Space-time is usually treated as an arena for dynamical theories, even when one takes into account the Einstein approach to gravity as geometry. To better.
SLAC - PUB July 1983
- 3156
T/E
SPACE-TIME
: ARENA
and
VADIM KAPLUNCVSKY+ Stanford Stanford
Linear
University,
OR ILLUSION
MARVIN
Accelerator
Stanford,
?*
WEINSTEIN
Center
California,
94 305
ABSTRACT
a-
This paper develops a framework
which allows us to treat the topology and
dimension of the space-time continuum
as dynamically
examples of quantum systems which are defined without which nevertheless undergo a transition
We present
a notion of space, but
to a space-time phase. The dimension
of the space is an integer valued order parameter
which characterizes
phases of a single system. We also show the interactions particles of the system are gauge-like.
generated.
distinct
between the low energy
Finally, we discuss the computability
of
Newton’s constant in this class of theories.
Submitted
to Physics Review D
* Work supported by the Department of Energy, contract D E - AC 0 3 - 7 6 S F 00 5 15 + Address after September 1, 1983: Department of Physics, Princeton University, Princeton, N. J. 08544
---.a,
1. INTRODUCTION Self-consistency
problems in many areas of physics have forced theorists to
try to achieve a better
understanding
scale. Such considerations,
whether
of phenomena
they have to do with the structure or with applications
early universe and the flatness problemf solving the hierarchy gravity
satisfactory
of quantum
field theory.
of the
of supergravity
necessitate that we learn to incorporate
problemf
into the structure
at, or near, the Planck
Unfortunately,
to
Einstein
to date, no
way of doing this has been proposed.
Many talented physicists have failed to solve the problem of renormalizing .--
conventional
field theory of gravity plus matter,
of “fine tuning” the cosmological constant. for a reappraisal sumptions.
and the concommitant
This paper presents the result of one such reappraisal. quantum
problem
For this reason we feel the time is ripe
of basic tenets with an eye towards eliminating
is a way of formulating
a
unnecessary asWe find there
field theory which violates our most cherished
beliefs and yet appears capable of describing physics as we know it. The notion abandoned
as superfluous
to a quantum
sional space-time continuum. out with a well-defined undergo a transition
field theory is that of the four dimen-
There seem to be quantum
systems which start
notion of time but no notion of space, and dynamically
to a space-time phase -
a phase in which the physics of the
low energy degrees of freedom of the system are best described by an effective Lagrangian
written
in terms of conventional
notion of the four dimensional
relativistic
space-time continuum
fields. In this sense, the as the arena within
which
the game of field theory is to be played is replaced by the notion of the space-time continuum
as an illusion of low-energy dynamics.
On the face of it the idea that one can abandon relativity 2
and the space*--.a,
time continuum
as superfluous
baggage seems ridiculous.
For this reason we
begin with a discussion of some of the ideas which led us to this approach. go on to describe the class of Hamiltonians
We
suggested by these considerations,
and present the methods we will use to analyze such systems. In particular, anticipate
results of the sections to follow and discuss the notions of dimension
and scattering time.
we
for a theory formulated
The next section introduces
without
reference to an underlying
space-
simple examples and argues that for these
examples low energy physics is correctly described by effective lfl-dimensional relativistic obtaining
field theories. This is followed by a section devoted to the problem of higher dimensional
space-times.
-I- lead to effective 2 + 1 and 3 + 1 relativistic this problem
is solvable.
time dmensions, internal Hamiltonian.
We present toy Hamiltonians
which
space-times, thus demonstrating
We also discuss the relationship “flavor” symmetries,
that
between the space-
and of possible phases of a given
The last section summarizes results and presents a list of caveats
and speculations about those aspects of the problem not yet studied in detail.
2. FUNDAMENTALS 2.1
MACH'S
PRINCIPLE AND THE COSMOLOGICAL CONSTANT.
A suggestive way of looking at the problem of “fine tuning the cosmological constant”
is that it might be related to the fact that the Einstein theory does
not successfully embody Mach’s principle:
which asserts that the local inertial
properties
by the other (in particular
matter
of matter
should be determined
in the universe.
a “test particle”
distant)
This assertion implies that in a really empty universe
should not know how to move; as a corollary, 3
an empty space ---.*
should have no geometry
at all. On the other hand, the Einstein equations allow
for a Aat space geometry
in a universe devoid of matter and radiation. This failure of Einstein’s theory to incorporate
Mach’s principle
is likely to
be related to the problem of “fine tuning” the cosmological constant, which arises when one attempts
to couple quantum matter fields to general relativity.
