrict

invariant

is

meant

generators it

is

the

Yang-Hills

hold.

The necessity of

p(j),

is

rule.

the

the

do not

what

equations

diagonalized.

a G(i)

Yang-Mills

of

transformations

be simultaneously for

gauge

G(j)‘s

for theory

divergence

that

gauge

eigenvalues

the

for

is

a super-selection

sectors

hold

the

sectors

observable

is

density, not

these

as Heisenberg

the

by

an irreducible

that

physical

equations

does

defined

span

fact

quantizing

charge

fact

The

different

relating

the

sector

invariance the

true,

time

non-zero

one

one

by any

of

sectors,

group.

gauge

meaning

is

any

another

that

of

of

gauge

equation

motion,

with

one

the

Abelian

of

the

statement

but

states

of

mapped

non-communicating

searching states

limit

expansion

for

greatly of is

the

of

complicates

a gauge in

vacuum

theory.

terms

of

-7-

free

the

the In fields,

the

theory task

of

in

the

discussing

g + 0 limit and

sector

eigenstates

the

the usual of

the of

free-field effects

way to to

Hamiltonian due

modify

to

gauge

any

calculate

with

operator

which

general,

an arbitrary

the

family

approximation

to

projects

state,

of to

or

onto

state

will

best

at

its’

alone;

trial

of

the

gauge

varies

vary

over

Hence,

track

There

namely,

as one will

keeping

so as to one theory

is

need

the

part.

In onto

wavefunctions,

need if

a

parameters

some submanifold

one

us

only

projection over

a

enable

by

invariant

a non-vanishing

states.

invariant

scheme

locally

have

hence

cumbersome.

groundstate

projection

invariant

gauge

is

Furthermore, its

gauge

invariant;

states the

it

gauge

variational

invariant

state.

the

attention

invariance

gauge

any

gauge-invariant

not

perturbative

multiply

defining

are

not

one

of

restrict extremizes

the

ratio

).

links

A general in

for

state

terms

of

I*>

=

the

in

the

Hilbert

eigenstates

X*

space

of

the

3.3

A SIMPLE For

variables

allow

us

establish

to

PQED3 and generalizes

where the

is

discuss

the

set

of

18)

I

we have link

be expanded

operators

as follows;

(3.7)

of

with

the

earlier

work of

link

X.

periodic

on the

1 L II this

class

This

subject

The class

Gaussians

wave-functions

analytically.

confinement

theories.

of

variational

computations

physics

exp(iCmg4g)

in has

the

Hamiltonian

in

a way which

of

wavefunctions

link

the

of

permits

readily we will

42.

variables

defined

The

form

r -mg

(3.8)

2

= exp

the

[

T(rng)

-Crn~bW,X’)rn~’

values

parameters. to

of

can

to

be the

generalized

Gaussian

function

of

variables

general

interest

all

with

non-abelian

the

=c

system

field

associated

on a family

out

contact

wavefunction

variational

form

carry

general

T(rnX)

in

to

to

consider

integers

we focus

us

to

this

for

CLASS OF VARIATIONAL WAVEFUNCTIONS

simplicity

which

most

rnz are

) and J?- stands

1

exp(iCm~4~)+(..,m~,..)

the

of

(J,a)

electric

EI where

= (

the

us

in

of

the

I

(3.9)

One form

subsequent

AM!,X'),

function of

this

discussions

wavefunction

-

12 -

can

function is

the

be taken which

single

will site

as be of mean-field

1

A,+(X,X’)

= lja, bb

(3.10)

Y where

we have

(3.10)

can

which

does

taken

be viewed not

represented

its

to

(L,a)

be

extreme the

ck

ck

are

F # m.

Fourier

(IJ,R)

are

limit

to

the

variables

constant, combination

of

-accurately

describe

As noted, of

gauge

invariant with

the

the

of

the

projection

(3.11)

the

the

links

(3.10)

general,

it

is

X and X’ are

corresponds natural

to

to

taken

the

assume

that

, whereas

extreme

sorts

that

the

strong

of

a good behavior

between

EXPECTATION given

in

easily

Pcj = l-l P,(i)

-

13 -

the

this

weak

coupling

coupling

limit

approximation. is

A

necessary

two

ck +

to

regimes.

VALUES are

project

accomplished

operator

in

these

(3.8)

we must

Hence, is

in

When we apply

find

becomes

interpolation

This

C-Vzlc/2.

PQED we will

form,

states.

(3.12)

operator

of

two

and

“frequencies”

wavefunctions

component.

be generally

l/2 1

case

the

invariant

can

function

-

Eq.

In

COMPUTING GAUGE INVARIANT

3.4

general

Eq.

as

respectively.

mean-field

these

a more

a function

parameters

ck + Rk

the

ie.

of

respectively.

Ck

of

the

Such

(m,R)

1

functions

approximation

and

part

1 exp(ik*(i-n-i))

= constant.

eigenvalues

(i,cr)

transform

2 1 cos(ka)

which

be

diagonal

variational

and

case

ck’s

are

the

for

6aa = V

A aom(X,X’l the

as

vanish

by

where

X and X’ to

not

the

them

wavefunctions

onto

by operating

their

gauge

on them

where

P,(L)

=

If

we define

by

(3.8)

through

the

[

&g link

One is This

is

k

the

class

Jda(i)

]

n [ sd4n

]

the

difference

between

the

a-parameters open

carry

out

the

#-integrations

2,

and

the

function

not

parameters,

L(a(j.1)

then

the

try

for

value

a classical

a-parameters

could to

expectation

of

useful

there

defined

..1 (3.14)

explicitly

It

wavefunctions

~(..,4~,..)*~(..,4~+~~

options

to

of

we have

two

by are

us

well

define

which

to

pursue

HP,

whose

define

to

in

the

point

in

be an advantage,

for

certain

action

th

ends

of

s stage. mg sums.

of of

gauge

two

do the

terms

degrees

this

an effective

us at and

of

system

on the

the

freedom

are

transformations.

this

paper

but

ranges

in

other

of

as

= log

to

evaluate For

approximation.

the

now it leaving

of

the

discrete

sums over

Carrying

out

the

integral,

will

4 and a-integrations,

(3.111

for

are

given

and

(3.11)

then

(3.13)

There

partition

cases

I

X = (i,al.

gives,

is

exp(ia(i)*G(j)I

2 = ,

z=n

where

!- .fda(i) 2n

(3.151,

be more us with

integer

by stationary

convenient

the

to

problem

of

phase

carry

out

evaluating

both

the

a set

variables’mx.

4 integrations

for

the

wavefunctions

defined

by

we obtain

z=n

!- .f da(i) 2a

I [ 1

exP[iC(m~+Vza)-1

-

14 -

mgA(X,X')mg'l

I

(3.16)

By explicitly the

carrying

configurations

(3.16);

ie.,

z = where

I[

r(rn%)

out

of

that

rng for

which

becomes

T(rnz)

n 6(9.1~(i)J

we only the

in

In

the

be removed

by

introducing

case

make

one

obtains

non-vanishing

constraints

contributions

on to

I

(3.17)

Eq (3.9)

include

lattice

a-integrations

which

(3.16)

vanishes.

and

(mx)

was defined

before,

the

.

The notation

configurations

Strum)

means,

the

variables

variables

theory

this

of

as

divergence,

of

the

P+l-dimensional

a set

of

integer

valued

condition

plaquette

can

variables

La

defining

where

the

plaquette

curl

of

the

L-variables

w

q

Ln(j.1

- Ln(i-67)

for

X = (F,2)

w

q

Ln(j.1

-

for

X = (j.,l>

is

defined

to

be

and

The by

sum over the

the

Ln(j.-iit) LX variables

being

(3.19)

unconstrained,5

we find

expression

z = 1 rctynxox3 and

Z is given

the

expectation

value

z

(3.20)

is

- - ’ g2

given

c

by

I

1 + -

(volume) g2

(3.21)

where



=

qnxp2 r r (ynxg3

I

and -

15 -

(3.22)

where

(4.2)

G(1) G(2) G (31

= Es’+Eg@ = E;Q’-E~’ q -(Exz+E~~)

G (41

= ExJ-Ez’+.

+ mxf3eX3

and

m~stJg~)l (4.3)

+

negative

plaquette

integers.

system

are

and

In

terms

of

these

transformation

(4.41 the

generators,

is

given

by

the

U(a(L),a(2),a(3),a(4))

It

follows

U to

from

a state

the

The most I$>

general

accomplished

state

= n

gauge

1

i c a(iI*GQ.) --i=l

commutation

(4.5)

relations

yields

of

the

state

that tU$>,

application

of

where (4.6)

this

the

Hilbert

space

is

of

the

form (4.7)

I..,mg,..> state

by operating

P,[I)P,(a)P,(3)P,(q),

dependent

X = (i,R>.

= 1 f(..ms,..) of

position

= 9~ ..,e~+a(i+g,l-a(i),..)

assumed

The projection

PslN

= expC

+(O~l,G~2,0~3,G~~)

we have

general

operator

canonical

(U-$1( ..,e2,..) where

most

onto upon

it

its

gauge

with

the

invariant projection

part

is

operator

Ps =

hence

!- lda(i) 2s

I [ 1

f(‘..,mz,..

-

19 -

1 expCimg(Og+V~a)l

I

(4.8)

The partition function

function

of

‘f’.

is

defined

to

norm

be the

Taking

the

of

Jda(i)

] 1 [ f (.,mf,

norm

(4.81,

of

Pgl99

recalling

and

that

is

Psz

a = P,

we

obtain Z(f)

E = n [ !-

.)*f

.I expCiza(j)T-m(j)1

(.,mg,

]

2n (4.91 where

the

lattice

V-lJ(ll

divergence

of

the

m’s

is

= m~l+rn~* = rnx2-rnxl = -(m~2+rn~3)

V-m(Z) V-m(J) and

(4.10) Carrying

out

mxl which

=

the

mX2

allows Z(f)

=

us

ati1

integrations,

-mg3

=

to

and

written

for

f(L)

follows

mean-field

from

relations (4.11)

Ln

Z(f)

as

‘f’.

