Lattice Theories of Chiral Fermions - CiteSeerX

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of emr2/g2 than the equivalent term in the denominator. Hence, as g --+ 0, I' vanishes faster than any power of g. This is then a compact equivalent of the.
SLAC March T

Lattice

Theories -*-

HELEN

R. QUINN AND MARVIN Linear Accelerator

University,

Submitted

* Work supported by the Department

Stanford,

to Physical

- 3905

Fermions*

-

Stanford Stanford

of Chiral

-PUB 1986

WEINSTEIN

-

Center

California,

Review

94305

D

of Energy, contract DE - AC03

- 76SF00515.

-cWe present a Hamiltonian ories on a spatial provides spectrum

lattice.

formulation

The formulation

exact chiral symmetry

-

for gauged non-anomalous

complicated

theories.

and non-local

in the mass +

calculations.

tory chiral lattice

0 limit,

with

that

it

a correct fermion

theory for QED. It can be gen-

The disadvantage

(on the lattice)

chiral the-

has the desirable properties

and correct weak coupling perturbation

eralized to non-Abelian

practical

ABSTRACT

of the formalism

is that it is

and hence not easily implemented

in

However it does provide an existence proof for a satisfac-

gauge theory in the Hamiltonian

2

formulation.

1. Introduction Lattice

methods an important

ories like quantum

chromodynamics.

quantum chromodynamics the extension

non-perturbative

tool for analyzing

Considerable

progress in understanding

has been made using lattice techniques.

of these techniques

gauge the-

to theories involving

truly

Unfortunately

chiral fermions,

for

example all grand unified theories, remains problematic. There is a widespread and Ninomiya’

, that

free chiral fermions sensible continuum

belief, principally

it is impossible

which&es limit.

to define a local lattice

n3 exhibit

spectrum

theorem

but which do contradict

has been given. The Nielsen-Ninomiya

Smit

fermions.

In addition

,2 which showed that

derivative

to their

may not satisfactorily

covariant

derivative calculation

which it

formulation

result,

there is the work

formulated

reproduce continuum computed

of lattice

with

theories of

of Karsten

a long-range

perturbation

in naive perturbation

and SLAC

theory.

Their

theory are not

are performed.

This

to be a consequence of the way in which the free fermion

was introduced.

Combined

has been taken as sufficient

with

the Nielsen-Ninomiya

theorem

evidence that no satisfactory

the existence of a problem, 3

the identification

this

gauge theory

of chiral fermions can be defined. We find that while the calculations and Smit indicate

that

the Nielsen-

the broader interpretation

even after all the usual renormalizations

result was interpreted

of the

theorem, however, is not the only reason

a gauge theory

result was that Green’s functions Lorentz

and which has a

which do not violate

for the belief that there can be no satisfactory chiral

even for

We will show for the case of free fermions

there exists a broad class of counterexamples Ninomiya

doubling,

theory

In fact, this belief is based on a misinterpretation

results of Nielsen and Ninomiya. _..

due to a theorem proven by Nielsen

of Karsten

of the origin of

this problem compact

lattice

Q.E.D.

of the compact This fermions

This

is not correct.

is easy to demonstrate

as was shown by Rabin

case has been presented

paper

is devoted

focusing

f but no comparable

at least one way of avoiding

of the problem

of these earlier difficulties

them.

We explain

In fact we establish only low-lying

theorem

introduces

theorem

is incorrect,

In 2+1 dimensions

namely

interacting

problems

exhibits

these difficulties

encountered

Turning

the problems problem,

spectrum

doubling.

by continuum

but are instead

from the question

theory

by Karsten

have prob-

of whether

the true origin

or not of the

understand-

manages to avoid these

gauge theory

due to the way in which

that

do not exist, how-

we find once again that

and Smit are not rooted

4

invariant

renormalization

we achieve a better

range SLAC-derivative

to the case of a lattice

encountered

when one

we are lead to the conclusion

In understanding

in the interacting the SLAC

of the

are not rotationally

chiral theories based upon short range derivatives

ing of how it is that problems.

that

required

ever the reason for this is quite separate the theory

their own difficulties

and higher such theories

Green’s functions

In analyzing

interpretation

and Smit for the long range derivative

even after all of the usual subtractions are performed.

a class

lie in the region near 5 = 0. While

encounter

to those found by Karsten

in a gauge theory,

and exhibit

the stLosger result that for this class of free field theories the

interactions.

lems similar

is a mis-

on lattices of finite extent for which there is no doubling.

states are those whose momenta

Nielsen-Ninomiya

chiral

in order to discover

these examples of free field theories prove that the broadest _-

of lattice

why spectrum-doubling

nomer for the consequences of the Nielsen-Ninomiya of short range derivatives

treatment

to date.

to a reexamination

on an analysis

for the case of non-

in the doubling

the gauge field coupling

to

fermions

is introduced.

not difficult

theory

compute

with

is considerably

the following

and hence much less easy to

gauge theories.

version of non-anomalous

symmetry

2. QED continuum

Abelian

Nevertheless,

weak-coupling

can readily

It is unfortunate

perturbative

undoubled

theory

be generalized

spectrum.

is reproduced

the usual subtractions -

that the prescription

to non-Abelian

theories.

which we have found is very unwieldy.

it is even more non-local

(on the lattice)

It will be obvious from the final form of our Hamiltonian tice calculations

using this formalism

in other approaches. computational

but rather

chiral gauge theories on the lattice

to show that

has a solution,

which

can render computations

same time retaining

the essential physics of continuum

lattice

field theories

studies local current-current shows why problems

which

correlation

of rotational

Section

non-invariance 5

task than

as the basis of a simple

of making

of constructing

approximations

more feasible while

at the

chiral symmetry.

2 presents

avoid the doubling functions

out lat-

albeit an ugly one. In the con-

to this Hamiltonian

free fermion

that carrying

the problem

section of this paper we discuss the question

The plan of this paper is as follows:

deriva-

at weak coupling.

will be an even more formidable

We do not offer this Hamiltonian

procedure,

in the

have been made. -

it does reduce to the usual gauged version of the SLAC long-range

tive at strong coupling

cluding

it is possible

gauge theories which has all

is exact for zero mass cases, with

a + 0 (g held small4 ),after -c-

3. The prescription

While

it is

desirable properties:

1. Chiral

_-

more complicated

than the usual lattice

a lattice

limit

is identified

to see that they can be avoided; however one pays a price in that the

resulting

to write

Once the true cause of these difficulties

a study problem.

of various Section 3

in these free field theories and arise in Green’s functions

for

short-range

derivatives

and how they can be avoided if the range of the deriva-

tive is taken to infinity. Q.E.D.

which

introduced bation

shows that

theory.

standard

reproduces

more intuitive.)

of non-compact

of the long-range

reproduces

(This section essentially is somewhat

derivative

can be

weak coupling

results

of Rabin:

lattice

perturalthough

In Section 5 we turn to the compact

and suggest a gauging which preserves the good results of Section 4. This

generalization parable

a gauged version

in a way which explicitly

the approach theory

In Section 4 we turn to a variant

to the compact

case is important

because it suggests that a com-

treatment

for non-Abelian gauge theories can be considered. -cSection 6 we review our results and discuss some open questions.

Finally

in

2. Free Fields

_If one attempts nearest neighbor mensions,

lattice

a Hamiltonian

derivative

limit.

dimension

This theorem

one finds that,

This repetitive

for a theory

doubling

of the spectrum

using the

in d-spatial

di-

interpreted

see have proven a the-

to show that any finite range lattice

doubling.

While the theorem

bling is a misnomer

for the phenomenon

it is simple to write

Hamiltonian

do not exhibit

as one goes up in

Nielsen and. Ninomiya’

has been discussed in several publications,

suffers from this spectrum

continuum

of free fermions

seems to be a quite general problem.

orem which is widely

tives which

theory

there are 2d-times as many states at a given energy as one expects in

the continuum spatial

to formulate

which occurs.

is correct

derivative

spectrum

We will demonstrate

douthat

free field theories based on finite range deriva-

spectrum

doubling

spectra.

6

and which

have entirely

sensible

:

NOTATION

2.1

In what follows of integers 3 =

we consider’.a lattice

whose points

are labelled by d-tuples

(jr,. . . , jd) where d is the dimension

of space.

spacing between adjacent points is given by the parameter dimension

of length.

-N

5 N for m = I,...

5 j,

physical volume,

In addition,

L3, with

Hamiltonian,

Hamiltonian

In addition

a, a quantity

with the

we assume that the integers 7 run in the range

, d , so that our theory

is set on a lattice

of finite

L = (2N + 1)a.