Quan-
-I- turn fields generally have a non-zero vacuum energy, whose contribution
to the
energy-momentum
tensor is equivalent
Unless this contribution mological a tightly
constant,
i.e. “fine tuned away”, Einstein’s
space-time curls up into
In contrast to this situation,
a theory which obeys
and yields no geometry at all for a truly empty space, i.e. de-
void of even quantum fluctuations, effects are taken into account. natural solution.
constant.
is precisely cancelled by an equally enormous bare cos-
curved manifold.
Mach’s principle
to an enormous cosmological
might yield a flat geometry tll when quantum
In this case the flatness problem would have a
However, now we have to solve another problem:
How to arrive
at such a desirable state of affairs? As a starting point we focused upon the fact that our ideas of space-time are derived by carrying
out measurements with clocks and rulers, the latter being
replaced by radar sets in more modern expositions
of the subject!
In other
words, our notions of space and time are the result of carrying out a large class tfl We mean, a flat geometry for the background space-time metric of the low-energy physics. 4
of scattering
experiments.
The necessary input to this constructive
approach is
the existence of some objects that can be scattered and a Hamiltonian describes the scattering process. Space-time, if it is an appropriate low energy physics, then emerges as a construct.
which
description of
This sort of argument naturally
leads us to inquire as to what role the coordinates
x,, play in conventional
field
theories. 2.2
WHAT ARE THE
x,'s?
Space-time is usually treated as an arena for dynamical
theories, even when
one takes into account the Einstein approach to gravity as geometry. 2- appreciate theory. written
this fact, let us reexamine the usual formulation
The Hamiltonian
for a self-interacting
To better
of quantum
field
scalar field in m-dimensions
is
as H=
I12(x) + pq2
What is the significance
of the variables
(x) + v (Q(z))]
(2-l)
*
5,’ which appear in this formula?
At the level of the canonical commutation
relations the xP’s are nothing but labels
for independent operators Q(x) and II(x) which define the theory. A priori
there
is no reason for xP’s to have anything to do with space and time. However, the way the gradient terms appear in (2.1) ensures that space-time defined by this Hamiltonian
leads us back to the same xP’s (up to a Lorentz transformation).
This rather non-trivial and diflerentiable
result implies that a preconceived notion of the topological
structure
is associated with the underlying
variables
and that
certain aspects of xP’s must be taken into account before we define the quantum problem.
In fact, great care must be taken in renormalizing
the theory so as not
to do great violence to this relationship. 5
*_ -.a,
At first glance Einstein’s theory takes care of these troubles.
In this case we
consider an action of the form
L =
which is invariant
/
dmx e
[ R(X)
+ matter terms ]
with respect to arbitrary
coordinate
(24
,
transformations
of the
variables xP. This invariance seems to reduce the significance of the xP’s to their role as labels for independent
quantum
degrees of freedom.
point of view this approach has another beautiful the background
From a heuristic
property
to recommend it: if
metric of space-time arises as a dynamical
effect, then there is
2a natural
reason why the space-time reconstructed
must come out intimately relativity,
from scattering
experiments
related to the variables xP. The equations of general
and therefore the Heisenberg equations of the quantum theory (assum-
ing that the theory can be quantized) form a set of hyperbolic equations.
It is known5
that for this set of differential
partial differential
equations the causal
hypersurfaces of the combined system of fields are determined persurfaces of the universally
by the causal hy-
coupled spin-two field, i.e. gPv . ( By the causal
surfaces of the fields, we mean the submanifolds the fields may be discontinuous,
of the space-time across which
i.e. the light cones of the theory. )
It follows
that if gPV develops a ground state expectation value, iPV, then solving the quantum problem for the matter fields will reconstruct those specified by iPV. If in particular by scattering
a theory whose light cones are
ijPV = qllV, the space-time constructed
the matter fields will be the desired Minkowski
to be a highly attractive theory; unfortunately
scenario, modulo difficulties
in defining the quantum
this heuristic picture isn’t completely 6
space. This seems
correct. ---.a,
Although
going over to the Einstein theory of gravity removes the geometri-
cal significance of the coordinates
xcL, the fact that they specify the differential
topology
In other words, although
of the system remains.