(4.12)

groundstate This

plaquette

system

variables

0~. it

quadratic approximation

in

that

form

and

of

the

fact

is

admits Since

the in the

Note,

we have

used

the

relations

(4.11)

f(L,L,-L,-L).

approximation

projection.

defined

the

(4.12)

function

the

to

= 1 f*f(L)

an arbitrary

to

q

rewrite

for

It

-mX4

leads

just

variables

partition

is

more

no difference

general

plaquette another

one

system

wr

it

follows

function

is

-

20 -

given

gauge that

combination approximation,

corresponds

to that by

the

approximation

after

saying

invariant

Hartree-Fock

section,

between

Hartree-Fock

way of

gauge

general

preceding the

the

single

only

the

there

choosing in

this

the

single of

the

as we a general

Z(Y)

[

=Cexp

Proceeding

in

Hamiltonian

in

4LZ

I

-Y

the

(4.13)

same way we obtain

an arbitrary



=

1 [ 2g2Lz

exp[

for

Hartree-Fock

-F]

the

expectation

value

of

the

state

-iexp[

-

4(L:t)‘]

exp[

-t]]

+>

(4.14) Given

(4.13)

and

minimizing

the

(4.14)

we can

determine

the

variational

parameter

Y by

ratio

(31 HPslW =

C(Y)

($1 Pgl-#> In to

order

to

a given

In

2 the

the

of case

operator

of

P SO”PCeS(l~2) Applying

this

factor,

one

z sources

to

one

‘= 1 f”f

into

sources,

a positive

interest

operator

assumes

independent

of

wavefunction

static

a sector

one multiplies

charge

at

1 and

corresponding by

a negative

a different charge

is

= p,+ ClJPs-

ZSOuPCeSr z SP”pceS

If

a trial

distribution

operator. at

project

the

obtains

(p)P,(3)P,(q) state the

(4.15)

I$>

and computing

general

the

normalization

result

(L+l,L,-L,-L)

that

the

Gaussians

in

= ev

[ - k

(4.16)

function the

‘f’

is

variables

]

C expE

a product

rnx (4.16)

-

(4L::)z

over

links

of

becomes

]

(4.17) If,

on the

wavefunction

other with

hand, A(X,X’)

one

assumes = bco

the

ACj.-j),

-

21 -

most

general

one

obtains

Hartree-Fock

(GAr,+A, 1

1

Z 5ources = exp

-

8

]

[ (-460+2A, )14:~1)1

1 exp

] (4.18)

where

we have

defined

Ato)

factors

differ,

normalization the

for

Since

both

the

ratios important

so

and

for

case

from to

it

the

completing follow

the

that

parameters

sources

are

square

this

is

the

same

argument

in

the

a general for

it

the

of

points

L is

shifted

exponent. result,

are

plays

mean-field

is

reveals

for

the

1 and 2; by

the

and

exactly

the

no role

(4.17)

We will as

with

Y-’

wavefunctions.

function the

overall

parameter

Hat-tree-Fock

partition at

While

functions,

L-independent

sources

the

the

Examination

the

with

identifies

general

is

between

= A(I?~) = A,.

partition

be ignored.

problem

with

and

factor

can

one

the

mean-field

difference

problem the

the

normalization

and

if

-4A0+201 then

combination

same

= A0 and ACti-

fact

general

in

only

source

one free

namely,

114.

see

in

This the

that

the

that comes

sections shift

Hartree-Fock

wavefunctions.

4.1.2

Evaluating

Evaluating situation

Z for

(4.13) in

and

which

straightforward

there

for

Examination

of

essentially

given

(4.141, are

(4.14) by

that

first

exp[

and

the at

so we will

reveals the

Strong

sources

g > > 1,

1 [ 2g2L2 &:(Y)

Weak and

for

term

in

Coupling analogous

formulae

1 and 2, begin large

the

is

with g2

the

for

the

particularly the

this

energy

case. is

ratio

-:]I

= c exp

[

4L2 - Y

I

(4.19)

-

22 -

and

so we would

the

parameter

can

be well

expect

that

Y must

in

order

be chosen

approximated

by

to

its

to

minimize

be quite

first

the

small.

two

energy

for

large

this

case

(4.13)

In

terms,

g

.ie

4

=

Z(Y)

[ I

1 + 2exp

-

-

+....

(4.20)

Y

and E(Y)

can,

to

leading

4g2exp[

order

- z]

in

exp(-l/r),

- >exp[

be written

1

- i] +-

=

&:(Yl

as

1 + 2exp

4 - -

g2

[ I

(4.21)

Y

We can

check

minimizing which

our E(r)

to

is

E;(Y)

in

the

(4.211

small.

size and

Taking

of

Y for

verifying the

large that

derivative

g by the

of

value (4.21)

Y

with

[ I

= 4gb (4.22)

Y

for

of

Y we obtain

2 -

exp

about

as defined

extremizes

respect

assumption

g > > 1. Given

sources with factor

the

relationship

we can static

sources

we have,

Z sources

carry

to

= ew[

between out

at

the

- !-J

g for

same exercise

1 and 2.

leading

Y and

order

for

Forgetting in

+ ew[

the

exp(-l/r),

-

t]

and

-

a situation

23 -

+

-..

the

case

overall

with of

the

no static situation

normalization

cl2

1 - 4Y

[ I

-exp

2 & sources

+ Fexp[

- t

]

- Sexp[

- t

]

= Z sources

(4.23) Minimizing

exp

which see

&(y)sources 1 Y

that

the

case free

happens

different

in play

of

from

the

and

for

one

the

case

parameter

we will

limit

expectation

Y yields

case

The

a single

of

reason

plaquette

either

mentioned,

must the is

we

we

parameters be the

values that

case, when

variational

sources

information

ourselves

values

the

with

In

case.

of finite

same,

for unlike

Y come out volume

role.

sorts

of

the

[

4L2 - Y

1 Lexp

lattice

plaquette. of

free

As already

log(g).

the

a significant all

source

an infinite

case

Obviously, but,

the

(4.24)

as g + [oI Y + 0 as the

effects

to

* 2g4

somewhat

source

what

respect

[ 1

differs

discuss

with

to

can

a discussion

operators

Ez.

be extracted of

In the

the

at

this

point;

groundstate

source

free

case

we have

I

= Z

and which

=

vanishes. -1

= -

However, =

(4.251

= - in

= -

the

case

of

= -

and

- 24 -

sources

we have

xLexp[

- 5

]exp[

-

(4:/112

1

= (4.261

Z sources from

which

it

follows all

corrections, sources.

that

the

flux

This

is,

of

We now turn

to

the

values

of

expectation the

limit

is

to

the

g + 0.

introduce

weak

coupling

presence

of

We see

from

tend

to

very

1 i ttle.

small

l/Y,

sum.

There

which

is

use

of

since

in

is

a way9

Substituting

that

shortest

plaquette

fields

in

and

see

to over

the

the

discussion for or

case

one

however,

to

in

the

confinement.

how the

appears

recast

keep (4.21)

of

small

sources in

to

in detail

evaluate

field

in

the

configuration.

as g + 0 ,

(4.17)

of

case

two

the

calculation

general

g exciting

to

&(Y) and

presence

small

limit

of

Coulomb that

has

version Z,

explicitly

to

the

this

more

small

joining

the

out

exponentially

line

of in

carrying

(4.13)

up to

evaluation

needed

because

evaluated

Poisson

the

for

preceding is

the

one

electric

goes

Evaluating

1 f(M) M

of

reason

sources

This

easily

the

the

coupling,

down the

study

tricks

the

zero.

goes

results,

weak

strong

course,

The the

at

high to

a great and l/r.

l/r

must

m-values

be difficult many (4.2)

terms into

To do this

also costs for in

each

a form we make

identity

= 1 .fd# N

f(Sa)

(4.27)

into

exp(i28N$) (4.27) (4.13)

and

performing

the

9 integration

we

obtain

Z(Y) =

(llYl"2 1 exp[

- F]

2 At

this

stage

it

(4.28) is

useful

to

define

-

the

25 -

more

general

function

4(L+ql)2 Z(r,fl)

q

1

exp

[ (llYl”2

=

1 exl[

of

Using

Hamiltonian

the

1 + g2

b -Z(y,O) bY

leading

g2Y

N = 0 and

can

be defined

result

coincides

as a continuous respect

One can

In this

get

in

terms

of

out

leads order

additional,

property

to

is

and

have

(4.14) the

.

weak

calculations same

a shift

&sources in

in

functional

the

gotten

by

Minimizing coupling the

form

but

treating

L

(4.31)

with

limit.

presence

continuous

the

limit

calculation small,

n2-4 exp g2 dependence the

in

= & in

the

-Ecr-

of

we would

both

of

sources.

Z(Y,O)

and

[ I

this

non-analytic

what

3 ztr,1/41 - 4Y

exponentially

sources

(4.31)

Y = 2gm2 in

analogous

terms

&

we find

(4.13)

the

(4.30)

by

= ew

order

therefore

that

replaced

To leading

in

I

I

with

retains

Z sources

leading

find

carry

case

Z(Y,~>

also

variable

Y we

to

1 terms

exp(-l/Y)

g2 [

4

1 - - )Ztr,fl Y

1 - exp(

1

& e-+--l-

known

the

as

2

This

]exp(-i2n9H) (4.29)

value

g2r2 = -

This

- y

2

The expectation Z(Y,vl

I

-

Mathieu

g + 0.

we find

term

ll2 [ I

so

variable

that

L and

Keeping

the

next

esources

has

an

that

- 2g2

upon

the

to

(4.32) coupling

constant

g is

If

rewrites

the

problem.

- 26 -

one

a well problem

as

that

of

a particle

are

study

ing

come from Let

the

a periodic

“tight

binding

“tunneling

case

electric

field

to

we see

gets

Z(‘Y,7))

the

region”

we can

to

from

when

as shown

7)=1/4

1 4

follow

from

-

8 b7) other

coupling

values

limit,

ie.



that

in

external

the



we can

vanishes

strong

charges

weak-coupling This

of

where

exp(-4q2/Y),

the

g + 0 we

non-analytic

electric

effects

expectation

sources

are

(4.26)

In the

fields.

the

in

ra

all

limit

value

present Using

.

of

the

the

the

expression

write

= - - -logz(Y,q)l and

these

the

energy.

that

However,

shifted

and

of

(4.25)

in

then

the

evaluation

vanishes.

L-distribution

potential,

corrections”

us now turn

source-free

for

in

use and

coupling

chooses limit,

to

to



1 ll = - - + -

4

exp

g2

(4.24)

+ 1.

limit

the

the

use

strong

directly,

This string

shortest

the

In the

.

L-expansion



have

one must

leads

the

(4.33)

Z +

is

the

expected

of

flux

joining

possible

N-expansion

in

the In

length.

given

result,

(4.28)

the

for

four

Coulomb the ,

four for

small links

g, is

ll2 I I - -

a2

that in

link

universe,

which

(4.34)

we see

problem

the

which

the

four

that one link

we mean

the

expectation

would

expect

universe. that

value to

obtain

By the

configuration

of

the

of

fields

by solving

Coulomb

the

problem

classical

on

in

fields

energy

Liz

ccoulomb = 2

the .