We use dimensionless physical

The physical

variables,

H;%as

except where otherwise

units of energy, we introduce

noted.

Since the

the dimensionless

U defined by

all fields are scaled by the appropriate

them dimensionless

and yield simple lattice

powers of a so as to render

commutation

and anti-commutation

relations. In particular,

lattice

fermion

fields are denoted $(j’, with

P-1)

The lattice

Fourier transforms

of the fields $(y) are given by

+h) = (2N ; where 2mt/(2N

2~

=

(ICKY,..., nPd) are the

+ l), for -N

5 n < N.

l)d/2

(2.2)

cf e 3

dimensionless

lattice

momenta

K,

=

The relationship terparts

between dimensionless

variables and their dimensioned

coun-

is as follows: (2.3)

The operators

which converge weakly

to the continuum

We are careful to work%finitGvolume levels and thus explicitly 2.2

field Q’(Z) are

because this allows us to-count

check whether

or not spectrum

doubling

energy

occurs.

1 + 1- DIMENSION : SHORT RANGE DERIVATIVES

Let us begin by studying case we use two-component

the simplest

case, namely

Dirac fields Consider

1+1-dimensions.

In this

the Hamiltonian

X(P) = -i~[(1~‘{Ijlt(j)~~~(j+l)}+(~4~1){~t(j)~~7L(j+2)}]+h.e.

(2.6)

KP

where we have adopted assumed that diagonal

periodic

$J(N + 1) = $(-N).

in terms of the Fourier

boundary

conditions;

i.e., we have implicitly

S ince (2.6) is translationally transforms

of the $(j)

invariant

it is

and has the form

w

where & (Q)

is defined by &(I$)

Note that the minimum

= y

sin(np) + (P4i1)

and maximum

sin(2mp)

values of tcP are &2rN/(2N

(2.8) + 1) and tcP

changes in increments First, spectrum

of 1/(2N

+ 1).

let us examine

the nearest neighbor

for this

In order

derivatives

case.

considered

to more clearly

about

the nearest-neighbor

l(a)

and (b) which

case is that

the important

near zero and 7r. Expanding

whereas for “N-p

of n. The important for every low lying

of all

of K is not symmetric features

thing

to

state near doubling.

for /A # 1 demonstrate

a short range Hamiltonian.

we see that the energy as a function

_-

function

plot the same function

this is not a general featur%of

To understand

the behavior

state near r+, = 7r. This is true spectrum

‘cP = 0 there is a degenerate Figures

exhibit

shows the

we only show the plots of E(K) for the region 0 5 K < T

since &(tc) is always an anti-symmetric notice

case, p = 1. Fig.l(a)

that

Here, for /..L= .I, .Ol, about n = 7r/2 .

of this spectrum

expand

(2.8) for n

in ~~ for (p( < N we obtain

and IpI < N we have

(2.10)

where & is defined to be 6

=

P

If we restore dimensions

GP + l> 2N+l

of energy to the problem

k, = ;K~

9

(2.11)

by defining

(2.12)

then we see that the physical

energy & (np)/a

for k, near zero is

a2k3 = kp + O(-$)

i&(ipa)

(2.13)

and for k, near x/a

i& (T - 6~) = /,.L-l$

63 a2pm1 $

+ 0 (

since (2N+l)a

= L Eq.(2.14)

energy Emin = T//AL. move off to infinite

we-take T/,uL + 00 as L- --+ 00 the spurious states

Hens&if

energy, and the only states which remain

their energy as a function At this juncture Ninomiya

-..*--

stated

E(n) is a periodic

of a continuous

The continuum

appearances

relativistic

variable

However,

function -oo

What

doubling.

5 K 5 00. The function

in finite volume

particles

with very different

Having

established

we now hasten to point

for tc = 0.

result

really says is

What

as

(2.8) clearly

IC is not a continuous

variable.

L + co and /.L + 0 are taken.

This is because the density

it is unity

the formal

with period 27r when considered

result depends upon how the limits

K M 7r is ,s, whereas

law for

to the Nielsen-

the theorem

Notice that even if L + 00 first, so that K becomes a continuous no true spectrum

dispersion

has happened

to the contrary

has not been violated.

satisfies this property.

at finite energy are

of momentum.

Despite

in their theorem

a function

with an ordinary

it behooves us to ask “What

theorem?“.

that the function

)

implies that the lowest state for K --+ 7r occurs at an

a single species of chiral fermions _-

(2.14)

variable,

there is

of states for the branch

one has are two species of

speeds of light.

that there are loopholes in the Nielsen-Ninomiya out that

theorem

this does not mean we are out of the woods. 10

While

the next

to nearest

neighbor

theory of free chiral fermions,

theory

successfully

it does not provide a satisfactory

for d > 1. The reason for this is that once interactions do loop integrals to loop integrals

(i.e. sums over intermediate coming

energy denominators

which

survive

_.

bling

with

region having

spectrum

but rather

energies on the order of r/a.

with

energy than at finite

are intrinsically

not rotationally

the Hamiltonian t-o 1~1< fi

invariant.

spectrum,

into account the

to do with

the number

of states in the neighbor

these states have features which

which is linear in R corresponds

which we refer to as the spurious

a --+ 0 and pL --+ 0, and the number

this region is of order (rr - 4)

L/ a. It follows

that

to mean a + 0 and pL + 0, then the majority

spurious

case

p -+ 0 many more states at

is all at high energy in the limit

limit

dou-

5 To establish this fact observe that for

(2.6) the region of the spectrum

. The remaining

there may be so many

For the next-nearest

energy; moreover,

by

has nothing

one sees clearly from Fig.1 that there are, in the limit infinite

rc M rr are damped

to Green’s functions which have been performed.

subtractions derivatives

one has to

as it does in more than l+l

contributions

short-range

theory

the contributions

the energy denominators

even after the usual continuum

of the low-energy

Although

When this happens,

it can yield non-covariant -&-

The real problem

states).

are on the order of n/pL,

sum over these states diverges.

interacting

are introduced

from the states corresponding

states at this energy that even taking

dimensions,

defines an undoubled

region,

of states in

if one takes the continuum of the states will lie in the

region.

In Section 3 we will exhibit to current-current that these spurious

correlation

the exact nature of the non-invariant functions

contributions

in more than 2+1 dimensions

states do indeed cause a problem. 11

Furthermore,

and show such effects

are unavoidable

for any finite

of the region in which taking

additional

that

this shrinkage

obtaining

2.3

& is a linear function

terms,

can make the spurious

range derivative.

of the spurious

a satisfactory

continuum

rather

it is true that

the width

of K can be increased somewhat

it is only by including region shrink

While

terms of infinite

range that

by we

than grow as ,Y, -+ 0. We will show

region as p + 0 as the crucial

ingredient

in

limit.

LONGER RANGE DERIVATIVES

One way to study

the+idth

6f the region in which the spurious

adopt a damped version of the SLAC derivative. tion of the chiral fermion

theory

states is to

Recall that the SLAC formula-

gives a Hamiltonian

of the form

(2.15)

*.---

where the function

S’(j - 1) is defined to be

(2.16)

In the limit finite

N + 00 this function

has infinite

range, falling

off like (-l)j/j.

N, note that S’(j - 1 + 2N + 1) = S’(j - 1)

which

For

means that the SLAC

i.e. on a ring containing

derivative

2N + 1 points.

describes fermions We observe that 12

(2.17)

on a periodic

lattice,

(2.15) can be rewritten

as a sum of terms with range 1 5 r 5 N;

i = -4 2 j=-N

As in the nearest neighbor where j’ is the integer modulo

+ +5’(r)

+ hc.

(2.18)

in the range -N

assumed that +(j + T) = $((j’)

5 j’ 5 N obtained

by reducing

(j + r)

(2N + 1).

~(~1

=

a damped form of this Hamiltonian

-i

ge

~++(j),$(j

j=-N

As before,

=.--

$+(j)a,$+’

f=l

case we have implicitly

We will now introduce

_-

5

U(p),

by a function

is diagonal

+

r)eep’_S’(r) + h-c.. _

a energy-momentum

&(K+), of the form shown

in Fig.2 for various

important

difference between this case and the next-nearest

the width

of the spurious

instead

of shrinking.

correct

relativistic

-2.4

HIGHER

momenta

Eventually, formula

for ~1 +

a dispersion

values of p . The case is that

E(k,)

= k, grows

law becomes the 7

in 1+1-dimensions.