tance has become a dynamical
concept, we still need to introduce
of ‘neighborhood’ and ‘deriuatioe’ as primitive, assume the xP’s to be coordinate a certain topological primitive
a differentiable
structure
the notions
i.e. undefined, terms.
patches on a certain manifold,
and differentiable
notion of differentiability
or a Hamiltonian,
the notion of dis-
When we
we establish
for the -space-time.
Since the
must be defined before we specify an action
the assumption that we are dealing with a theory defined on manifold carries non-trivial
content.
.- 2.3
BASICS.
Motivated
by these considerations, we explore the question of whether formu-
lating a theory so that the analogs of the xP’s are reduced to labels for quantum degrees of freedom6 Mach’s principle
will allow us to progress towards the goal of incorporating
into field theory. We begin by introducing
J =
We then introduce
(l:l=l,...,N)
a set of quantum
“harmonic
.
oscillator”,
ables II(Z) and Q(Z) , satisfying canonical commutation
and multicomponent
an indexing set
or “bose field”, vari-
relations
“Fermion field” variables \E, (1) and X4!/:(1) , satisfying canon7
*--.a,
ical anti-commutation
Intuitively,
relations
these quantum degrees of freedom are to be thought of as playing the
role of site variables in some sort of lattice field theory. In addition to these site fields we also introduce link j&Z&, which for the sake of definiteness we choose to be harmonic oscillator variables Pap and XaP (lm) ’ The Um) notation ‘(Zm)’ is introduced to stand for an arbitrary pair of integers I, m E J, ordered such that 1 < m. These link fields are assumed to satisfy the canonical .-- commutation
relations
The role of the link fields is to provide dynamical
variables which allow us to
write the analogue of a kinetic term for a Hamiltonian notion of differentiation
or even neighborhood
without
to be defined.
Starting with these objects we consider a Hamiltonian
H
=
c
[$ H”(Z) + Xfm@(I)
- iXl,a
a(m)
of the form
+ V(Q(Z))
{ Q+(Z) K@(m) -
‘E+(m) Q(Z) }
+ f P”r,, Note, to simplify
requiring that the
the discussion we will temporarily
+ v (- * - X(1,) * * *) ]
.
treat the fermion fields as
one component objects and drop the Dirac indices CYand /3. We also introduce a 8
---.a,
matrix
notation
for the variables X(lm):
Xl,
and define X”,
=
- Xml
X(1,)
for
1< m
,
Xtrnl)
for
m < 1
;
1
=
(2.4)
to be Xfm
=
c &k kcJ
As it stands, (2.3) is the Hamiltonian
Xkm
-
(2.5)
of a complicated
quantum mechanical
system all of whose couplings, for want of a better choice, are assumed to be of order unity. .--
The idea we wish to pursue is that, for some dynamical
subset of the variables X(1,) expectation
acquire non-vanishing
_ the system and, to the extent that fluctuations map the matter
the derivative
sort
of lattice
forms?“.
The terms involving
values are identifiable
term in this effective lattice Hamiltonian.
X(lm) ?“, and “What
fields” in
in the fields X(1,) can be ignored,
expectation
drives the dynamical generation
in the fields
for the “matter
problem into some sort of lattice theory.
variables X(lm) which have non-vanishing
are: “What
ground state (i.e. vacuum)
values X(lrn) . When this occurs, the terms quadratic
6(l) and K@(Z)yield a solvable zeroth order Hamiltonian
reason, a
of expectation
as
Two main questions values of the fields
While the meaning of the first
question is self-evident, the second merits a brief discussion. If the ground-state
of a Hamiltonian
to the vanishing vacuum expectation geometry (or effective lattice theory) -
of the form defined in (2.3) corresponds value of the fields Xtlrn) , no zeroth order reasonable or unreasonable -
forms. In
other words there is no sense in which the matter fields scatter in a background determined
by the vacuum expectation
acts on an equal footing
value of the link fields; everybody inter-
with everyone else. On the other hand, if a pattern 9
*--.a,
of vacuum expectation
values does develop, then there is a zeroth order matter
field theory whose arena of definition non-vanishing
expectation
values.