= 1 - Hence,

two

~ + ~ + ~ + ~

- 27 -

I

(4.35)

is

minimized

subject

to

the



+ - +

= 1 = -1 = 0



-

= 0

conditions

and

Eq.

(4.36)



is

(4.36)

satisfied

if

Il

= Il

Il

= 11-xl

we let ll

=

= x

and

This

allows

us

three

the

route

and

the

between

From

this

essentially charges,

that

con fined

to

and These

are

exhibited sketch

by the

along

whereas

terms the

the

x yields

xmin

straight the

line

flux

flows

the the

to

zero

which

are

this

the

up to

field

field

coupling

line

configuration,

of

the

strong

straight

P+l-dimensional

treatment

to

of

at

The effects

weaken the

respect

quarter

tends

strengthen

slightly

as

= l/4;

hence,

joining along

the the

longer

1 and 2.

coupling

Cou lomb

slightly

one

vertices

expC-(constant)/g21. to

flows

remaining

as the

energy

3

with

flux

we see

appropriate

+ 3x2

(4.38)

the

the

Coulomb

(4.38) of

of

the

C(l-x12

minimum

quarters

charges

rewrite

g2 = 2

&coulomb

Taking

to

(4.37)

of along

which

the

the

flux

spreads

theory

iine

in

- 28 -

the

to the

flux two

is

external

out

into

the

order

joining

out for

at

of

exponentially

spreads

responsible

problem

joining

terms

these

the

the linear

weak

coupling,

next

section.

small the

terms

two

other

charges

links.

confinement and

we will

is

4.2

PARTITION

4.2.1

The String

To treat one

FUNCTIONS AND STRING

in

more

complicated.

a1.3

using

could

not

can

be directly

treatment

can

interested

concentrate

to

be drawn

from

to

for

projection

operator

et.

al.

refer

to

their

establish

the

in

first

of

to

of

bring

it

to

so,that for for are

PQED.

this

establish of

the to

how the

the

problem enable

same

out

et.

by Drell

PQED but this

which

section at

we

which

it

of

the

the

case

of

QCD we are

couplings, enough the

to

case

the

we will

to

reason

the

establish

detailed of

connection

can

transition

methods

used

reader

be used

for

PQED in

between

interested

technique

a deconfining

are

rest

and the

computations

the

strong

For

3+1-dimensions,

a stage

all

be restricted

detail

existence

In

Since,

however,

see

carried

theories.

methods

quickly

sufficient

our by to

to

the

theory. aim

an equivalent

unconstrained

our

all

for

3,

there.

formulation

work

-3+1-dimensional

and

joining

The general charge

will

however,

confinement

case

We will,

Drell

string

the

follow

2+1-dimensions.

reference

that

LATTICES

2+1 and

was suitable

PQED and

establishing

on showing

discussion

which

of

both

was already

non-abelian

compared

in

phenomenon

into

calculation

to

in

same way;

approach

FOR INFINITE

Field

lattice, the

This

our

calculation

Our

infinite

be generalized

develop

Radiation

essentially

a different

will

this

Its

PQED on an

proceeds

most

and

TENSION

distributions

the

is

to

recast

form

in

external

the

the

sums

in

the

over

presence link

representing

appears

which is

which

problem

variables

charges

problem

the

of

sources

variables the

are

appearance

of

explicitly.

confronts

us

in

evaluation

of

the

- 29 -

the

presence

partition

of

arbitrary

function

the

where

the

valued

charge

distribution

function.

situation,

The special

and

the

or ~-9 describes external

source

to

free

can

case

us consider case

the constraint

everywhere

except

that

if

constraints

then

mz

= Estrins

where

along

of

chosen),

~1

lying

chosen for every

to to

every

the

lie

configuration

two

sources

can

we now wish

constraints

the

to For

~1 and

one

that

~1 and ~2 where

it

m ’s

be rewritten

of

in

the

do is

show

In this

~2. zero

+l

and

-1

in

the

sum we

satisfying

these

as

(4.40)

zero

for is

way

in

the

all

either which

links +l the

we choose

~1 and ~2

the

X lying of

~2

then

-1

line

configuration

of

except or

the

left

that let

(X1 +m’x

it

# ~1

constrained

1.n~ is is

free

simplicity

at

appearing of

al 1 i

constraint

sources.

is

source

pair

from

What

at

the

charged

the

with

is

integer

= 0 for

passing

m-configurations

always

i

eliminate

one

a configuration

along

link

situation

we have

where If

the

the

the

= 0.

could

points

all

= 1 and p(i)

in

(4.39)

be an arbitrary

an oppositely

To eliminate

it

to

= 0 for

observed

one

the

which

and

o-m'(i)

of

at

is

~2 and

El,

with

on the

Estrins(X)

and ~9~

p(j.1

2+1-dimensions.

case

the

observe

that

in

taken

P(JDD) = -P(E~)

same for

respectively.

case

We already

(3.201

do the

is

.

I

fl stY+m>(i>-P(i)

p(i)

situation

sources.

sum (3.17)

one

case

the

I[

rnglA(X, ,Xt)rnX’

-1

= 1 exp

Z gen

the

(depending

upon

of

links

be two

line

joining

between

~1 and ~2, satisfying

- 30 -

case and the

joining

satisfies

to

In this

‘s

a line

rn’z

x-axis.

rn’x

along

joining the them

the

points

two (X1

is

otherwise.

source

is

l-axis

points

Estring zero

location

condition

on the

these

~1

is

conditions

with can

plus

one Since

can be

be

written

as a given

configuration

it

configuration,

follows

Estring

from

our

plus

a source

discussion

for

free

the

source

free

case

that = 1 exp C-UJxg+E -s t ring)%

Zsources(El*QZ)

A(~,~‘)(~xL+lstring)X’3

(4.411 where

now,

the

Equation in

ref.

sum over

(4.39)

3,

that

gradient

of

function

f(n);

can the

the

which

appeared of

the

the

energy

the

question

proven

as

the

lattice

of

a plaquette

(4.42)

into is

(4.411,

no cross

using

term

the

fact

between

of

case

generalization of

the

function

of

field

the

is

the

state.

linear

one

which source

of

the

that

for

gradient

of

a

and

on the

Coulomb

The presence

of

does

fractional

Z’ is

scalar term

this

not

the

expression

plays

any

the

function

in

term

have

(4.431 shifts

universe.

depends the

(ynx(~+~IIgtl

the

plaquette

of

confinement

pair

static

of

the

the

= p(i). charges

radiation of

is

#string

A’#string

field

the

This

problem

the

curl

is

no role

in

for

us

interest

time.

The

of

the

as

(L&xgl~

volume

partition

for

this

be written

plus

unconstrained.

we observe,

= zt c expC-(~~x(L+bj~~A(X,X’)

in

alone.

at

is

we obtain

infinite

#string

+

there

a curl,

is

if

UP) can

+ s t rins(iI

expression

= ba,A(i-j>

which

L(i,a)

simplified

Estring

function

Zsources(PlrlZ2)

part

be further

= -V&(i)

this

and

variables

ie.

Substituting

scalar

integer

function

a scalar

Lstring(X)

A(X,X’)

the

field pair

of

unique Since

the

solution

plaquette

in

is

Vf+string function

configuration charges

to

which order

-

to

31 -

the the

lattice Coulomb

en is must

focus

it

Coulomb

the

field vector

of

the

potential

be added

to

the

into

a string

Coulomb joining

them.

For

Estrins

a given

is

Estrins

introduced

on a sum of

integers,

location

the

of

function

the

convention

the

sources

string

we can

Such

plaquette

is

of

it

the

‘string’ of

function

along

to

around

the

at

stay string

take

the

the

straight

the

problem

remove

lattice,

Since

constraints

will

so

long

fixed

at

~1 and ~2

et

in

the

we will

al,

we will

the use

C-i,$>.

assume

path)

the

changing

range

also

(shortest

as

to

Drell

values

link

defined.

corresponds

Following

en will of

uniquely

artifice

integers.

on an axis

drawn

up is

shift

a shifting ew by

the lie

function

as a mathematical

endpoints

respectively.

the

that

joining

the

If the two

charges. We can in

the

the

presence

energy

vacuum two

now restate

of

state

static

know

of

above. in

PQED, and theory.

the

arbitrary system

the

terms depend

is

a general is

given

obtained

of

no straightforward

the is

simple not

cx

by choosing

the

which

allow and

form

of

so simple

generalization

present.

-

32 -

minimizing

Z(e).

= 0,

us

Coulomb

the of

of

while

The the

case

of

en-configuration

Gauss’ for

that

function

by setting

AtZ!,X’l

parameters

as

partition

an en-configuration

upon

the

distributions

The manipulations

The situation of

determining

charge for

theory

charges

discussed problem

of

of

the

to

rewrite

field

Law for non-abelian discussion

are the

the unique

to

abelian case we will

and

we now

THEORETIC TECHNIQUES

FIELD

4.3 For

simplicity

we restrict

2+1-dimensional rewritten

theory,

for

which

to

the

the

case

partition

of

the

function

can

be

as

Z(E)

Using

= 1 exp

the

where

V stands

Poisson

for

the

volume

we can

by continuous can

(3.11)

and

fields

Z(E)

[

of

replace 9(n).

be written

the

(4.44)

notation

of

(3.12)

we have

I

ik*(i-j)/c.k

identity,

function

of

= -

I

AW!,X’l(~x(L+~ll~’

-(qxQ+f))~

representation

AM,X’)

L,

discussion

(4.45)

the

lattice.

the

integer

In

terms

Now,

of

by

valued these

invoking

plaquette

fields

the variables,

the

partition

as

I

-~(~+~)~A~(n,x’)(~~)n’

1 + 2~cos(2nN~9$)l

(4.46) where

we have

A,

(X,X’)=

This

is

the

3 .