AND MASS TERMS

of the preceding formulae to d+l

mass terms is straightforward.

given

and so in the limit

0, the dispersion

for massless particles

DIMENSIONS

The generalization

with

dispersion

neighbor

region is only on the order of fi,

of physical

(2.19)

r=l

in /cP with

p -+ 0 the region

as follows:

dimensions,

The generic Hamiltonian

with fermion

is of the form

J/= C -i [$+(j-t~fi~)a,~(j’)D,(~) - h-c.1 + mc 7Cl+(j’,P+(j’ , (2.20) f

f

3,PlV

3

where m stands for the dimensionless v = l,...

,d can be chosen either

mass m = mu.

The functions

as in (2.6) or (2.19). 13

I&(T)

The matrices

for

QI and

P are oV = 707~ and p = 70, where the r’s are the Dirac matrices for d+ldimensions.

Because we wish to study

four component truly

massive fermions.

Q.E.D.

to purely

left-handed

The Hamiltonian ergy momentum

is manifest

fermions

the projections

can also be used for (1 f 75)/2 and setting

in the m = 0 case and so the restriction

is not a problem.

(2.20) is easily diagonalized

dispersion

theory we consider

Clearly, this formulation

chiral gauge theories by introducing

m = 0. The chiral symmetry

perturbation

relation

in momentum

space. The en-

is

(2.21)

where

is the discrete Fourier transform

The 2d doubling

of the nearest-neighbor

the region of momenta where all components the spectrum

of the function

theory has been lifted in (2.21). In K, are close to zero (2.21) describes

of a massive fermion with speed of light 1. When the form (2.6) is

used for D the states for which all components species which

propagates

with

regions where some components not correspond

Dl(r).

speed of light

ICY are near 7r look like a second l/p.

there are 2d - 2

of Z are near zero and others near 7r which do

to any sort of relativistic

spectrum.

near zero but ICY = K/U - R, we have a low-lying behaves approximately

However,

For example for /cZ and ICY region of the spectrum

which

as A2

& M tcz+ tci-I-s •j-m2 P2 ( ) which can at best be interpreted in the x and y directions

l/2

(2.22)

as a species of fermion which propagates freely

but pays a price of at least r/pL 14

for motion

in the

z-direction. spectrum

It is the existence which

the discrete

leads to non-covariant

rotational

the contributions Ci @n limit

of such intrinsically

invariance

of the lattice

is retained

functions.

While

when one sums over

qf, which

are not invariant

with

respect to small rotations

even in the

a -+ 0.

survive,

provided

SLAC

derivative,

In general, operators

for a free field theory

ture usually

encountered

The operator

show that

in the limit

Correlations

Functions

the continuum

whose Green’s functions in the discussion

no such terms

N + oo.

there is an ambiguity

should be chosen to represent

We will choose an operator

theory.

(2.19), we will

we choose ,U = 0 (1/N2)

3. Current-Current

-.---

of Green’s

regions of the

from all such regions, we find they lead to terms of the form

For the damped

_-

behavior

non-covariant

as to which

currents

mimic

Qt(Z)rV?(Z).

the divergence

of weak coupling

lattice

lattice

struc-

perturbation

we study is the charge density operator

(34 which has the Fourier

where Z = (nr,...

transform

nd) is a vector whose components

p(y) is the time component

of a four-vector 15

operator

are integers. which

The operator

satisfies the lattice

equation

wherecr=

I,...,

d and p is such that -N

of 7 are intrinsically

non-local.

however, since it is sufficient of the current

(Explicit

expressions for them are given in Ref.8,

to study the behavior

of the time-time

we will not need them here.) The correlation

will be the time ordered products

-&-

-

5 i, + p 2 N. The spatial components

where the operator

functions

(&?,t’)P(.Lt)))

-

-

(3.4)

p(j’, t) is defined by

P-5)

p(j’, t) are normal ordered products

As defined, the operators

formalism

of interest

of the free field currents

p(j+, t) = eeiWtp(i)eiyt.

have vanishing

components

vacuum expectation

these Green’s functions

of free fields, and so

values. Since we are working can be calculated

in a Hamiltonian

using familiar

Feynman rules

except that 1. Time is continuous, 2. Space is discrete variables

so Q integrations and finite,

ICY= 27rp/(2N

3. The free fermion

so momentum

+ 1) for -N

propagator

run from --oo to 00. sums range over the discrete

5 p 5 N

is given by

1

16

P-6)

where j&o,

$7) = r”no + c

P-7)

r”Zi,(ICv) v

where

E,(lc,)

is the general function

Hamiltonian

spatial

We want

conserving

b-functions

of the zero components

components

THE

3.1

in the definition

of the

(2.20).

4. The overall momentum b-functions

appearing

of momenta,but

WF!JH LQOPS

to examine

the Fourier

For d-spatial

the charge density-charge

A;,P(4’,qo) = (zn

are ordinary

periodic b-functions

for

of momenta.

PROBLEM

two charge densities.

at each vertex

density

transform dimensions

correlation

6(r’-

+‘l)d,d

.

of the time-ordered the unsubtracted function

p’- s’-

product

of

expression

for

is given by

(2N + l)n’)l(Zp,

Zr, Zs, ko, qo)

p,r,n (3.8) where 6

27rrr’

27rp’ = 2N+l’

&=

2N+l’

27rs

@=rc,-=

2N+l

(3-g)

and 1 (q. + ko)70 + ~17~5~(1~~) Our problem the limit is tedious

- m7’ kor” + a-&i(t~~)

is to estimate the contributions

results and note certain

to the appendices.

salient features. 17

1

(3a1o)

of the spurious regions to (3.8) in

a + 0. The detailed discussion of this calculation, and is relegated

- m

while straightforward,

Here we will

simply

summarize

For the case of l+l is naively

logarithmically

conservation needed.

the vanishing

derivative

limit

of the most singular

current are

contributions

from

If we take the limit

L--+00

contributions

00

with

vanish,

(3.11), and so the inequality

negative

however,

p, L and a for the case of the

6 = constant,

as required.

-

It is clear too that,

)q) < &Ii/ a is satisfied for all finite physical

restriction

to (3.8) - (3.10)

and no subtractions

(2.6) for 0 < q < &i/u.

i.e. -IT-k all spurious

limit;

of that divergence

regions of the K sums on the parameters

u--,0,

_-

in the continuum

1 lists the dependence

next-nearest-neighbor

then

the expression corresponding

divergent

guarantees

Table

various

dimensions

external

momenta

holds for all cases of interest.

the

in the

The results for

q are similar.

For the damped SLAC derivative can be well approximated ~(1 - ,/jZ).

The analytic

this point

is complicated,

value of E(K)

we see in from Fig.2 that the function

by the linear behavior however

behavior

of the function

D(n)

= n up to tc of order

in the spurious

we show in Appendix

in this region occurs at IcN = 27rN/2N

cP(s)

I that

region beyond the minimum

+ 1, where it is bounded

by

(3.12)

Thus,

using this derivative

results in the limit In higher and requires

in l+l

(3.11) provided

dimensions subtraction.

dimensions,

we can also obtain

satisfactory

6 5 1.

the current

current

In the spurious 18

correlation

function

is divergent

regions for the next-nearest

neighbor

.

derivative

the slope of the function

tion introduces

a factor

(qu/p).,

E(K) is of order l/p,

Th ese inverse powers of p are dangerous; they

can serve to enhance the spurious of limit

can be found.

therefore each subtrac-

contributions

The most singular

so that

no satisfactory

choice

contributions

come from the regions

is in the spurious

region near ?r, and the

where one component

of n, say y,

remaining

are in the linear region near K = 0. In this case, for the

next-nearest

components

neighbor derivative,

we obtain contributions

to the Green’s function

which behave like

-k-

da:cr(l - o);

--

derivative

.

problem;

region has a width

(3.13)

,U + 0 so these contributions

be avoided ? It may appear that

suffers from a similar

case the spurious

.

1 + (1 - ctt)q;L

In

Clearly, there is no way to take the limit How can this problem

=:--

F

-2

disappear.

the damped SLAC

however, one must be careful.

of order 4.

In this

Thus we can choose to take

the limit 2

p=&( ; > ; and

L -+ 00

for

a + 0 (3.14)

with

6 = constant.