is specified by those link variables with
Our problem is to determine
if there is any
reasonably simple way of choosing terms for (2.3) so that the resulting becomes equivalent proximating
to a Hamiltonian
that of a relativistic
lattice theory with dynamics closely ap-
continuum
theory.
to show that for a large class of Hamiltonians second possibility
The aim of this paper is
of the type specified in (2.3), the
is realized. In addition, we will show that simply by increasing
the number of degrees of freedom the resulting approximate
theory
a relativistic
continuum
lattice theory can be made to
theory arbitrarily
well.
.a2.4
BISONS
VERSUS FERMIONS AS FUNDAMENTAL
Before discussing specific calculations,
BUILDING
let us see if we can intuitively
derstand those aspects of the theory which control the dynamical ground-state v(*‘*X(lm)
expectation values for link-variables. *.‘)
9
can determine
BLOCKS un-
generation of
Obviously the potential terms,
whether the theory develops non-zero expecta-
tion values of the link variables; however the vacuum energy of the matter fields can play a similar
role.
The possibility
generation of non-vanishing
that matter
link-field expectation
spective of the details of the link Hamiltonian, we will argue this possibility tem are fermionic.
is very appealing.
In a moment
matter fields in the sys-
we will find that fermionic
stabilize the resulting space-time structure link variables.
values by themselves, i.e. irre-
is realized if the dominant
Unfortunately
fields cause spontaneous
fields alone cannot
against quantum fluctuations
This will lead us to consider generalized plaquette
of the
terms in the
--
link-field t
potential
(we will define them later), which can produce stable space10
*--.*
times. These generalized plaquette terms will play a crucial role when we address the question of obtaining
3+1-dimensional
space-times, and when we discuss the
way in which these theories generate effective low energy gauge theories. Despite the fact that the matter fields cannot, by themselves, play the decisive role in determining
the structure
of space-time, it is pedagogically
advantageous
for us to begin our discussion of the general theory with an analysis of the role they play. This will allow us to introduce, important
but unfamiliar
in the simplest context, most of the
concepts we will need in the discussions to follow. Let
us begin by focusing on the effective quadratic problem defined for the fields 9(Z) in (2.3). Since we are dealing with Bose fields, their contribution
to the vacuum
-I- energy is & boson = -
--
fc+:
7
P-6)
where by 6~ we mean the eigenvalues of the matrix familiar
. ( This is just the
sum over the zero-point energies of a set of independent harmonic oscil-
lators. ) The contribution expectation
coming from (2.6) is a positive function of the vacuum
values xflrnj , and is minimized when all EA’Sare equal to zero. This
occurs if and only if the ground-state vanish.
x:m
expectation
values of the link-variables
Hence, the Bose site fields play no interesting
role in our problem and
we will ignore them in all discussions to follow. The situation
is quite different for fermions because their contribution
vacuum energy is negative. Computing energy reduces to diagonalizing -
I.
By definition,
iXl,
the matrix
the fermionic contribution
to the
to the vacuum
the quadratic form
- [Q!+(Z) Q(m)
of expectation
-
‘E+(m) Q(Z)]
.
values Xlm is antisymmetric; 11
(2.7)
hence, it
has an equal number of positive and negative eigenvalues (it may have some zero eigenvalues as well).
Since the ground state of the theory is obtained by filling
the negative energy sea, it follows that the fermionic contribution
to the vacuum
energy is the sum over the negative eigenvalues of Xlrn , or
&fermion =
From (2.8) we see that even if the potential ical generation
of non-vanishing
(2.8)
-&q.
terms in (2.3) do not lead to dynam-
vacuum values for link-variables,
the fermions
perforce generate such an effect. What we have to study is whether the fermionic -I- contribution
alone is sufficient to stabilize the effect against quantum fluctuations
of the link variables.
2.5-
LINK MEAN FIELD THEORY.
Assuming that our generic Hamiltonian we turn our attention ing the properties
only includes link and fermion fields,
to the tools we will use to analyze the theory.
of the ground-state
the Bose-fields, is an extraordinarily proach is to adopt a variational
of a Hamiltonian
problem.