Since

ranges

of

integrated

by parts

and

defined

the

new plaquette

function

V~xA(X,X’)XV%’ representation the

which

variables

+a are

integration

Z(E) =

d&

and

[

absorb

coincides

with

continuous these

equation

fields factors

we can to

rewrite

I [

1 + 2~cos(2aN~~9p-~~))l

exp

(4.32) shift

of

ref.

the

Z as

-~~~A~(~,~‘)~x’

I (4.47)

The

Hamiltonian

can

now be rewritten

I: 1 + 2x cos(2nNx(+-~)x1

where

-

33 -

as

I

‘#*H’g (4.48)

‘PE and

exp(

where

-fl9~A,(n,n’19~’

the

operator

1

H’ is

(4.49)

defined

g2 H’ = Ecoulotnb

1 1 -

c (vx+)2+-

+ 2

as

tCexp(b/bdn)

I

+ exp(-b/b&)1

g2 4

(4.50) It

shou Id

of

notation

not

stand

order

in

the

In

notation

the

weak

series

the

its

not

in with

carry

one

keeping for

H’;

this

of

only

expand the

since

of

next

ref

of

sections magnetic

the

cosh(b/b+n) two

they

do

done

this

3 and

the

freedom.

the

first

an abuse

We have

degrees

the

saturates can

3.

notation

many

to

commiting point,

section the

over

one

that

rather

9 and 9 at

a system

limit

means

H’ = &coulomb

we are

introduced

coupling

form

that

symbols

of

argument

an approximate

point

connection

will

which

in

the

mechanics

of

Hamiltonian

th is

objects

establish

quantum

abuse

at

using

for

to

normal

be noted

of term

in

This the in

paper. the

as a power

terms.

This

leads

to

ie

+ H1

where 29 HI

Hl

is

= gtZtoas,z 2

the

natural coupled

as

in

-

of

a free

Hamiltonian to

of

- -l_c 2g2

take

‘P as

harmonic

(4.49)

A,(n,x’)

the

(4.51)

b&2 massless

wavefunction

oscillators.

scalar of

This

the

means

field, ground that

V stands

labeling

the

Ok = ftk

state $ can

hence, of

this

it

is system

be represented

with

= -

1 exp(ik.(i-j))ok

I

V where

#;

for

the

volume the

of

the

lattice,

plaquettes

and

centers

of

= ( 4 -

2 C cos(ka)

and j

are

integers~

(4.52)

1”’ -

i

34 -

Comparing weak

this

result

coupling

limit

The reader have

the

one

This

g.

show

is

plaquette

that

d+

4s

all

the

allowed

(4.281,

N=l

In

they

b,,,q,,, we see

preceding

(4.461

of are

when

that

well

oscillator of

the

effect

was very

term

in

because

that

in

discussion

the

the

to

we explicitly

case

the

we

terms

negligible

leading

carried

out

corresponding

approximated

by

techniques,

it

contributes

a term

~*(~)cos(~~(~~-Ex))~(#)

I

for

contributions

saw this

harmonic

any

form

that

calculations.

Eq.

Using

general

realized

We already

N-expansion term.

have neglected

cos(Pi~Nn5$). in

the

ck + R/g2.

will

implicitly

order

with

is

the

N=O terms

straightforward

= Zocos(P~ren)

of

the

to

form

.* - -

1 Wk

c 1 vg2

exp [

(4.53) which

is

exponentially

expression

one

small

can

write

compared

the

to

the

ground-state

N=O term,

energy

1 E(E)

The

= ecouI

last

term

in

kinetic

and

effect

on the

energy the

Starting

from

-&

& 5ources

the

the

x-axis.

this

problem The

CJk +

density

existence the

two

the

but of

equivalent

vacuum

of

-

4) bE2

represents of

= ~coul

static

the

N=l

Hamiltonian. it

the

plays string

of

an

(4.54)

correction It

has

important

the

role

in

are11

et

al

show

that

(4.55) separated

v2,

-

both

tension.

(4.54)

sources

to

a negligible

lT2-4 + --v2D 4a3

"mass"-parameter,

this

as

8a2g2Z(c)

terms

Using

b2Z(E) chT2

equation

potential

establishing

for

+ -'1 2

Zo.

35 -

is

given

by a distance by

D along

w2 = 4n2exp

It

[

by

existence whose

of

v2

persists the

effect

the

field

factors

turns

the

the

is

The

zero

order

for

that

in

the

dissipating

and hence,

electric

4-field

flux

theory

ie.

of

the

thus

it

joining

only

4-field

at

the

is tension

the

even

the

the

string

The screening

from

in

represents

a non-vanishing

limit.

of

u2

induced

confinement;

exp(-const/g2);

a string

factor

effect

be crucial

coupling

to

there

to

e-parameters

persists

ok + Wk + u2/g2

screening

from

weak

to

cos(2nN4).

out

different

of

coupling,

(4.56)

a non-perturbative

is

in

I

as a correction

the

existence

because

c “k

g2v

may be understood

#-propagator

1 -

n2 - -

keeps

focusing

of

weak

two

static

charges. At adopt

this

point

the

general

persisted

in

coup1 ing.

using

then

occurs

g,

phase

transition

function

Znlf

in

to

which

the

ok of

the in

for

values

the

string the

mean-f

y Rk2

tension

of

ck

-

if

but

but

region

of

value

for

also

there

vanished.

The

the would

reason

leading

to

w2 = 0.

The

the

existence

of

36 -

in

weak

and CJk y Qk2.

results

ion

we didn’t

rather

(4.56)

1

(vX(L+eI)2/r

happen

= constant

a poor

approximation.

approximat

the

g < 1,

implies

mean-field ield

case

tension in

exponent

into

obtained

string

would

wavefunction

down

the

only

what

our

form

not

the

ask

of

choosing

one

inserting

of

form

density

divergence

disappearance

to

mean-field

that

below

that

infra-red

interesting

amounts

energy

of is

the

find

state

a value

is

Hartree-Fock

This

We would ground

it

this

a logarithmic

Indeed,

an apparent

the

partition

is

(4.57)

be

and

this

is

just

dimensions.

This

transition

at

Although

string

ask

in in

tension

is

also

that

parameters

be arbitrary,

we see a A(if!,X’)

we obtain

a sequence

interpolate

between

approximation. also

(or

in

to should

function

A(X,X’)

increases.

job

should the

in

computing

be able

to

carry

in

the

is

to

deal

able

problems.

case with

However,

string

if

the

general

as we will

phase

Hat-tree-Fock

-

the

37 -

the

full

ck)

of

range the

links

location

of

the in

doing

then

one

expansion function.

uninteresting without

section,

to

theories

interested

a cluster

case next

the

1 >> g2 >> got,

is

two

Hartree-Fock

finite

partition

real

which

range

the

these

number

g2 = 0 as the

for

the

approximation

evaluate

only

basis

parameters

but

is

is

string

we allow

these

an approach

in

the

transition,

to

see

as the

between

and

one

it

transition

a finite

by performing

PQED, such

it

mean-field

to

phase

approximation of

the

tension

out

phase

the

for

towards

Hence,

this

spatial

approximation,

calculation

calculation

a K-T

use

difference

1, we expect

move

the

Kosterlitz-Thouless

course,

give

two

mean-field

we compare

K-T

functions

Fig

the

Hartree-Fock

we generalize

mean-field

transition

If

be non-vanishing

As shown

in

P+l-dimensions,

can

alternatively

partition

the

one

the the

A(X,if?‘)

of

if

and

below

that

PQED in

problem.

Hartree-Fock

this

good

the

analysis

and

Since

if

model

a Kosterlitz-Thouless

for

the

to

be expected

this

is,

region

that

X-Y

r.

it

small.

the

exhibit

mean-field

the

in

of

answer

of

the

to

from

wrong

treatment

is

of

how wrong

that

calculations

will

the

we see

by allowing

known

conclude

to

variational

is

one must

as computed

1,

function

value

a systematic

Fig

of

model

gives

interesting

tension

partition

a finite

approximation

for

the

for

about Of

since any the

a

one

serious case

of

the

non-abelian

gauge

approximation outlined

might

theory prove

may prove

to

quite

be the

5.

5.1

directly

with

difficult,

only

and

feasible

NON-ABELIAN

the

the

Hartree-Fock

procedure

just

one.

GAUGE THEORIES

A REVIEW OF THE GENERAL FORMALISM Extension

of

non-abelian

value

any

of

of

as easy

the as

theories

the

it

is for

in

treatment

of

gauge

we will

for

going

in

the

the

treatment

pointing

out

do in

and

general

Our

the

the

abelian

this

from

of

the

formulation

clear and

of

finite,

treatment

will

non-abelian

differences.

38 -

this

to

not

time,

the

give

string

is

theory,

and

outline

the

variational

between

the

of

present

the the

SUC2)

considerations.

projection

operator

non-abelian as opposed be brief,

by emphasizing

-

at

is

section

versions

simple

the

sources

differences

very

to

expectation

relating

Hartree-Fock significant

the

not,

an SU(2)-gauge

emphasizes

notions of

of

a discussion

a way which

intuitive

we will

case

chapters

static

We will

What

show,

transformations.

keep

theory.

and

without

theories

PQED emerge on to

and

gauge

the

preceding Unfortunately,

SU(3)

mean-field

and

discuss

theories

or

the

functions

with

abelian

As we will

theory

Before

partition

etc.

the

calculation.

the

in

straightforward.

energy

SU(2)

formalism

presented is

resulting

spectrum,

general

gauge

formalism

ground-state

results

tension,

the

gauge

evaluation

the

dealing

the theories,

lattice to but

formalism gauge

infinitessimal, we will

parallels as well

try between as

to

Reformulating

5.1.1 The space

Hilbert of

space

periodic

functions

in

abelian is

is

the

Under

this

link

of

into

the

generated

by

has this

gauge

the

a gauge

by

G in

the

each

link

of

G, is

the

U(1)

with

the

Lie

group?

G into to

the

we have next

step

we see

transformation

that

the

defined is

each

to

upon

the

effect

and

Hilbert

space

of

full

these of

in

based

of

measure

on the

Hilbert

space

any of

the function. the

arbitrary

by a gauge

function

-

In

ie.