In this case the number of states in the spurious region remains finite as a -+ 0 and so their contributions rather than as integrals.

to Green’s functions Furthermore,

must be treated as discrete sums

as can be seen in Fig.2 the slope of (n) is .

never of order l/,~ in the allowed region of K. Instead,

the function

discontinuity

Appendix

at n = 7r which must be carefully

analysis of the 3+1-dimensional of the subtraction

current-current

terms required by continuum 19

treated. correlation

II details our

function,

renormalization.

develops a

including

all

Table II defines

the regions of momentum shows the dominant

behavior

other contributions discrete lattice

space which must be treated of sample contributions

can be found by exploiting

rotations.)

In the limit

separately

and Table III

from various regions.

the symmetry

of the theory under

(3.14) only the usual contributions

4. Non-Compact

using a damped SLAC derivative,

0. Yet Karsten

contradictory

Q.E.D.

couples equally Karsten

perturbation

results?

are well behaved lattice

behave satisfactorily

and Smit found u_nsatisfactory -&-

in weak coupling

theory.

functions,

in the limit

results for the SLAC . -

How can we reconcile

considered

by Karsten

to all n-modes

the photon

and Smit.

these seemingly functions

couples in the version

The current

of the fermion,

whereas the vertex

obtained

r/u

(4.1) describes

a coupling

and so, in the limit

to photons

spurious regions to loop integrals

strong.

(4.1)

As with spectrum by the work of Rabin,

is proportional

to fermionic

states with

Thus, the contributions

and not spectrum

first discussed by Karsten

doubling,

to the slope of

are enhanced by factors of order l/p

a + 0, ,X + 0. It is this behavior

at the root of the problems

which

,X + 0, the coupling

but IC < T/CL becomes arbitrarily

the limit

by

and Smit behaves like

sin(qiu/2)

E(k)

of

which we discussed

E,(k+q)- E,(k) Eq.

a -+

derivative

The answer is that the current whose correlation

is not the one to which

survive.

QED

The previous section proves there exists a current whose correlation calculated

(All

doubling

20

from the and survive which lies

and Smit.

there is a way out of this problem.

who gave an answer which

k + q >

is applicable

It is suggested only to the case

.

of non-compact

QED. Let us review the substance of Rabin’s solution

turn to a generalization

of it which is more suitable for a compact

and thus for generalization It is helpful compact

to a non-Abelian

to begin with

lattice

Q.E.D.

gauge theory

theory.

a review of some elementary

The Hamiltonian

and then

properties

version of non-compact

Q.E.D.

of nonis de-

fined in terms of link fields At and their conjugate variables El which satisfy the equal time commutation

relations [El, Al,] = -iS,,,t -)-

(4.2)

-

where f! and .@ stand for two links on the lattice. the Hamiltonian Hamiltonian

For. the pure gauge theory

can be chosen to have one of two forms.

is the one which trivially

of photons simply

transcribed

.

-

The first form of the

reproduces the ordinary

free field theory

to a lattice.

W) Here, e denotes a link variable, electric field variable associated with

and P denotes a plaquette

the plaquette

form of the Hamiltonian different9

i.e., El is the

associated with the link !Z, and Bp is the magnetic P. If we denote the links bounding

P by er, &, es and e4 respectively,

slightly

variable;

then Bp = AtI + Al, - Aea - Al,.

is more complicated

copies of compact

Q.E.D.

H = c

- &

GE;

and produces

variable

the plaquette The second

a large number

of

; i.e., +cos(Bp).

(4.4

e The important interacting

difference between (4.3) and (4.4) is that (4.4) defines a theory qc

degrees of freedom for g2 # 0 whereas (4.3) is a free field theory. 21

k

this reason the theory based upon (4.4) h as some of the features of a non-abelian theory

(at least for g2 >>

1 ).

,

Both theories have an invariance

under time independent

gauge transforma-

tions wherein ;

6Ee=0 where

X(T) a function

the points

6Ae=

defined on points

V&i

of the lattice.

y and ? = y + C then VLX = X(3)

transformation

is the unitary -)-

(4.5)

- X(y).

operator -

If f! is the link joining The generator

of this

.

s

u (A) = ei Cl ELVLX _-

which can be rewritten,

using the lattice

U (A) = e

The lattice

divergence

be identified

analogue of integration

(4.7)

Et is defined as the directed

the point j’. Equation

as the generator

the Maxwell

gauge transformations.

equation

V-E(;)=0

is equivalent

to the statement

gauge transformations.

The generalization

ter fields is straightforward. formation

that physical

not only shifts

(44

states are invariant

under arbitrary

of this result to include charged mat-

In this case a general time independent the fields Al by the gradient 22

sum of

(4.7) shows that V . E(T) can

of local, time independent,

From this it follows that imposing

by parts, as

-4 C;(V.E)(j)X(;)

of the link variables

El's over the links touching

(4.6)

gauge trans-

of a function

X(y), but

i

it also multiplies

the matter

the time component

fields by a phase factor

of the conserved electromagnetic

gauge transformation

is generated by the unitary

U(X)

We identify

= e

If we let jo (T) be

current,

then an arbitrary

transformation

ic;[-V.E(~)+jo(j7]X(~)

(4-g)

.

as the generator of arbitrary

-V.E(T)+jo(T)

and once again the statement

gauge transformations,

that physical states are gauge invariant

to imposing tizing

eiqx(;).

as a state condition -&in A0 = 0 gauge.

that Maxwell -

corresponds

equation which is lost when quan. -

The next question which arises is, “ What does the pure gauge theory Hamil_-

tonian

look like when one restricts

attention

One way to answer this question

states?“.

the operators

Et and Al can be written

to the sector of gauge-invariant

is to observe that even on a lattice

as a sum of a gradient

and a curl; i.e.,

Ee=E,L+ET

where Ef

is a function

whose lattice

(4.10)

curl vanishes, and ET is a function

divergence vanishes.

(For details of this decomposition

therein.)

a similar

Obviously,

decomposition

tonian

this decomposition

for non-compact

see Ref.9 and papers cited

holds for At . Furthermore,

clear that the transverse part of Al must be a function With

it is

of Bp alone.

in hand it is a straightforward

Q.E.D.

whose

to write

which has a well behaved continuum

a Hamillimit

as

follows:

H = HO - Z’ C Q’t(J~pKll(~+ r$,)D,(r)eigCtl T,P 23

AS + HT;

(4.11)

c;

stands for the set of links along the line joining

iso = $ c

[Ef”

+ Ea”]

- &

7 and i+

r& where

ems

e

(4.12)

P

and HT is given by

HT= C at (J eigAf Q (j’ +

I;> (eigAT _ 1) .

(4.13)

;+p It is now easy to answer the question just posed, “What look like when projected invariant

iriSthe

does the Hamiltonian

sector of gauge invariant

states?“.

Since a gauge

state is one for which the equation

_.

V~EL(;)-jo(;)(r#)

is satisfied

and EL, by definition,

can be written

Ef

For gauge invariant

Ho = -g

The first term

(4.14)

as

= -VJ.

(4.15)

= -$Vjo

(4.16)

states Ef

so we can rewrite

=O

(4.12) as

c jo(T)$jo(T) j’,i’, is easily recognized

+ $ c

ET2 - -$ L

which 24

(4.17)

P

to be the Coulomb

charges at the sites T and ? (a quantity

xcos(Bp).

rapidly

interaction

between

two

passes to its continuum

I

.

form)

and the remaining

the remaining

terms in the HO are explicitly

be dropped since the only effect of this operator

completely

As for

(4.11) we see that the factor of eiAL can

terms in the Hamiltonian

EL for a gauge invariant

gauge invariant.

is to change the eigenvalue of

state and we have now chosen to rewrite

this change

in terms of the change in the charge density je(y) which is caused by X&t(;) and !I!(;).

the operators

Hence, without

ever fixing gauge, beyond making

the choice A0 = 0, we have obtained the Coulomb-gauge in the sector of physical states. The important 1. The free fermion

form of the Hamiltonian

features of this Hamiltonian

part of th@Hamiltonian -+-

is arbitrary, -

in particular .

are: DP(r)

can be taken to be the damped SLAC derivative. 2. The coupling _-

teraction

to transverse

constitutes

the lattice,

photons, which together with

the entire interaction

that is it is independent

Hamiltonian,

the Coulomb is strictly

in-

local on

of the form of DP(r).

It is easy to verify that with a coupling of this form the results of the previous section for the dependence of the current-current a are retained

and the limit

it introduces compact

theory.

invariance D,(k)

fractional

to the treatment

which

introducing

a straight

QED

flux on links of the lattice and hence cannot be used for a

does not dictate

relationship

on CL,L and

presented in the previous section is that

However an important

and the form

function

(3.14) is satisfactory.