Our ap-
technique which has proven to be quite effective
this method, a variant of the familiar
Rayleigh-Ritz
mean-field theory.7 While variational
successful when applied to the calculation
the order of a phase transition,
like (2.3), even without
difficult non-perturbative
when applied to lattice systems, namely, Hamiltonian
particularly
Determin-
technique, is not
of critical
exponents or
it is usually quite successful in determining
what
phases exist. Since this is the only way in which we will apply this method, we feel comfortable
using it in its simplest incarnation. 12
---.w
The approach, as we will implement
it, is to choose as a trial wave-function
for the ground state of (2.3) ( minus the Bose fields) a product state of the form
I’,)
= n exP (Vm)
The variational
parameters
a ( x(lm)
appearing
C(lm) , which determine the expectation
-
-
’ [*fermion)
C(lm))2)
in the wave-function
P-9)
are: the variables
values of the operators X(lm) ; the vari-
able 7, which determines the width of the gaussians, and therefore the expectation value of operators such as P$,,
and ( Xflm, - C6m, ) ; and the unspecified state
I*fermion ) . To determine whether or not the link variables have non-vanishing expectation
values in the ground-state
of the system we simply compute the ratio
.---
&effec the ( C(lm) 9 7 9 I*fermion) ) = _
and minimize it as a function 2.6
(Q’varlH 1%x) (*varlQvar)
’
(2.10)
of the parameters C(lrn) , 7 and I\kfermion) .
SCATTERING PROBLEM.
What do we mean by discovering a scattering problem hidden within the theory specified by Hamiltonians
of the form (2.3) ? Our approach is quite straight-
forward and not necessarily the most general one; nevertheless, it will suffice for our purposes. We will show that for the cases of interest there exists a function F(Z,S) of the E UCl’1d ian coordinates 5 and points 1 E
J,
which allows us to define
,
(2.11)
the effective fields @I(5) as Q(Z)
= c
F( Z,Z)xlqZ)
IEJ
such that they have canonical equal time anticommutation of infinite .-
number of quantum
degrees of freedom. 13
relations in the limit
Furthermore,
we will show
that the quadratic
part of the effective fermionic
Hamiltonian
can be written
in
terms of Q’(Z) as H fermion =
Given these “continuum” Q(
.
(2.12)
fermionic fields, we define the time dependent operators 3,
and use these operators
dxXD+(Z) r;-a’,Kl!(Z) I
t )
=
e-itHfermion
\zl(,jZJ
(2.13)
eitHfermion
to define wave packets and scattering
states, etc. The
fact that the Heisenberg equations of motion for these fields comes out to be the usual relativistic
equations guaranties that the space-time reconstructed
using
.-- them will be of the desired type and that its dimension will be (d + l), where .‘J
Edis the dimension of the variables 5, provided the residual interactions the effective low energy degrees of freedom look approximately -
x-coordinates.
interactions,
it is worth
answers to the question of the
noting that there are in principle
in which they can be approximately continuum
local.
IG-/11dxi i
@t(xI)
*(x2)
where the support of the function
!& =
{(~1,x2,-~,
for interactions
two ways
In any lattice approximation
theory there appear non-local, non-linear
One possibility
local in the same
It remains to be shown that this is the case.
Despite the fact that we have no definitive residual
among
interactions
. . . \E(x%) - A(x1,x2,.
. . ,x,)
to a
of the form ,
(2.14)
A (xl, x2,. . . , xn) extends over a set
x,):
(xi-xj(
16
for i,j=l,...,n)
(2.14) to look approximately
local is to include
only terms with small values of 6 (as measured in the physical units), 14
so that ---.w
all violations
of locality occur at separations too small to have been resolved to
date. The second possibility have very small magnitude
allows for large values of 6, but only for terms which IC. These terms are potentially
more dangerous, since
if the support of A(xr, . . . , xn) extends over the whole set, then the magnitude of K must be very small.
3. TWO-DIMENSIONAL 3.1
THE SIMPLEST SPACE.
To get a feeling for the variational .--
SPACETIMES
discuss the results of such a calculation,
procedure and the way in which we will let us begin with the simple Hamiltonian.