39 -

is

then

order

to

Lie

groupI

the

lattice

square

way

U(1)

the

an arbitrary

where

numbers,

from

valued

lattice.

upon

each

theory

complex

non-qbelian

space

for

spaces.

definitions;

the

exp(iG).

where

the

The U(1)

functions

the

space

the

numbers.

the

of

of

group

complex

set

be the

j..

number

of

link

establish

acts

defined

links

vertex

complex

the

preceding

the

each

the

space

theory

Haar

of

the

U(l)xU(l)x.....xU(l),

complex

usual

of

as the

to

transformation (4.6)

it

notion

group

Now that

think

over

of

case

in

for

G with

for

upon

respect

think

group

Hilbert

product

set

of

factor

based

and

the

which

as being

The

This

8%.

multiplication

we can theory

to

modulus

a U(l)

associated

gauge

variable

wBs defined

unit

product

space

theory,

the

Language

variables

of

direct

G, we replace

with

usual

Abstract

theory

group,

numbers

numbers.

the

link

a U(1)

the

gauge

taking

from

the

just

in

gauge

the

of

complex

we can

functions

.from

by a gauge

lattice

Alternatively,

extend

upon

identify

complex

product

abelian

identification,

the

Theory

of

of is

we can

Hence,

the

a product

set

multiplication

of

Abelian

functions

acted

theory

group

the

Hilbert

gauge

integrable

theory functions

integration

is

done

group. for

the

in

which

arbitrary an arbitrary

Referring

back

position a(i)

gauge

is

to

(4.5)

dependent to

take

$(.,0x,.)

to

~(.,e~+a(i+~R)-a(i),.),

conventions

just

+(.,0x,.)

established

as $‘(.,ux,.),

follows

that

where

shifting

corresponds

to

the

we can

the

which

rewrite

ua: is

rewrite

the

U(1)

element

element

itself for

gauge

the

function It

exp(iQ).

then

transformation

ux by a phase

a product

formula

Following

periodic

8% by a gauge

group

is

the

X = Ci,R>.

the

argument

multiplying

expCi(a(j+h~)-a(i)11 Hence,

we can

for

of

group

factor,

elements.

transforming

a wavefunction

as $‘(.,ugp,.) where the is

g(a(iJ) case

stands

of

the

for

but

not

for

the

to

drop

g(a(i))

the

define

a label

the

group

entirely.

g(i),

for where

under

a gauge

order

case

to

continue

given

of

vertex

Rg stands

which

we write

the

order

is

to

think

no loss to

i.

In

for

the

transformation

the

group

and

one if

tends we

by specifying

the

a

an arbitary

element by

it

gauge

generality

case*

group

specified

the

a(i),

be given this

factors

gauge

of

of

for

matters.

a non-abelian

by a function

transformation each

in

the

Obviously,

exp(ia(i11.

theory

There

gauge

$(.,Rz,.),

tranforms

to

as being

arbitrary

element,

function

the

convenient

transformation

element

non-abelian

(5.11

particularly

group

theory

When we generalize is

the

abelian

irrelevant,

(5.1)

‘(a(i))u~g(a(i+~a)),.)

+ $(.,g’

of

the

g(i)‘s

link

X,

as

follows; $,(.,Rx,.) Now

the

fact

multiplication forced a gauge

by the

(5.2)

+ 4(.,g-‘(i)R~g(i+~R),.) that of

the

gauge

Rg is

condition

transformation and

by g-’ that

the

right

product

transformation.

-

40 -

associated multiplication of

two

gauge

with is

the

left

by g is

transformations

is

We should theory

of

observe

group

functions

that

the

the

group

SU(2)

representation

of

the

group

representation

of

the

group.

group

groups?

it

a number

times

The

reason

the

group

that

SU(2)

can

act

or

right

left

(L,f)(h) and

the

right

the

for

f to

complex

square

is

the

regular

lies

in

dimension

of

acts

on the

regular

functions

a given

in

function

in

a

the

fact

representation

the

of

role

integrable

numbers

called

irreducible

to

space

of

an important

the

that, of

for

the

representation. representation

two

ways,

f(h)

is

either

we can

by

define

left the

be

= f(g-‘h)

(5.31

translation

(R,fJ(h) Given

equal

ie.

plays

importance

every

on the

of

the

It’s

SU(2)xSU(2)

translation;

translation

to

SU(2)xSU(2),

contains

of

(5.21

The space

representations.

from

compact

formula

of

f(h)

can

be defined

as (5.4)

= f(hg) definitions

(5.3)

and

(5.41

it

is

easy

to

check

that

q Lg Lt = R, R+

L9t R9t and L,

R+

for

arbitrary

that

every

the

basis

(or

GxG for

that

every f(h)

where

the

spin-j transform

q

Rt g,t

A general

E SU(2).

function functions the

(5.5)

L,

over of

the the

general

function

group

can

can

of

harmonic

be expanded

irreducible the

special

be expanded

analysis

uniquely

representations

For

case).

f(g)

result

case

of of

in

is terms

of

SU(2)xSU(2)

SU(2)

this

means

as (5.6)

= 1 aj lmD( j) I,(h) D-functions

representation

are of

nothing SU(21.

but

the

Under

as -

41 -

left

representation translations

matrices the

of

D-matrices

the

and

under

right

translations

(R,D( j) IJ From

these

(h)

= 1 D(j$,(h)D(j),,(g)

equations

representation these

)D( j) pm(h)

= 1 D’ j) lp(g-f

(L,D(j)I,,,)(h)

we see

matrix

(2j+lJ2

that

D(j)Im

functions

(5.7) each

(h)

is

transform

matrix

element

of

over

the

group,

under

the

a function

among

themselves

the and

that

action

of

SU(2)xSU(2~. This

concludes

definition

of

of

states.

for

the

the

the

Let arbitrary arbitrary

over

links

of

to

the

and

mean-field

the

our

gauge

mean-field

the

general

Hilbert

space,

transformations

question

of

act

was defined each

link;

and

the

our

space

our

wavefunctions.

wavefunction for

upon

generalizing

Hartree-Fock

same wavefunction

+,,,f(.rRg,.)

the

of

way arbitrary us now turn

that

For

definition

formulae Recall

to

be a product

ie.

= n \I(Rg) x

specific

case

of

(5.8) the

U(1)

theory

we chose

for

$(8x)

the

gaussian

form -k(e) In

order

group

= 1 exp( -m2/2r> to

generalize

we need

Since

the

to

xve)

are

z exp(imD).

functions

representation

U(1) one

is

We have

the

traces

matrices

(5.9)

the

case

form

in

abelian,

of

all

noted

by a phase of

the of

of

and

already

group the

the

slightly

dimensional,

matrices (ie.

to

this

by multiplication

representation characters

this

rewrite

group

representations

exp(im6)

more its

given

that

U(1)

themselves,

coincide.

-

42 -

general

by

the

acts

In

Lie

notation.

functions upon

other

1x1 matrices

representation

compact

irreducible

are

factor. are

arbitrary

Moreover

words,

and

matrices)

these

so

the

and each

the group then

representation x”(g)

are

of

U(1)

an orthogonal

eigenvalues

of

Adopting function

the

this in

*(RX)

in

rewrite

m; the

group;

characters

and

the

L2 = (-ib/bG)‘,

our

single to

link the

are

m2.

mean-field

non-abelian

wave case.

as

x( m) (RX) (5.10)

of

the

which

wave

case

function

(2 j+l)x(

- -

immediately

generalizes

to

we write

4’3 [ I

= 1 exp

U(l),

the

generalizes

$(RaI

[ I

over

of

immediately

L2 - 27

SU(21,

functions

we can

form

by an integer,

operator

we write

this

of

of

Casimir

J/CR%) = 1 exp

case

set

way which

Obviously,

characterized

notation

To be specific,

the

is

j ) (RX)

2r

I

(2 j+l)x(

j) (g) (5.111

where

3 stands

Casimir

for

operator

normalization

of

where and

generators The write the

is

the

final

notion

operators

,as

has

of

SU(21,

eigenvalues

are

to

later

simplify

wavefunction to

do is

define

indicate

is

j(j+l),

it

the and

quadratic (2j+l)

is

a

case

is

formulae. the to

non-abelian be

~C(2j+lI~‘j’(R~)l

an arbitrary

‘m’

J-3

to

J,,,(X)A(X,X’)J,,,(X’))

before,

subscripts

of

down

one

= 1 exp(-1

At%,%‘) where

whose

Hartree-Fock

All

4c.rR;~I.l

generators

introduced

the

simple.

three

SU(2)

factor

Generalizing equally

the

set that

of

one

(5.12)

variational is

to

in

order

parameters,

sum over

the

three

SU(2). concept

the

which

Hamiltonian

of

the

U-operators

are

the

lattice

needs of

to

be defined

an arbitrary associated

equivalent

with of

-

non-abelian

43 -

the

a link. path

ordered

to

be able

gauge

theory

These

link

exponentials

to is

[“(PathqaB = [P(exp(i$dx@z-A,)) and

like

(n,n)

these

operators

transform

representations

Inr

is

determined

use

in

(5.13)

paragraphs

it

of

Given

is

easy

an irreducible

and

g1 and

U(j)aa Adopting U’s

of

in

Xl,.

the

92 the

with

representation

of

the

generators

is

link

in

the

operators

like

L we

preceding which

under

transform

arbitrary

as

gauge

define

the

matrix

elements

an arbitrary

gauge

D(jlas(RX)

transform

D(j)vP(Rs)

same

of

the

spin

transformation

by

as (5.15)

D(j),o(gz)

(5.161

D(j)kB(gt) form

of

j

tranformation

u(j)yD

links

the

(5.14) are

the

the

gauge

invariant

a plaquette,

X,

expressions by forming

in

the

the

ordered

form

.,X’

are

the

operators

when

on a functions that

that

function

(5.17)

U”~V(R~J)U-‘Va(R~5). four

we used

we see

multiplying

the

of

do

we can

D-matrices, acting

where

presented

Drj)aV(g~-l)

q

definition

the

the

GxG),

SlJ(2)xSU(2)

functions

under

same order

Replacing

to

Under

1 Ua~(R~OU~~(R~2) where

has

SU(2).

that

associated

products

of

+ D(j)av(g,-l) this

(or

representations

a set

D(j)ae(RX)

‘RX921

so we see

transformations

= D( j)as(g)3(R~)

of

D(j)ao(g,-

gauge

discussion find

one

functions

representation elements

to

all

[UC J’aa$lCRg) the

the

representation

transformations;

where

matrix

(5.13)

under

SU(ZIxSU(2)

by which .

ao I

links in

associated

the

Uap(Rg)

example by

a sum over of

the

of

their

the

the one

plaquette

of

variables,

x taken

plaquette

definitions

products group

with

in

operators is

universe.

terms of

of this

equivalent

form, to

by (5.18)

- 44 -

where is

X(j)(R)

is

equivalent

character

= trace

The

in H =-

Hamiltonian

form

of

which

of left

definition

gauge

is

believed

if

to

the

gauge

the

that

of

Theory

SU(2)

gauge

theory

can

be

PQED; namely,

that

is

ordered

notation

product

Et% stands

for

that

generators

link.

the

form

the of

same as the

elements the

the

squares

Note, the

group

a sum over

from

that

is

of for

constructed

transformations

Hence,

it

sum of follows

sum of that

of

Hamiltonian

links of

from

our

the

squares

of

the

right

handed

appearing

in

general. term

be given of thing

fundamental x(~/~)

Non-Abelian

for

operator

gauge

character

to

the

generators

interaction

general

expression

the

gauge

quite

Susskind,

most

General

plaquette

and

generators.

the

and

(5.20)

Casimir

handed

The

the

parallels

the

(5.191,

of

(5.20)

by

for

quadratic

left

SU(2)

9=

handed

the

of

X(“=‘(Rn)

in

the

j representation

9=

Rx stands

defined

spin

(5.19)

Hamiltonian

2 where

the

[D(j)(R)]

The Kogut-Susskind written

of

to

x( j ) (RI

5.1.2

the

the

(5.20)

as the the

one

involve

in

operator

fundamental can

do;

representation

been

chosen,

which

following

multiplies

for

This

example,

one

sum over

characters.

as defined (ie

in

j=t 1 but

by x(l).