5. Compact The major objection

correlation

physical

a relationship

of the vertex

TP.

leads to a problem

point has been learned-gauge

between the form

of the derivative

In (4.1) we have seen that

when we gauge the SLAC

line of unit flux between the fermion 25

fields.

it is this

derivative

by

i

Although

.

the discussion

us how to modify that

the Lagrangian

there need be no tight

derivative fields.

of the non-compact of the compact

connection

terms and interaction

theory

does not directly

theory,

it points out the fact

between the gauge invariant

between the fermions

a compact

fermionic

and the transverse

Consider then the most general form for the gauge-invariant

tell

gauge

derivative

in

gauge theory:

-A-

-

.

-

where the sum over paths runs over all possible paths from j to j + nj2 and upath is any suitably _-

chosen weighting

factor.

to the choice upath = 1 for the straight

The usual gauged derivative

corresponds

line path and wpath = 0 for all other paths.

Let us instead consider the weighting

Wpath

,-s”+)

=

(5.2)

c lEP

where n,(e) = number of times the path P traverses the link I? in a positive -number

direction

of traverses in a negative direction.

At strong coupling this weighting however for weak coupling examine the coupling Consider any plaquette

(5.3)

is clearly dominated

the string

to any plaquette

by the shortest path-

becomes very non-localized. variable

Now let us

Bp given by this prescription.

and isolate the contributions

to the sum over paths com-

ing from terms which carry flux (p, q, r, -p - q - r) into the four corners of that 26

plaquette.

The coupling

I-p = -;$

2

to BP is

w(p, q, r)

2

(4n + 3p + 2q + ?+-(4~+3p+2q+r)ag2

(5.4

n=-co

(PM)

where Z

=

C

w(p,

q, r)

2

(5.5)

n=-co

(ww)

The factor

e-(4n+3p+2q+r)2g2m

w(p, q, r) is the sum of the weightings

of the parts of the paths which

do not touch the referenc@aqueIke

P (modified

the reference plaquette

For small g2 it is instructive

weighting).

by an n-independent

term from

to rewrite

this

sum using the identity

2

f(n)

=

n=--00

2

7

dx e2amzif(x)

m=w

--co

.

(5.6)

One then finds

1 L,q,r

r(g) = -

g

We see explicitly

W (P, q, r) Cz=,

CPA,? 4P,W)

(2nm) sin 7rm (%+iq+r cos Tm

(1 + c;=1

( 3P+iq+r)

that every term in the numerator

of emr2/g2 than the equivalent

term

) e-m27r2/$2 em2*2/g2)

(5’7)



is smaller by at least a factor

in the denominator.

Hence,

as g --+ 0, I’

vanishes faster than any power of g. This is then a compact

equivalent

result

for rendering

given in the previous

derivative variables

gauge invariant

section;

i.e., it is a prescription

which does not introduce

BP to any finite order in g. 27

a coupling

of the the

to the plaquette

Once again the coupling neighbor

term.

contributions fermions

can be reintroduced

The perturbation form

nearest

theory so obtained will not acquire any spurious 7r, since the coupling

the regions near K +

in this region

using an additional

are not significantly

enhanced

of photons

to

by such an interaction

term. Although

this weighting

of the calculation

leads to a very attractive

just given to a non-Abelian

made use of the Abelian

nature

result,

the generalization

theory is non-trivial.

of the gauge field.

Our counting

Each string

was weighted

by exp(-g2E2)

w h ere E is the field created by the string operator

flux-free

The state created by the action of successive U operators

state.

acting on the on a link

is not unique achieved.

in the non-Abelian case so an equivalent weighting is not readily -). Fortunately, we believe that the strong result achieved above is not

necessary in the non-Abelian --

Consider

instead

case, because such theories are asymptotically

free.

a weighting

(5.8)

I.---

where Lpath is the length to a given plaquette

of the path in question.

variable

Bp, similar

to that

A calculation

of the coupling

given above for the weighting

(5.8) shows that I’ vanishes at least as fast as g 5. In an asymptotically such non-leading

contributions

to perturbation

in the a -+ 0 limit.

28

theory

are presumably

free theory irrelevant

6. Conclusions Although

the calculations

ofthe

preceding sections of the paper were carried

out for a theory of massive fermions a chiral

theory

projecting

from

all fermion

the outset

it is obvious that we could have dealt with

by setting

the parameter

m = mu = 0 and

fields onto fields of a single definite

assert that we established

chirality.

Hence, we

that there do exist schemes for defining purely chiral

gauge theories on a lattice. Unfortunately,

we learned that in order to get results free of lattice

not only does the fermioni?deri&,ive introduction derivative --

of gauge fields has to be written we sum over an infinite

results will be unmodified.

so that for each term in the fermion Obviously,

for

which do not correspond to the straight

of the flux between fermions

or not one uses the Hamiltonian

range but also the

number of string configurations.

strong coupling the string configurations line routing

=.---

have to have infinite

artifacts

are severely damped;

which we have proposed,

The problem,

hence, whether

the strong coupling

as always, is what happens as we try

to approach the weak coupling region of the theory. We will now argue that for couplings on the order of unity there to be a significant written

down and a straight-line

of derivative. multiple

difference between the behavior

Furthermore,

gauging of a theory

we will

string configurations

argue that

of the theory we have

based upon a SLAC-type

the necessity for summing

over

reflects an aspect of the problem which transcends

the desire to get weak coupling perturbation the desire for a renormalization

or less we expect

theory to work out correctly;

group invariant

29

formulation.

namely,

6.1

EXPONENTIAL

We will

DAMPING

show -that starting

simple renormalization just

discussed.

spacing

wave-function

is essentially

_-

degrees of freedom corresponding

carrying

problematic,

Q.E.D.

constant.

factor

transformations.

by an amount

with

exp(-g2rlpath)

non-Abelian

theory

the results obtained lattice

artifacts

theory

kinetic

spacing.

is somewhat

term changes

It is easy to see3 that one

where r is some calculable lines and zero otherwise the fermion

damped SLAC derivative

,U = g2r . This, however, is a disaster since although

g2 + 0 as a -+ 0 it does so only logarithmically; in the previous

sections, in this sort of theory

will survive the continuum

now we are faced with 30

in a

thus, given the fact that

limit would appear to be unavoidable.

In the case of a more general upath the situation reason for this is that

for gluon

for the-remaining

group transformation

term goes over to the form of the exponentially factor

state

straightforward.

For a choice of tipath which is unity for straight

a damping

ground

modes of the gauge field is to modify

this result means that in one renormalization kinetic

a trial

and a larger lattice

for a non-Abelian

out the high-momentum

for a small value

near ~/CL. Integrating

we wish to ask is how the gauged fermion

effect of integrating the weighting

to momenta

the process is entirely

under such a renormalization-group

in the manner

free field wave function

i.e., a theory with a new coupling

for compact

derivative

Since, we are dealing

obtain an effective Hamiltonian

out such a calculation

The question

-.---

that we begin calculating

a Gaussian

GROUP

a SLAC

free theory, we can imagine writing

which

degrees of freedom;

based upon

and a small value of the coupling.

out these degrees of freed-we

While

a theory

group ideas force us to gauge the theory

an asymptotically

and fermion

,from

To see this, imagine

of the lattice with

AND THE RENORMALIZATION

can be quite different.

the question

of what

The

is the mean

i

length of a path joining

terms in the fermionic

question since in computing petition

this quantity

between the damping

factor

paths, and density of entropy factor, given pair of points, paths.

Clearly,

kinetic

term.

one has to take into account the com-

e-g2rL, which

tends to favor the shortest

the number of paths of length L joining

which grows with

a

the length of the path and favors longer

it is quite possible that

as one varies g2 the entropy

win out and the mean path length can grow rapidly. will argue that this will happen for g2 r as follows: some lattice

direction

Of the remaining without

(6.1)

by underestimating

any path joining

the number

two points separated

by distance r must have at least r links along that direction.

(L-r)

Kiiks~ (L9~)/2

any risk of retracing

can be chosen ih any of 2(d 2 1) directions

the same path.

The remaining

(L - r)/2

be chosen to bring the path back on axis and so are not completely follows that there are at least 2(d - 1)(L-r)/2 points separated

by a distance r. This is a gross undercounting to show that

some finite g. For a weighting

such as (5.8) we find

(L)

_ xL,r

r>

random.