,i
H = C [ : P{,)+ c Xbrn) - ix(lm) -{ Q+(l) Q(m) -
.
Q+(m) *l(Z) }]
_ W (3.1)
The virtue
of this Hamiltonian
using straightforward uninteresting.
is that we can completely
analyze our problem
analytic techniques; the drawback is that the results are
Using the general trial wavefunction
-ic(lm)
specified in (2.9), we obtain
‘( \k fl Xl?+(Z)Q(m)
-
KP+(m) Q(Z) I!Pf) ]
.
(3.2)
The variation
over 7 is easily carried out, yielding rrnin = p, which leaves us
with the problem of minimizing J
over the class of all purely fermionic
IKPf). Since the fermionic term in (3.2) is purely quadratic Iqf)
is obtained
in the fields, the best
in the usual way, namely by diagonalizing 15
functions,
the matrix
Ctlrn) , ‘--.a,
expanding
the fields Q!(1) in terms of the eigenfunctions
UX(Z) and filling
the
negative energy sea. To be specific, if the normalized functions UX(Z) satisfy
where icx are the eigenvalues of the matrix Cl,, combinations
we introduce orthogonal
linear
of the operators Xl?(Z)
xP(X) = c
u;,(Z) Q(Z)
In terms of these operators the fermionic Hamiltonian
where the variables *t(A) as the original
.
can be written
and @(A’) satisfy th e same anti-commutation
fields. If we define IO) to be the state annihilated
operators Xl!(X), then the ground state of the Hamiltonian applying to it all of the operators *t(A)
as
relations
by all of the
is obtained from IO) by
f or which the eigenvalue EA is negative.
The ground state energy is
& vacuum
Returning
=
-:c,&
(3.3)
’
to the discussion of (3.2), we observe that the matrix
Cl,
is an-
..tisymmetric -
and can, by means of a real similarity 16
transformation,
be brought ---.a,
into the form
It then follows that the variational
energy can be written
as
(*,,I H Iflj’,) = “‘“s- ‘) - (7 + %> + $ c c”x---f c fi x x
- c3s5)
From (3.5) it is clear that not only does the variable 7 come out independent N, but that the minimum *-
mon N-independent advertised,
is achieved when all of the eigenvalues have the com-
absolute value 1~1~1= fip2.
the fermionic contribution
This analysis shows that, as
to the ground state energy forces at least
some link variables to develop non-zero vacuum expectation tion “to what space-time
of
does the resulting
values. The quescan be partially
theory correspond?”
answered by saying that to the degree this theory has a space-time, its dimension is 1 + co. Let us explain this rather cryptic remark. Following the discussion in the preceding section, we extract the space-time dimension from knowledge of the energy spectrum by inverting transform
procedure.
fermionic
Hamiltonian
so a d-dimensional
the usual Fourier
This works because it is in the momentum (2.12) is diagonal.
Hamiltonian
For a massless theory
basis that the c(k) = fl,
has a density of eigenstates which behaves like
P(E) 0s: ft‘-’ - Hence, if we order the eigenvalues of the fermionic Hamiltonian
Cl,
in ascending value and plot them versus the integers j = -n, . . . , nf2 we can read off the dimension from the detailed shape of the plot:
in a d-dimensional
case
-112We have chosen an odd N = 2n + 1. .-
17
‘--.a,
c(j) is proportional
to fi.
It follows that as d + 00 the energy c(j) tends to
the same value for all positive j ; hence this simple system is infinite-dimensional. There are two major lessons to be extracted behuue us expected and cause the vacuum
from this exercise: first, fermions
expectation
to become non-zero;
and second, jermions
to high dimensions.
The real problem is to understand
value of the link variables
by themselves tend to push the system
how to get a reasonable
system of low space-time dimension.
3.2
MAKING
A ONE-DIMENSIONAL
SPACE.
The results to be presented in the rest of this paper were obtained by means of a mixture
of numerical and analytic techniques. Although
the results of analytic
methods alone would suffice at this stage of the discussion, we feel that comparing them with the results of computer calculations may lead to some useful insights. Moreover, some of the speculations which are presented in the last section of this paper are based solely on the computer
results obtained for small N.
For this
reason we have chosen to present the material to follow in semi-historical
fashion.
