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).
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12. T. Banks, J. Kogut,-S&abyTD.
S. D. Drell
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1304 (1983).
and L. Susskind,
Phys. Rev. D15,
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D. Horn and M. Weinstein, Doe1 and D. Horn, submitted 14. Marvin
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Rev. D30,
51
2589 (1985).
1256 (1984). C. P. Van Den
to Phys. Rev. D.
Phys. Rev. D26,
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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+$