It

of the number of

the average path length

Le-g2TL2(d

- 1)(L-r)/2

t: L,,. e-g2rL(2d)(L-r)/2

diverges at

(6.2)

for g2 < (l/27-) ln(2(d - 1))

this

links must

d’1st’mc t paths of length L between

long paths but it is sufficient

Clearly

along

expression

diverges

for all r.

This

establish.

32

is of course the result

(6-3) we wished

to

6.2

SCALES AND CONFINEMENT

PHYSICAL

For a non-Abe&an fixed physical important

gauge theory

weak-coupling

in such a theory.

perturbation

It is possible,

theory

in QED, may in fact

even intuitively

plausible,

average length of the flux strings does not diverge in physical some finite

multiple

by the discussion Imagine

of the physical

of the previous

a recursive

increase the lattice

spacing.

recursive

steps.

finement

means that

large distances I..--

confinement

Eventually,

successively

Starting coupling

with

is suggested

thinning

degrees of freedom to

a form such as (5.8) at small lattice

as the lattice

at sufficiently

spacing increases over many

strong coupling,

in units of the confinement

above the critical

units but is in fact

scale. This notion

there is an energy cost for having

at this point the argument

that the

section.

pdureof

spacing we evolve to stronger --

occurs at some

length scale. This suggests that the fact that (L), + 00, which was

to reproduce

be modified

we believe that confinement

length.

given in deriving

value and so we expect

we expect that con-

gluon flux stretch

Presumably,

this means that

(6.2) applies but with (L),

over

a value of g2

M Lconfinement; hence we expect

the effective form of the fermion

derivative

set by the confinement

Since we believe that this form of the Hamilto-

length.

nian will apply to the study of the motion analysis would suggest that the turning K M 7r means the appearance Although

go over to the damped form on a scale

of quarks within

down of the spectrum

of states with

sions, because they did not appear to have a rotationally invariant

lattice

artifacts-they

spectrum,

for momenta

near

in our earlier discus(or at least approxi-

the states of the recursed Hamiltonian

can be finite mass hadronic

33

our earlier

masses on the order r/Lconfinement.

we referred to states of this sort as spurious

mately)

a hadron

states of &CD.

need not be The reason this

can be true is that coupling

many recursive

and the SLAC

involves off-axis ally invariant. obtained

will,

along the way, evolve into a form which

terms as well as on axis terms and which is essentially We raise this rather speculative

of a strong

coupling

not require an infinite a weighting were that

version

of the SLAC

range for the SLAC derivative

derivative

the existence

of a massless multiplet

massive multiplets multiplet.

of the theory

of particles

The relevance

states are written

of pseudoscalar

of this calculation

in the

.lo Our results

described

is spontaneously -

which we identified

breaking

did

and would be valid also with

such as (5.8). The result of the calculations the chiral symmetry -&-

rotation-

point because of results which we

in an earlier paper where we analyzed chiral symmetry

context

_-

derivative

steps are needed to go from weak to strong

in that paper

broken, .

Goldstone

implying

bosons and also

as a p, p’ and a possible

to the present

one is that

X’

if these

in terms of the “free” quark modes of the coarse grained lattice

then the massless states contain no quarks from the region K M 7r but the massive ones all contain

at least one quark from this region.

The combination

of these earlier results with our admittedly

speculative

ment suggest that those features which are essential for obtaining spectrum

and weak coupling

perturbation

argu-

the correct free

theory may be less important

when it

comes to the study of the hadron spectrum.

6.3

PRACTICAL

CALCULATIONS

Given the discussion which gauging

attempts

to deal with

of the fermion

long strings.

of the previous sections we would argue that any theory

Obviously,

truly

derivative

chiral term

theories, which

must eventually

contains

this is not the sort of term

34

deal with

a sum over arbitrarily

which

lends itself

readily

a

.

to carrying

out practical

implement

our results within

First,

we would

the framework

like to point

out that

then arises as to how to

of a practical

computational

the formalism

scheme.

we have introduced

is

explicitly

Hamiltonian

in nature

and we do not know how to generalize

Euclidean

path integral

approach,

except in the obvious way as a theory of a finite

spatial

lattice

with

continuous

for a spatial

lattice

formulation.

The introduction

time direction

difficult

which

introduce

an asymmetric

so cannot

make any statements limit.

Monte

Carlo techniques

of Blankenbecler

Yet another

couplings

in the

which would seem to be .

formulation

computed

possibility

the Hamiltonian

A more attractive

Naively,

this would

We have not studied

decoupling

of the spurious

alternative

is to adopt

and Sugar”

who work

directly

the with

from the Hamiltonian.

is to eschew Monte-Carlo

theory

methods

about

of a Euclidean

than the space direction,

of the of all propagators.

in the continuum

perturbative

.

terms for the time direction.

states

tack

Hamiltonian

takes a finer mesh for the time direction

matrix

down a Euclidean

of more than nearest neighbor

appear to lead to a single doubling

the transfer

to write

to understand.

and use only nearest neighbor

this theory

it to a

It makes no sense to take our prescription

and use the same prescription

One can, of course, theory

time.

in L lead to a non-hermitian -6.

extraordinarily

_-

The question

calculations.

by strong

coupling

like the t-expansion

techniques perturbation

:3 or whatever

and directly theory

at-

,‘” non-

other method

comes

to hand. In any such approach and study

one’s results

one could work

with

it will

be necessary to truncate

as a function

of the truncation

the nearest-next-nearest

35

neighbor

the full Hamiltonian scheme. fermion

For example, derivative

and

include sums over strings quantities they

which

depend

which

are expected

upon

of approximation

to scale early, i.e. ratios

the parameters

are insensitive

Reliable

p and Lo.

to the choice of these parameters.

and presumably

obtained

can obtain

on the approximations.

believe, from the strong coupling

calculations

that the masses of the true physical -&p and LO.

_-

of masses, and see how answers will The virtue

control

of this sort

over the dependence

Furthermore,

of

we have reason to

which are have studied

states will indeed be relatively -

.

previously,

insensitive

to

A WORD ABOUT ANOMALIES

6.4

Up to now we have studied the question of how to write gauge theory -:--

be those

scheme is that we know the specific form of the Hamiltonian

being approximated the results

of length up to some length Lo. One could then study

for the case of a non-anomalous

we know that chiral

in the continuum

symmetry.

chiral symmetry.

Presumably,

although

carry out the entire construction what

goes wrong

a range of couplings a satisfactory study

which

continuum

of the dynamics

lattice

This is because

it does not make sense to gauge an anomalous it is not apparent

said in this paper, the same is true for our lattice

Probably,

a satisfactory

models;

from what

we have

even though

one can

we have just described for theories of this type.

is that will

theory.

in such a theory

it is not possible

allow us to take the limit Analysis

of this question

a t

0 and define

requires

of models of this type and goes beyond

to find

a thorough

the scope of this

paper. There

is another

the discussion

question

of fermions

related

in lattice

to anomalies gauge theories,

36

which

usually

and that

comes up in

is that

there is

I

a widespread

belief that

symmetries.

It is therefore

correctly

in a lattice

believed that

theory.

which our discussion theory

such theories cannot

chiral current theory.

-.---

The first fact is that for any

there is, even for a current

theory,

an exactly

corresponding

be solved of notions

which would be

conserved lattice

chiral

then goes that this means there must exist an exactly

for a lattice

upon a SLAC

to this charge in the continuum

version of the l+l-dimensional

derivative.

this lattice

theory

continuum

limit;

limit

cannot

limit

charge.

conserved

of the lattice

In an earlier paper by one of us14 it was shown that this argument -rc. -

fallacious

_-

in any global

This belief is based upon a collection

fermions

conserved in the continuum

anomalies

the U(1) problem

has shown to be fallacious.

of massless lattice

The argument

exhibit

Applying

discussion

conserved chiral charge in density

which has a finite

which does have a continuum

by a scale factor

in the continuum

supports

model based

our present analysis to a 1+1-dimensional

the effects of damping

additional

a parallel

of a current

moreover, the only axial current

produce lattice artifacts provide

there is an exactly

it is not the integral

is not conserved.

shows that

While

Schwinger

proportional

theory

to g2 do not

limit of such a theory; hence, our results

for the arguments

presented in Ref.14.. Obviously,

of theories in higher dimension

will require more care and

remains to be given.