Inspection of the simple Hamiltonian come out equal in absolute magnitude: of the matrix matrix
Cl,.
the vacuum energy is a sum of invariants
The term multiplying
Cfrn , and the fermionic
determinant
(3.1) reveals why the eigenvalues tend to
the variable p2 is the trace of the
energy is proportional
to the logarithm
of CFrn . This leads to the conjecture that the situation
if one destroys this property
of the Hamiltonian,
of the
will change
and a more connected pattern
..of vacuum expectation .-
values will develop. In order to test this hypothesis we 18
---am
study a slightly more complicated
H
=
C
[i’l$m)
+
VxT[m)
Hamiltonian,
-
ix(lm)
‘{
namely
Q+(Z) Q(m)
-
Q+(m)*(Z)
} ]
.
(14 P-6)
We are interested (3.6).
in this problem for r > 1, since the case r = 1 reduces to
Our early attempts
the parameter
to understand
this problem began by assuming that
7 in our trial wavefunction
was large and constant.
studied the vacuum energy of the system as as function alone. In this case finding the minimum
We then
of the parameters C(lm)
of the function
P-7) is no longer a procedure which can be carried out analytically: _
unlike the pre-
ceding case, the term
cannot be expressed as a function
of the eigenvalues alone. For this reason we
turned to a computer analysis of the problem for values of 11 < N 5 121. This analysis led to the surprising imized (3.7) h ave non-vanishing
result that the configurations
vacuum expectation
which min-
values of all link variables
Xtlrn) ; moreover, absolute magnitudes of the expectation values tend to be equal:
lvaclx(lm)
Ivac>=
&x.
On the one hand, this means that our original conjec-
ture was correct and breaking the invariance of the Hamiltonian mation of more interesting these configurations able lattice theory.
highly connected configurations;
forced the for-
on the other hand,
seem to have nothing in common with any easily recognizOn the contrary,
these systems appear to describe a set 19
of fermions living upon what a mathematician
would call an “N-simplex”
i.e.
an object for which every point 1, is a nearest neighbor of every other point 1’. Naively an object of this sort appears to have dimension (N - 1). At this point, it is important
to recall that the dimension of the corresponding
space-time picture can be obtained only from an investigation eigenvalues of the matrix
of the spectrum of
I vat). In order to better understand
(vacl X(1,)
what is
happening we must study the spectrum of eigenvalues for fermionic Hamiltonians of the form H*elmplex =
t-Ix
c(lm)
’ { *+Cz) *trn)
-
*+Crn)
*Cz) }
9
(3.8)
(14
*-
where the variables C(lm) are randomly carrying
assigned the values f 1. The result of
out this eigenvalue analysis on a large sample of matrices
is the re-
markable fact that over most of its range, the spectrum of eigenvalues is linear when plotted
versus the integers i = -n, . . . , n.
our intuition,
theories defined by Hamiltonians
equivalent to 1+1-dimensional Understanding
of the type specified in (3.8) are
It is obviously impossible to diago-
matrices of type (3.8) f or arbitrary
be absolutely sure that we are dealing with 1+1-dimensional basis of computer
calculations
to
field theories.
the Pattern of Eigenvalues.
nalize all antisymmetric
This means that, contrary
large N, so we cannot object solely on the
for N 5 121. However, Dyson’s results’ con-
cerning the behavior of eigenvalues of random matrices enables us to understand our computer
results and to explore the pattern
of eigenvalues for almost all
matrices C(lm) . Our problem is to find the distribution matrix,
whose [independent]
matrix
of eigenvalues for a real antisymmetric
elements are essentially 20
random.
Let us *--.a,
follow F. Dyson in our definitions:
a random antisymmetric
element of an ensemble AG of all real antisymmetric with a gaussian probability
* * * A(2n,2n+l)
PC 41,2)
1 =
matrix
(2n + 1)
x
is a random
(2n + 1) matrices
distribution
c
exp
[ -
c
( A$,m)b2)
] dJ4(1,2)
3
- - - dA(,th,,tL,l)
(km)
where a and C are constants.
The major result which we will need for our
discussion is the following Theorem
I:
whose probability
A random element of AG has eigenvalues distribution
1
.dEl...dE,
i=l
distribution
.