-

45 -

(5.20) that

Kogut

any

representation.

an arbitrary

Hamiltonian

replaced

has

could

wavefunction is

not

the

generalize

this

It

commonly

confines this

and

is

no

is

charges

in

longer

true

5.1.3

Defining

Having the

defined

space

of for

invariant)

part,

(usually

its

Using

the ajIm

Defining

it

our

theory;

remains

of

a state

a state

having

These

general spin

Dj-functions; f(g)

of

formulae

the

for

onto

us

its

are

to

give

source

sources

too,

Hamiltonian

at

an explicit

free

a finite

analagous

and

(or

gauge

number

to

the

formulae

case. the

onto

functions

onto

points.

To derive SU(2)

space

projection

and

abelian

Operators

Hilbert

wave

the

two)

the

the

trial

expression

for

Projection

form

for

the

projection

we first

j part,

expand

of

the

a function

function

over

in

terms

of

eg. = 1 ajl,

(5.21)

ojl,(g)

orthogonality = (2j+l1 the

relation jdg

f(g)

projection

of

for

Dj-functions

we obtain (5.221

D*(j)lm(g) f(g)

onto

its

spin

j part

to

be

j = C ajlm(g)D(j)lm(g) m=- j

f(j)(g)

(5.23)

we obtain f (91

where

dj

discussion onto

its

function, it

against

projection

q

(2j+l)

c I

dh f(h)

= dj

J dh f(h)X’j’(g”h)*

= dj

$ dh’fcgh’)

is

the

gauge and the of

in

order

invariant to

project

appropriate a function

D(,j)I,,,(g)

(5.24)

X’j’(h’)*

normaliztion

that

D*(j)I,(h)

factor to

It

(2j+l).

project

a function

part

we integrate

onto

a configuration

is

-

of it

46 -

the

with

from link

against

To be exact,

character. $(.,Rx,.)

follows

the sources the

this variables

constant we integrate gauge

invariant

[ 1

(P,+)(.,Rg,.1

= n jdg(i)

~(.,g“(i)RXg(i+~),.) (5.25)

and

the

projection

P sources

onto

a state

with

spin

j sources

at

~1 and 132 is

=

(5.26)

5.2

SU(21

MEAN FIELD

Equation theory.

This

which of

(5.11)

one

and

exact of

weak

gauge

coupling

parameters,

as

must

in

the

If

approx

mation

To derive this

class

the of

VW

one

effective

variational

the

in only

Hat-tree

statistical

we must

both

the

obtains

the

constant

of

limit over

all

the

X(1’2)(R)

The small

treated

as well

approximation,

variational

gauge

family

a 6-function

not

mechanics

wavefunctions,

one

value

Fock

a single

in

constant.

are

SU(2)

parameter

coupling

expectation

point

the

~0,

the

wavefunction

one

obtains

coupling

field

behavior

takes

takes

coupling

as

having

one

vanishing

weak

reasonable

infinite

of

f or

has

a simple

the

limit

mean-field

a price

provide

to

mean

for

maximizes

the

general

If

which

around

pay

and

one

approximation

most

does

wavefunction

vibrations

must

it

field

limits.

theory.

group it

but

the

wavefunctions

groundstate

the

a mean

by no means

choose,

variational

strong

defines

is

can

THEORY

but

in

the

one

parameter. problem

equivalent

compute

the

to

partition

function

= n [.f dR%] n [I

dgCi1]

J’(.,Rx,.

)*!I

1 (5.271

-

47 -

Since

the

mean

wavefunctions,

field

wave

carrying

straightforward.

function

out

Each

is

the

Rs integration

gives

I [ exp

-

[

of

independent

is

quite

RX integrations

jC j+l) 3% = ,f dR% 1 exD

a product

2Y

link

a factor

1(1+11

I

2Y

X

(2j+1)(21+l)x(j)(RX)*xcl)(g-l(~)R~g(i+~)) (5.28) Substi

tut ing

functions

the

definition

, using

functions

of

the

the

D(j)-functions

of

fact

group,

and

character

the

D(j)-functions

using

the

as a trace are

orthogonality

of

D(j)

the

representation

relations

among

the

we obtain jC j+l)

$8 = exp

and

that

the

[

I

Y

(2j+l)x(j)(g’l(~)g(i+~a)) (5.29)

so Z = n .f dg(i) Having

carried

of

the

partition

of

the

mean-field

group the

lattice.

Z

where lattice

The

and

is

out

Z into which

specifically,

is a function

of of

sums

each

an integer the

group

of

for

function by giving

be shown

the

to

have

evaluation

the

projection

integrals

over

integrand jg

specified

Z will

for

integrations

a partition are

reduced

factor

evaluation structure

now carry for

we have

normalization

the

explicit

c n Ci[(j,m,m’)~l

CiC(j,m,m’)xl

or

by specifying

formula

More q

to

configurations

link.

RX integrations Z,

state,

determined

this

sums over

the

function

We will

convert

per

out

variables. sum is

(5.30)

n $;a

appearing

each

link

explicitly defined

the

of

associated only

those

-

48 -

the and

in

three

in

in

terms

of

integers

general

form (5.31)

fl Q(jz,rn~,rn’X) a factor

the

with integer

every varibles

vertex jx,

of

the

rnx and rn’g

belonging are

to

link

Q will

links

attached

to

dependent

factors.

be a function

of

jg

exp

-

the

For

vertex the

alone

= (2j+l1

[

where

we have,

link

label

which

for

on the

expresses

different

for

factors

rhs the

the

for

Deriving

the

remainder form

for

three

cases

3~.

is

given

notation, .

from

the

weighting

and

nothing

the

of

this

section

the

case

of

factors will

two

is

is

the

factor

dependence Ci

is

by gauge

invariance

spatial

dimensions.

factors

appearing

three

Ci

more

needs

a bit

more

be devoted

spatial

dimensions

wavefunction

dropped

imposed and

of

mean-field

by

The vertex

two

(5.291,

the

Q(jX,m~,rn’~)

(5.32)

of

form

spatial

(5.32)

constraints

come directly

expression

of

of

of

functions

I

Y

simplicity

The

case

and

j(j+l) Q(jx>

1.

to

to

dimensions.

a factor and

is

The in

be said

about

and

their

will

not

them.

and

the

explicit

The generalization

straightforward,

b

the

complicated

deriving

on the

to

be presented

here. Since lattice

the they

partition

can

in

over

expressions vi

where

(5.27)

which

of

(i-t?2,2)

taken

respectively.

3x = Y(j) 31(j)

is

Tr

the

associated by

for

the

with

fixing

involves

we find,

= $ d9U.J we have

are

be derived

function

term

where

factors

Ci

the

a single

attention

the case

upon

variable d=2,

vertex that

gCl>.

that

of part

the of

Isolating

we need

to

the

that

evaluate

a sum

form

Sa’Sx2

(5.33)

3a33x’

X1 = (i,l>,

X2 = (i,2),

We now proceed

D(j)Cg-lCi))

weighting

by rewriting

DCj)(g(d+fia))

factor

in

-

X3 = (i-~?q,l)

(5.291,

49 -

and Xb =

3~ as

I and

(5.34) we expand

the

trace

as

Tr(D’j’(g-‘h)) Note

that

we have,

fields

per

always

ordering

the matrix the

in

link,

inverse

ie. the

of

by giving each

one

expanding

the

m and m’,

where

indices

a group

associated order

With

three

integers

these

integrations.

the

is

is

all

adopted

that

group

each

it

Dropping

we have

the

each

link

the

to

associated

more

the

convention

order is

in

(5 .33)

each

carry

with

integer of

associated

always

to

matter

two

matrix

term

attached

a simple

factors

in

element

conventions

for

introduc~ed

subscripted

second

these

terms

trace

m and m’ so

element

with

m’,m.

of

(5.35)

= 1 D’j’,,‘(g”)D(j’(h),‘R

with

m,m’ and subscripted is

vertices

in

specified

vertex, out

the

and

the

for

group

other

than

L we define Ci[(j,m,m’lxl

= .f dg(i)

D(jt)ntm’l(g(L)‘l)

D’ jd&

n3(g(j.)l

D(jz),t,‘z(g(i)-I)

x

D(jbQ’&9(i))

(5.36) We can

now use

the

D( jjn’,,,(g)

identity

D(k)p’P(g)

= c

D( ‘)i’i(g)

(5.37) and

its

complex

conjugate

to

rewrite

(5.36)

as

1

CiC(j,m,m’)~l

= 1 -





2J+l



(5.38) It

follows

coefficients m’l+m’z which

can V-m’

immediately

from

the

properties

of

the

Clebsch-Gordan

that = m’3+m8v be rewritten = 0 and

and ml+mz

= m3+mb

as

V-m = 0

(5.391

- 50 -

Hence,

the

two

divergence in

two

the

sets

and In

two

The only

for

encountered

constraints

case

integer

non-trivial

valued

separately

in

on their

removing

these

the sums

L and

satisfy

treatment can

constraints

variables,

L’,

of

PQED

be removed

in

amounts

to

each

plaquette

for

j’s

not

completes

for

a mean-field

for

our

general

under

problem

requires

the

to

Monte-Carlo

go to

one

has

use

reduced

Hamiltonian j-values allow

us

the

the

to

work

Introducing

problem and

the

to

fact to

the

on significantly

and

this

coefficients

the

W(2)

as easily

as

problem

is

vanish

it

is

is

not

is this

dimension

of

the

lattices

this

clear

that

if

one

the

fact

with

the

of

Y only

groundstate, than

one

of

how much

even

ranges

sort

how much

and

function,

most

function

evaluating all

This

partition

by working

for

energy

the

function, theory.

methods,

evaluate

larger

gauge

at

clear

that

partition

of

by analytic

by one

contribute

of

d=2

What

the

form

the

To date

a computer.

has that

a few

should can

deal

with

formalism. sources

it

way i namely,

the

general

be handled

j-fields,

constraints.

be treated the

on the

Clebsch-Gordan

to

techniques

link

Euclidean

which

of

one

the

of

study. can

the

appropriate

and

formalism, per

since

approximation

theory,

is

is

derivation

cannot

abelian

function

of the

function

the

the

care

satisfying

This

(5.40)

constraint

taken

partition

for

sort

m and m’,

and m’ = VnXL’

automatically

in

the

the

this

fields,

defining m = VnxL

of

of

dimensions,

introducing

2,

integer

conditions

same way.

and

of

into

introduces vertex

factor

our

problem

a limited is

more

-

only

number

modifies of

complicated.