Nevertheless,

any non-perturbative

effects which will change the fact that computations

present formulation continuum Another anomalies contributions

of lattice

form if the limits misconception in theories which

with

was

there is no reason to believe that there are in our

gauge theories of massless fermions go over to their a -+ 0 and g2 --+ 0 are taken properly. which leads to the conclusion lattice

fermions,

cancel the ordinary

37

is that

fermion

that one can have no

species doubling

contribution

produces

to the anomaly

calculation.

Obviously,

from what

has been shown in this paper this argument

applies only to the case of the nearest neighbor logical case of the nearest-next-nearest density

neighbor

derivative. derivative

of states in the doubled region is very different

region and no detailed

cancellation

of contributions

Even in the pathowe have seen that the

from that

in the normal

is possible for general values

of /L.

6.5

FINAL

REMARKS

We would like to conclulue byemphasizing presented

In addition,

manner. theories

and the dynamical Hamiltonians familiar

attack origin

of spontaneous

of non-elementary

gauge theories,

the fact that

allows for a more direct is possible

Goldstone

attack

on questions

breaking

Admittedly,

where one explicitly

symmetry

It could

symmetries breaking

breaks these symmetries

the symmetry

is restored.

the

to deal with than the more

of dynamical

to discover how to take the continuum

on physical

bosons.

preserves all chiral

then attempts

In addition

symmetry

however this may be illusory.

this class of theories

in theories

gauge

in a way which preserves global chiral symmetries

which we propose appear more difficult

well be that

than

allows us to discuss vector-like

on the problem

versions of lattice

lattice

of chiral fermions in a straightforward

this same method

of massless fermions

and allows a direct *.--

which we have

in this paper show that it is in fact quite possible to formulate

gauge theories which allow the introduction _-

that the - arguments

limit

and

in such a way that

to this remark we would also like to point out that while we believe

grounds that the complexity

aspect of our discussion

is heuristic

of our Hamiltonian

is unavoidable,

that

and should not be taken at face value. Since

38

this

paper

impossible problem

shows that

something

which

was heretofore

is, in fact, possible we encourage

formulation

of a satisfactory

-)-

theory.

-

39

believed

others to have another

in the hope that they will find something

at a simpler

widely

we have overlooked

to be

look at the and arrive

APPENDIX l+l

Dimensions-Next

A

Nearest ‘.Neighbor

The results of Table I are obtained from the current-current tion given in (3.8). In If1 tions in the continuum

expressions.

performing

= 1) this quantity

theory because the apparently

terms cancel by current lattice

dimension(d

evaluating

the Ice-integration,

A&(&q,) =n C

the traces,

func-

requires no subtrac-

logarithmically

divergent

Hence we make no subtractions

conservation.

After

correlation

Feynman

on our

parametrizing,

and

(3.8) yields

1 b(Kc, -

up -

~8 -

2rn)

2N+1 p,r,n

da

s 0

J’(nr,np)

(A-1)

where [g(G) F(Kr’np)

=

[OZ2(lCr)

-

s(Kp)]

[(l

a)fj2(Icp)

+ (1 -

-

Q) c(Kp)

-

a.zi(rC,)]

(A4

+ m2 - ~((1 - (-r)qiu2)]3/2

and m=mu

,

27r.s 2N+1

P=Ks=

*

(A-3)

We wish to study fixed finite external momenta q. For the sake of discussion we examine

q

> 0, which means 0 < s < N.

the n-sum to contributions contribution

from n = O-the

Then the delta function

non-umklapp

when p + s > N. Thus we can rewrite

restricts

terms, and an n = -1

the sums in (A.l)

as

N-s CGp(nr-np-n,--2ms) P,n

We can further

=

C

6(r-p-s)+

6(r-p-s+2N+1)

. (A.4)

p=N-s+l

p=-N

divide the contribution

5

of the first term in (A.4) by looking at the 40

behavior

of E(K).

We have, for the case of the next-nearest

zj(K,)= yp)

sinn,

In the region of small rc, this function

ii(K)

= lc

+ (’ GP)

SiI12Kr

is well approximated

for

K < fi

neighbor

derivative

(A-5)

.

by the linear behavior

.

(A-6)

We will study the limit

( In this limit

any finite

-i

:

-

-

-&-

P = 6

= 2N~1

for finite 6,

N-t

00.

>

physical

q

(A-7)

satisfies

(A.8)

Thus we can further

divide the sum over p into the regions Ip, < @W + 1) 27r >

In each region of the sum there are an infinite

number of terms in the limit

a -+ 0

and hence we can make the replacement

(A-9) The three contributions

are thus

/“i ARegion I = 2l J‘-4

d/c F(n + WC) 41

(A.lO)

dtc F(/c + ~5, KC>

@rN-s)/2N+l +

and finally

the umklapp

dn F(n + b,~)

(A.11)

dtc I’(% i /cs - 27r, n) :

(A.12)

s fl

contribution,

2rN,‘2N+l

-A ARegion III = G

J

2a(N-s+1)/2N+l

--

All that remains to obtain the results given in Table I is to estimate contributions

from each of these expressions

in the limit

(A.7).

the dominant

To do this, it is

useful to note that

&c +n,)= E(K)+ &I(1

+ /J) cos K - (1 - p) cos 2rc] w

-lo ( 2 1

(A.13)

= E(K)+ n,B(n)+ 0 T4 . ( 1 In Region I we use the linear approximation

A Region I = 3 &l da J-5

Shifting

the integration

variable

(A.6).

This gives

(qa){(l-2cY)rc--aqa} dn I( n+crqa)2+a(l-a)(q2a2-q~u2)+~2]3~2

and exploiting 42

*

the K +-+ -K symmetry

(A.14)

this can

be rewritten

as

2a(lARegion I =

a)qV - q2)a2 + 7TAZ2)]3/2

[n2 - a(1 - a)(qi

fi+aqa +

dZ

J

qa[(l

- 2a)Z - 2a(l-

[E” - a(1 - a)(qi

a)qu]

(A.15)

- q2)u2 + ?&22]3/2

lp-crqa = 2

s

da

m2 - a(1 - cr)(qz - q2)

where the second integral&as tire range. remaining -

2

a(1 - ct!)q2

The first

term

contributions

The contributions

been estimated represents

by setting

2 = &

the usual continuum

vanish in the limit

over its en-

contribution.

The

(A.7).

of Region II can be treated

using (A.13).

We find

(2rN/2N+l)-(I--a)qa

1

B2 (4

4q2u2cx(l - 2a)

ARegion II = 2;rr

dK R3/2 (n, gu)

J fiaq”

+

dn quB(n)[(l

- 2@(n) + 241R3i2 (n, qu)

s &-ffv

a)quB(n)]

(2nN/2N+l)-aqa

dK qaB(n)[(l

+

- 2@(n) + 2a(lR3i2 ( IC, qu)

s (2nN/2N+l)-(1-a)qa

a)quB(K)]

+0 (A.16) where R(~,qu)

= Z”(K)

+ a(1 - a)(q2u2B2 43

- qzu2) + m2u2 .

The second two integrals integrand

within

second integral

are estimated

the range multiplied is bounded

gives contributions

by taking

the maximum

by the range.

The contribution

by a term of order q2u2/p2 while the third

of the integral

of order /J.

The first integral

is best treated

by dividing

tC13

I-cos(fi-qu)

< %]dy

(--$)

+higher

order in p.

(A’17)

= 0 q2a2 (3 CL m In the second region we set cosn = -1 + y. The contribution

of this region is

then

J

479&L

' dy L2c2 - d212 --) o(p) dcY (2N + 1)2 s Y2(l - y)5 E

(A.18)

where

E = 1 + cos

Finally,

27r(N - (1 - Q)S) 2N+l

in Region III, the umklapp

27+(1-

(y.) + ;

(2N + 1)

1 2

= O(qu)2 . *

region, we can expand both E(K) 44

and D(n +

KS - 27r) about their values at n = 7r. This gives

+=+-~~+l)]=;

.I

5(/c + tc, - 274 =

(,,:I)

(-2s+a) P

7r 2N-+-1

(A.19)

1 < 0 < 2(s - 1). Thus the contribution

of the umklapp 2(9-l)

1 ARegion III = F

f 0

region is

-da

&j -I

1

(-d[(’ - do + dS - d] [Q(S - a)2 - (1 - Q)cqW

Hence we have shown that only the usual continuum limit

(A.7).

45

contribution

survives in the

APPENDIX The Damped

SLAC Derivative

B



In order to evaluate the loop contributions we need first

to derive certain

Hamiltonian

properties

using the damped SLAC derivative of the spectral

E(Q).