51 -

things

distinguished This

comes

in

a simple

vertices about

5

because

one

x(l)(g(j)I

projects for

the

any

vertex

against

the

constant

derived

for

Z by saying

m”s

which

the

partition

add

Although

no calculations

expected

that

our

to

hope

by going

abelian

what

we know

5.3

over

difficult

theory,

and

about

this

Hat-tree-Fock

defined

in

(5.12)

perturbation

differential

operator

characters,

we already

group

is

compact.

formula and

contribute spin-f,

to

only

locations

and

zero

been

carried

PQED in

phase

it

to

the

transition,

in

It

P+l-dimensions. PQED, this

will

is

be

approximation.

the

Hartree-Fock

for

the

wavefunction

non-abelian

close

by us,

approximation

a deconfining of

out

this

theory

chapter

is than

with

it

is

for

a discussion

of

problem.

Note

theory

source

P+l-dimensional

approximation .

of

of

j’s,m’s

vertex

mean-field

Hartree-Fock

we will

each

have

the

THE HARTREE-FOCK APPROXIMATION The

the

of

the

with

more

d=3

case

case

to

dealing

considerably the

the

the

the of

sources

the

sort

predict

for

instead

functions.

this of

located,

summarizes

at

of

at

partition

case

a character

configurations

case

spin-g

of

the

as for

Unfortunately,

up to those

happens

those

the

is

loosely

momentum

in

against

a source one

only

incorrectly

what

that,

remedied

for

will

which

angular

add to

wavefunction

If

then

contribute

analogy

the

zero

which

theory

at

that

function,

otherwise

SU(2)

i

function.

up to

configurations

is

mean-field

for

that form

and

even of

While

the

it effects

gauge

gauge in

act

the

gauge

related

theory

projecting,

solely the

of products to

is

by using

definition

on a sum of

projecting

- 52 -

GAUGE THEORY

non-abelian

before

AM,%‘)

letting

include

TO THE SU(2)

the

perturbative

the of fact

that

wavefunction

would

force

this

anyhow,

perturbation

theory

into

our

formalism

manipulations evaluation

of

Hartree-Fock

out

this

this

end

of

this

out1

exh ibit

the

approach

the

discussion

show

that

gauge

will of

As in

energies,

energy,

str ing

for

the

very

between

the

we carry

in

can

encountered

in

to

carry in

to

be derived

in

evaluating

the

that in

We will non-abelian of

the

no explicit

we will the

the

techniques

an evaluation

for

at

be true

and

since

in

Hartree-Fock

alluded

abelian

formulae

etc.

the

interesting.

section,

the

function

may well

out

for

discussion

partition

be presented

Analogous

which

computational

is

preceding

in

the the

than

way

It

by better

will

tens ion,

energy)

procedure

theory.

before the

(and difficult

the

force

analysis

etc.,

alone.

function

the

even

tricks

limit

of

brute

abelian

this

these

encountered

difference

function.

partition

the

uith

best

We will

be superceded

emerges

of

the

derivation

be the

a qualitative

evaluation

study.

the

more

finding

incorporating

subsequent

function

new difficulties

however

theories

partition

ning

will

future,

of

of

even

considerably

i s under

Our

near

is

The problem

to

technique

the

of

partition

eva luation

to

case.

the

case.

section

order

Unfortunately,

equivalent

method

simplifies

considerably.

approximation

mean field

this

focus

on the

expectation exactly

value the

same

manner. The principal function

for

differential commute same

problem the

Hartree-Fock

operators with

local

appearing

gauge

link.

This

fact

integrations

in

closed

wavefunction in

the

comes exponential

transformations,

makes form.

it

nor

impossible However,

-

53 -

to one

from

partition the

in

fact

(5.12)

do not

with

one

another

carry

out

the

can

evaluate

this

that

on the

gauge expression

the

as an expansion what

we will

about do

in

+hf(.,Rg,.)

the

this

mean-field

partition

section.

function,

We begin

by

and

rewriting

that

(5.12)

is

as

= U~d~xAWl

1 exP[-C

expC-x

J,tX>At%,%‘>J,t%‘11

fi X( j) (RX) (5.41)

where

we are

because with the

able

the

operator

al 1 of

the

product

break J-J

the is

operators

of

we expand

to

second

a Casimir

J,(X).

characters

the

exponential

of term

product SU(2)

in

mean-field

of

and

(5.41)

two

terms

commutes acting

upon

wavefunction,

hence

if

as

J,Cf)A(X,X’)J,U?‘)

exp(-1

first

the

exponential

the

operator

The

produces

into

=

1 - c Jm(X)A(X,X’)J,M’)

+ (1/2!)(-&.I*

+... (5.42)

then

we can

derive

Hartree-Fock

average

procedure,

and

this

in

discussion long

the

of

the

of

by analyzing

Using

in

the

is

of

reason

corresponding

to

of

believe

that in

order

of to

between

of

which

This

is

of

one

can

obtain

some

the

abelian

the

course

in

that

theory

is

do much

seen

the

will

the

each

already

it

3+1-dimensions,

differences

to

P+l-dimensions,

physics

in

that

we have

PQED in

the

terms,

operators.

However,

analyzing

the this

a product

physics

confinement

function

as a sum of

future.

We believe

expansion.

theory

of

there

way towards

physics

partition

representation

mean-field

than

the

better

our one

by means

can of

given

in

(5.42)

- 54 -

we can

go a such

intuition

for

the

be fruitful

to

begin

and

non-abelian

gauge

approximation. expansion

a crude

rewrite

(5.41)

as

an

+hf(.rt?Xr.)

+ x(x

=

A(X,~‘)Tr[T,(j)D(j)(R~)]TrET,(

l)(l)(R~‘)l)

fl x(~)(Rx”)

+ ... .... . (5.43) where

we have

simply of

the

used

trace

of

SU(2)xSU(2) the

to

the

zhf

upon

to

the

character and

that

of

compute

overlap

the

D-function

generator

order

compute

the

that

D=function

appropriate In

fact

acting

multiplying the

the

the

of fact

that

representation the

left

a representation

is

by

the

the

left

generator

equivalent

matrix

is

to

representation

of

SlJ(2).

Hartree-Fock

partition

function

we need

to

integral

= .f dg(i)

l

dRf

+hf(.rRz,.)*

~hf(.,g-‘(i)R~g(i+~~)r.) (5.44)

Substituting first

different is

involving

the

variables no

sums

for

longer over

Of course,

correct. configurations the

vertex

in

the

of

that

that

It

must which

factors

have

(5.43); appear

when

one

only

angular

momenta

be replaced add for

up to the

-

55 -

which One

that

these

link

factors

the

links.

The

at

angular

different

a structure

with

over

by

we have

approximation.

computing sums

The

terms.

which

namely,

a I uV on some of is

the

function

mean-field

function [T(j)

which

the

form

partition

condition

as a sum of

contributions

obtained from

Zhf

partition

remaining

difference the

we obtain

mean-field

elements

interesting,

link is

to

sort

the

those

obvious

matrix

this

(5.44)

The

from

contributions

more

is

discussed.

difference

of

into

contribution

already is

(5.43)

the

the

effects

second, of

configurations

a vertex condition momentum configurations

terms of

add

up to that

zero

zero

one or are

one.

different.

We close

this

section

by outlining

general

structure

the

derivation

of

this

the

expansion

of

case,

we only

result.

In order

to

Hartree-Fock to

focus

The

derive

partition on two

first

the

of

function

typical

these

1 &(X,X’)

about

terms

terms

is

which of

Tr[Tb(j)D(j)(Rz

the

of

mean-field

appear

in

the

expansion

the

form

)I*

Tr[Tbck)Dck)(RX’)l*

the

of

need

(5.44)

.

X

1

n X’j”(g-‘(i)R~“g(i+~~)) [

(5.45) Now one the

can

do the

mean-field

(5.45)

Rg integrations.

partition

As we already

function

this

?b(%!)

Sb(x’)

leads

to

saw when Kronecker

we evaluated

b’s,

and

so

becomes

[

n j- dg(i)

where

the

I

1 Atx,%‘>

factors

rbCf!I

are

defined

to

be

= Tr[D’j)(g-‘(1))Tb(j)D(j)(g(~+~~))l

T),(g)

(5.46) for

X = Cj,RI.

over

the

this

variables

expanding

out

elements at

vertex

factors

factors

are in

the

one

As in

expressions over

derived modified .

in due

proceeds

the

for

products

In the

a time.

(5.461

point

g(j).

as sums

vertex

traces

At

case the to

One then

of

of

D-functions,

of

(5.46)

fact

this case;

that

sums over

56 -

all

the

case,

characters

-

evaluate

mean-field

mean-field the

to

we do this

products and

then

leads

to

of

by

group

we focus the

however,

of

on one

same set

two

Tb(j)-matrices insertions

integrals

of

of

the

link

appear

in

the

this

sort.

The second

sort

wavefunction

of

term

one

as an expansion

encounters

about

in

evaluating

a mean-field

the

Hartree-Fock

configuration

is

of

the

form

n[.f

dR,]

n[I

dgti)]

n[%[j)tns1*]

the

factors

one

[

n s dg(i)

where

the

I’b(RX)

are

defined

now carries

out

I

7’b(%)

link

T’b(%!)

c AK+!‘>

factors

by

the

T’b(X)

(5.47)

RB integrations

(5.48)

difference

in

with these

appear

in

equations

and

variables

it

carry is

out

we see

places the

that

to

Tr[Tb(j)D’j)(g-l(i)g(i+~~))l -

in in

the

explicit

convenient

D”‘(g(i))qb

(5.48)

-

expressions,

different

obtains

are

(5.46) two

and

T’b(x’)

= Tr[Tcj)bDcj)(g”(l)g(l+~~))] -

Comparing

T6tk)

7’bu-Q’)

= TrCTb’j’D’j’(g-‘(1)R~)g(~+~~))l

7’b(RX) As before

T’b(RZ)

I

Xcj’)(g”(i)RXg(i+~p))

where

1 AW,3!‘1

there that

is the

matrices

trace.