The

defined in (2.19) gives

Q-4+ 1 2jpp> = 2N

eev4m

where r+, = 27rp/2N + 1 &r-&the -normalization of E(n)

function

+ &-)m

P.1)

C(,U) is chosen so that the slope

is 1 at tc = 0. We are going to study the limt 6 = (2N + 1)2

’ = 6 (;I”

In this limit

/JN is of order l/N,

; N -+ co

6 fixed.

(B.2)

and hence vanishes as N -+ 00. A little

algebra gives n

~$c~)=cK~

I+.LN+~~~~

c

{

r=l

Fr 0)

(B-3)

P- ; -

,F-!‘,

$s2

F(N

+ $ - s) + 0(/-b (pN)2)

where F(r)

=

1 - (-1)’ cos 42 2 sin2 n,/2

(B-4

Now 2N1+ 1 fl:

F(r)

= ““,:

r=l 46

’ + o(1)

(B.5)

i

and 2N1+l

(B-6)

CF(ti++-s)=e+O(l).

Hence i(n)

=

n -

,,,“”

n>

+

P-7)

0 (PU, p2N2)

Thus,

dE .-I-

2iJ 7r(7r - rc)2

dn -&-

The maximum

- O(/w2N2)

(B.8)

-

-

-

occurs at

(71. - K)= E

.

(B-9)

--

For K greater than ris of width

the slope of E(K) is negative-this

and hence contains

+ 1) even in the limit

2rm/(2N ‘.I-

l/N

dm

only a finite

N +

00. The minimum

spurious region occurs at KN = 27rN/(2N 4/..+N 5(&N)

=

KN

-

The current-current cally divergent We will

r2

+ 1)

correlation

in the continuum

+ O(p,p2N2)

function theory

contributions

(B.2).

K,

=

in the

Eq.

(B.10)

.

(3.8), is quadrati-

and hence requires three subtractions. and study

in Table III, we adopt the following

1. evaluate the traces, Feynman parametrize

47

value of Z(K)

> RN - 2pN

of interest,

I n order to define precisely

grations;

of momenta

+ 1) and is given by

make the same three subtractions

in the limit

number

spurious region

what

the resulting

expression

we mean by the various

procedure: and perform

the (finite)

~0 inte-

2. divide the sums over each of the spatial

momenta

into the regions given in

Table II; 3. treat each such region as a separate expression to be three times subtracted at

(q0,G)

=

0.

This yields the results of Table III. (Because the boundaries are q-dependent

the sums in the subtraction

than those in the original correctly,

terms may run over a different

Of course, provided

the subtraction

the answer does not depend on how the contributions

between

entries

subtraction

on our tsbJe.

procedure

a one-dimensional --

expression.)

of the various regions

Consider

However

it is important

treats the umklapp

contribution

example of a twice subtracted

the unsubtracted

is done

are distributed

to note that separately.

a correct

Let us outline

sum just to make all of this clear.

expression

B’(s) = xF(

where q = 27rs/2N

range

~2,K~)~~(Kz+ h + 27rn - K,)

+ 1

N-s

N

-N

N-s+1

(B.ll) + c F(~p,~p+~.s-2~) . B’(q) = c F(IC~,K~+K~)

Now subtract

twice.

The contribution

of the first sum in (B.11) is

N-s Bgegion

I = c -N

-

F(~P>"P

+ d

-

f&'bp+p)

-N

27rs 2N+l

N d c-Ndn,

F(np,

up +

+ 2FJ1

G>

F(KN,KN)

r=O

(B.12) 48

While

the second sum gives

B~gion

11 =.

~

F(~~p,np + US - 2~) - 2~~

.

~F(~N,K-N)

(B.13)

N-s+1

Since the umklapp

region is of width

of order q before any subtraction. term which is different

q the contribution

of Region II is explicitly

The second subtraction

for the umklapp

and non-umklapp

introduces

a boundary

contributions

and hence

does not cancel out when the two regions are added. The results by-step

given in Table III are then obtained by a straightforward step-)procedure. In the spurious regions the denominator E(n) is replaced

by the bound _-

upper bound subtractions

(B.lO) (Ifi

w h’1l e numerator < r).

factors

of E(K)

In both the normal

can be replaced

and the umklapp

the sums can be replaced by integrals

by the

regions, after

using the prescription

2rm’/2N+l

1 2N$1

m’ 1 c p=m+2?r

J

dnp .

(B.14)

2rm/2N+l

However, the number of states in the spurious region remains finite in, the N + 00 limit,

hence these regions must be treated

49

as discrete sums even in this limit.

REFERENCES 1. H. B. Nielsen-and Masao Ninomiya, sen and M. Ninomiya, Ninomiya, Nucl.

Nucl.

Phys. B193,

Phys. B185,

2. L. R. Karsten 3. J. M. Rabin,

Phys. Lett.

Phys. Lett. 105B,

130B, 389 (1983); H. B. Niel-

219 (1981); H. B. Nielsen and M.

173 (1981); H. B. Nielsen and M. Ninomiya,

20 (1981)

and J. Smit, Phys. Lett. Phys. Rev. D24,

85B,

100 (1979).

3218 (1981).

4. As usual in this statement we ignore the fact that the renormalization -). group applied to Q.E.D. implies that g + 00 for a + 0, that is we are talking -.

of a naive a +

in formulating

0 limit.

asymptotically

This is because we are actually free theories for which

a +

interested

0 and g + 0 is

indeed the relevant limit. 5. Note that 1..-

infinite

in the limit

a --+ 0 any finite

value of the dimensioned

what we call the spurious 6. Paolo Nason, Nucl. . 7. An alternative

value of &(Ic) corresponds

energy &/a; hence, in this limit

Phys. B260,

269 (1985).

procedure for making the range of the Hamiltonian

sharp cutoff causes oscillations

of the form shown in Fig.

will limit

3. This procedure

of the lattice

exponentially all further

finite is

because any

which have to damp out. Nevertheless,

procedure is possible and leads to an energy-momentum

smoother,

states in

region indeed have finite energy.

to cut the sum off sharply for Ij - 21 2 Lo. Th is is undesirable

in his treatment

to an

Schwinger

discussion to this case. 50

dispersion relation

was discussed by P. Nason

model.6

damped derivative,

this

For our purposes the

is easier to deal with

and we

8. S. D. Drell,

M. Weinstein

and S. Yankielowicz,

Phys.

Rev.

D14,

1627

(1976) 9. S. D. Drell, H. R. Quinn,

B. Svetitsky

and M. Weinstein,

Phys.

Rev. DlO,

B. Svetitsky

and M. Weinstein,

Phys. Rev. D22,

619 (1979). 10. S. D. Drell,

H. R. Quinn,

490 (1980) and Phys. and S. Gupta,

Phys.

11. R. Blankenbecler

Rev.

D22,

Rev. D26,

1190 (1980); H. R. Quinn,

3689 (1982).

and R. L. Sugar, Phys. Rev. D27,

12. T. Banks, J. Kogut,-S&abyTD.

S. D. Drell

Sinclair

1304 (1983).

and L. Susskind,

Phys. Rev. D15,

1111 (1977). 13. D. Horn,

M. Karliner

and M. Weinstein,

D. Horn and M. Weinstein, Doe1 and D. Horn, submitted 14. Marvin

Weinstein,

Phys.

Phys.

Rev. D30,

51

2589 (1985).

1256 (1984). C. P. Van Den

to Phys. Rev. D.

Phys. Rev. D26,

Rev. D31,

839 (1982).

FIGURE

CAPTIONS

1. (a) The spect rum of the nearest neighbor derivative .I

spectrum

doubling.

(b) The spectrum

(L = 2N + 1 = number of sites) of the next-nearest

the larger slope and correspondingly

neighbor derivative lower density

The energy of the last state is approximately (c) The spectrum

of the next-nearest

(Note the change of vertical 2. (a) The spectrum (L=2N+l=

(CL= 1) showing true

The

for p = 0.1. Note

of states for tc near 7r.

r/pL.

neighbor

derivative

for p = 0.01.

scale.)

dXmped SLAC

derivative

for the case p = 0.01.

of the damped SLAC derivative

for the case p = 0.002.

number of sites)

(b) The spectrum --

Note the shrinkage of the spurious region. (c) The spectrum

of the damped SLAC derivative

3. Fermion loop graph encountered in the computation correlation

for the case ~1= 0.002. of the current-current

function.

4. fermion loop encountered in computation tion

52

of current-current

two point func-

i

Table

-Region



I

Contribution

.I

-

Usual continuum

-?ru+$