In

integrations

write

the

an important

order over

traces

T@ to the

appearing

and

compare

the

gCi> in

(5.48)

as

=

-

TrCD’j’(g(~))T’j),D’j)(g(~+~~))i (5.49)

where

the

factor

generators the

transform

group

(5.49) factors

D(‘)(g(i))

under

into and

(5.48) the

the

as the

comes

from

basis

vectors

transformation we see

‘vertex

that

factors’

the

fact of

that

the

a spin-l

representation

sort

which

-

57 -

of

term

one must

of

Substituting

Ta + D(g)T,D”(g). this

SU(2)

modifies use

in

both evaluating

the

link this

contribution are

to

changed

in

(5.45)

and

is

change

the

because of

the

the the

Hartree-Fock same way as

(5.46);

of

in

the

namely, the

extra

preceding

by

of

that

only

J,m,m’-

angular

momenta

at

a vertex

(5.48) the

we see, vertices

link

plus

momenta

this must

non-abelian parallel

in

the

case

in

wipe

spin

mean-field

1.

of

the of

charged

it

is

the

our

matrix

newt

and

it

zero

could out

is i

(5.49)

the

soI

in

order

of

the

expansion

that

to

the

the

mean-field has the

at

the of

part

which

the link

the has

effect

of

for

zero?

effective

3+1-dimensional

AND CONCLUSIONS

58 -

the

effect

transition

for

changed

feature

into

the

is

up to

It

the to

add

its

the

additional

and m and

about

all

In

same procedure

condition one

It

about .

contribute

to

phase

-

comes

momentum

hope

SUMMARY

Tb(j).

sum of

medium

approximation

in

which

theory.

deconfining

6.

the

the

there

This

abelian

of

for

this

wavefunction

and out

angular

I;rhen we evaluated

derived

that

vertices

factors

and

we carry

case

the

vertex

up to

problem,

be to find

at

a new kind

mechanics

to

add

this

link

integrations

up to

when

The

appearing

gCj.1

added

In

Hat-tree-Fock

introducing

will

extra

D(‘)(gCi))

(5.491,

appearing

is

configurations

substituting

D(l)-function

which

the

However,

1 and m.

changed

introduction

factor we did

function.

were

the

factors

cases

function.

they

vertex

condition

partition

partition

no

of statistical

this

medium

one gauge

expects theory.

6.1

PURE GAUGE THEORIES In

this

paper

we presented

non-perturbative

gauge

non-abelian

abelian it

is

that

and

any

of in

invariance the

the

although

our

real

to

one

allowing

tension has

reason

carrying

out

described;

ACf,X’l seen,

for

that is

this

is

that

wavefunction in

the

weak

is,

kind

of

of

and

of

calculations.

order

reasonable

While

results

Hartree-Fock

set

calculation

of

would the the

gauge

exciting

As we indicated

59 -

this

out

holds of

of

be obtained

at

the

that by

function As we have

results

physics the

the

we have

for at

perturbation the

as

the

seen

possibility

projected

an understanding

-

such

already

to

a simple

procedures

which

reasonable

outside

methods is

parameters.

give most

these

wherein

variational

without

be used

behavior

might

calculation

theory

mean field

it

numerical

the

particular,

analytically,

into

both

gauge

can

In

we have

of

regime.

requirement

out

for

abelian

other

unity,

a calculation

coupling

from

dimensions,

this

problem

of

minds,

to

differs

and of

same

the

the

calculation

lead

to

calculate

To our

will

the

Hamiltonian

the

a Hartree-Fock the

the

is

accurately

that

as

applied

techniques

generalized

treated

2+1-dimensions. time

hope

when

be written

couplings

to the

to

abelian

both

was on application

group

us

for

approximation.

paper

these

renormalization

method

incoporates

mean-field

could

out

One virtue

the

method

space-time

this

incorporate

of

string

in

which

space

promise

the

focus

approximations matter

of

all

carrying

theories.

of

This is

in

framework

and

formulations

it

exactly

of

theory,

results.

that

gauge

formulation

previous

approaches6

of

the

for

calculations

lattice

non-abelian

simplifies

losing

invariant

Hamiltonian

technique

a formalism

end

PQED in this theoretic

of of

confinement the

last

section,

we believe

proved

to

be true

expansion theory the

of

the

is

6.2

mean-field

actively

function

one

the

of

a c~ompari son

for

PQED and

significant

the

most

statement

of

the

the

SU(P)-gauge

differences

perturbative

the

this

between

wave-functions.

important

Hartree-Fock

things

to

calculation

do at

for

the

in

either

FERMIONS fermions

field

sites

of

the

reduction

of

statistical

the

of

one

the

uses

While, freedom,

combined

mechanical

chooses

of

methods.

In

the

wavefunction

to

be a

wavefunction,

whereas

a generalization

in this

the

case

leads

to

boson-fermion

partition

these

fermion

same single-site

approximation

degrees

no new difficulties

formulation

one

wavefunction.

of

presents

Hartree-Fock

approximation

Hartree-Fock

6.3

of

because

because

level,

pursue

of or

over

number

crudest

that

introduction

product

second,

first,

theory.

mean-field

free

the

reasons:

wave

structure

INTRODUCING The

the

at

to

non-abelian

two

PQED and

we believe

time

for

Hartree-Fock

projected

Obviously, this

for

revealed,

gauge

this

of

of

systems

additional

problem

function

is

to

quite

in

the

fermionic

with

a large

complexity

the

an effective manageable.

COMPUTING THE HADRON SPECTRUM Since

the

extension

of

interesting is

that

of

quark

introduction the

techniques

aspect it

permits bound

of

states

fermions we have

of

this

sort

of

us

to

formulate

in

a concrete

into discussed

scheme in

non-perturbative the way.

-

this

60 -

problem

this

seems

to

paper,

one

be an very

computational of

The key

computing point

to

the

scheme spectrum

be made

is

that

having

consider

computed

carrying

states

of

a variational

out

form

a variational

non-trivial

flavor;

of

the

groundstate

calculation

ie

one

could

for

one

the

can

lowest

gauge-project

lying

states

of

the

form l*hadron>

fas(il-i2)

= 1

The variational

parameters

wavefunction

fco(iI,

wavefunction

I+vacuum>.

part

of

vacuum

the

wave

energy

problem

is

the

calculation fixed principle without two

function

gluons

in

color

orthogonal

to

energy

Obviously, related

to

upper

bounds,

this

vacuum

momentum

determined

as a function be able

of

these

calculations

but

variational one

can

to

of

the

estimate are

subject

calculations only

try

and

- 61 -

the

and see

in

original

theory

taking

how well

the

or

state

of

three state

to

be

calculation. minus

the

mass. usual

differences the

this

in

with

momentum

glue-ball to

the

for

theory

this

three the

the

is

non-vanishing

the

vacuum

of

calculation

forces in

the

an equation

of

of

trial

variational

perturbation

three

should

yield

state”

parameter”

the

spectrum the

the

in

“free

A similar

state

state

spectrum

doing

will

take

in

computation

performing

state

“bound

appearing

only

singlet

one

all

the

hence,

the

appearing

by the

that

could

first

parameters

“glue-ball”

one

The non-vanishing

vacuum

the

are

determined

of

the

a global

the

parameters

wavefunction.

momentum.

By plotting

the

energy

since

the

then

fas(j);

for

fermions,

problem

follows

“bound-state” possible

this

are it

the

in

Since

function

for

time

and

density,

(6.1)

9*(il)a$(iz),l~va,,,,>

method

caveats of works.

CONCLUSION

6.4 This

discussion

presented There

in

will

this

to

effective

statistical

numerical results with works

us

one

to

will

in

one

Euclidean continuous complicated

of

the

say

work

be able

the

to

case

always

problem, integrals systems

of for

can

obtain

and

doing

that

should on larger

the

most

in

learning

in

the At

we say will

Even

if

one

one

dimension

make

doing it

time

62 -

it

to

the

the

to

the

compare

end

sumsI the

sums

the

problem to

more

than

use

fact

corresponding

rather

with

the

and

way this in

than

deal

in

not

that

forced

discrete

time.

of

is

enough by

less

this

analytical

certainty

is

at

how best,

mechanics

possible

lattices.

-

with

formalism

functions

between

heartened

statistical

is

this

the

study

be accurate

the

one

of

partition

balance

we are

PQED.

working

merit

of

best

Nevertheless, the

aspects

problems.

nor

be,

those

involved

evaluation

what

will

methods is

deal

we feel

mechanics

experiment. out

which

out

methods

Monte-Carlo that

carry

for

summarizes

paper

be a great,

practice,

possible

briefly

REFERENCES

1.

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such

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now only the

oriented

in

of one

the

this

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with

possible

to

going

L’s

is

gauge

the

becomes L’s

over

for

to not

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Lectures

dual

L for

lattice. even

variables

essentially

the that

faces

of

L.

Susskind, 1975

3+1-dimensions,

unconstrained

condition six

and

variables

satisfies

the

Kogut

PQED in

the

invariant

the

J. Erice

integer

automatically

case

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introduce

amounts

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sum of

sum of

m,

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thing

dealing

Maxwell

which

395

which

case

Rev.

condition

each However, though

in

we are because

0.8

~~0,

for

each

cube

the

the

cube

must

vanish.

This

point

is

completely the

points

lattice J.

Greensi

Drouffe,

in

discussed.

of

in

th te,

Phys

case this

ref.

Since

2+1-dimensional

fine

6.

explained

3 and

we are

from

this

discussion

and

the

only point the

way of

dealing

with

inter~ested

in

discussing

on ue will

not

go into

introduction

of

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64 -

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M.

it

is

the

FIGURE

1.

Graph

of

as

would

it

the

Hat-tree-Fock approximations

string

tension

be calculated (dot-dash (dashed

in in

curve)

CAPTIONS

2+1-dimmensional

mean-field and

finite

curves).

- 65 -

(solid range

PQED curve),

perturbative

Hartree-Fock

% 1-82

COUPLING CONSTANT Fig. 1

4251Al