P , we can compute the function of inter-
est, namely the eigenvalue density p(E)
which tells us the expected number of
[ E, E + dE ] . Th’IS eigenvalue density (i.e. density of
states) can be found by Dyson’s method and in our case of an imaginary symmetric
f”
1 -g(E:/4z2)
eigenvalues in the interval
, . . . , GE,
is given by
P(El,...,
Given the probability
0, fiEl
(2n + 1) X (272 + 1) matrix,
anti-
all of whose entries are 31, it comes out
to be
and vanishes for 1E 1 2 fi
. ( See the appendix for a simple derivation of p(E)
due to Brezin, Itzykson and Zuber. ) The spacing between eigenvalues is given fl3 This is the case for any real antisymmetric (2n + 1) .-
21
x
(2n+ 1) matrix. ‘--.a,
1 &GFTF
= -.47rn
A(E) = 1
2n+l
P(E)
’
which in the large n limit becomes
A(E)
=
&.
(1
- ?J-1’2
.
We see that near E = 0 eigenvalues are spaced fairly uniformly, endpoints
of the spectrum
the spacing becomes thinner.
but near the
These results are in
complete agreement with the numerical results. .s- Why Is This a One-Dimensional
Space ?
The Dyson calculation
tells us that
our computer result is not an artifact of small n, but a behavior one expects to hold for n + 00. For energies small on the scale of fi -
eigenvalues are uniformly
spaced; therefore the index j = -n, . . . , n , which labels them, can be interpreted as a momentum
variable for an effective one dimensional theory. In order to make
this interpretation
more physical, let us multiply
scale factor A =
a& ’ having the dimension of a mass. Then we can write the
spectrum of eigenvalues which are relatively
the Hamiltonian
(3.8) with a
close to zero as
,
(3-g)
where k=?!
L’f
* ,
j =
and
-n,...,n
Momenta defined by (3.10) are characteristic defined in a spatial volume L. Equation 22
L=&Q
for l+l-dimensional
.
(3.10)
lattice theories
(3.10) allows for maximal
momentum
w h’lc h corresponds to the lattice spacing a = r/k,,,
2*n = ffi-A, k moz = 7;ucd-.
As we take the limit n +
the equivalent volume.
00, the ratio of L to a grows like n, and
lattice theory passes over to a continuum
theory defined in infinite
This limit is only unusual in that the lattice theory simultaneously
both finer and larger, as measured in units of the scale parameter The explicit
unitary
mapping
of the N-simplex
theory
given: we define transformed Q(x)
= Ali2
c
gets
A.
into an equivalent
lattice theory whose kinetic term is given by a generalized SLAC-derivative now be explicitly
=
can
fields Q(z) as
eizk(j) U;(Z) X4(1) ,
(3.11)
1EJ -n 1 the energy of the highly connected configuration grows like n to a power between unity and 3/2; whereas for the simpler configuration
it only behave like n. This explains why the numerical
select the highly connected configurations not important
if we allow C(l,) *a-
over the less connected ones. It is also
do not vanish. Numerical
that all C(i,)
calculations
computations
show that
to take the values &C and 0 with varying probabilities,
one-dimensional
the
nature of the spectrum remains intact when as many as forty
percent of-the variables C(lrn) vanish. What About Fluctuations -
-
in the Link Variables ? Up to this point we have been
interested in establishing the relationship
between numerical studies and the anal-
ysis based upon Dyson’s theorems for random matrices.
Having seen that these
techniques reproduce all of the essential features of the computer results obtained for small values of n, we now return to the question of how quantum fluctuations in the link-field r-dependent
variables affect our conclusions.
This means restoring all of the
terms to our expression for the expectation
nian. For the case r = 2, i.e. the case of a Hamiltonian in the fields X(l,),
& =
n(2n+
this expectation
1).
;+-+-
where cy is an undetermined Inspection
value of the Hamilto-
having a quartic potential
value has the form
671
6t-j c2
4r2
2-t
+ qC-
1
constant of proportionality
-
aCn3/2
,
(3.16)
of the order of unity.
of (3.16) reveals that in the limit of large n, the variable 7 tends to 25
a constant value and ?? tends to zero. This means two things: case reduces to an effective “infinite fluctuations
dimensional”
theory; and second, that the
in the field dominate the vacuum expectation
to an n-independent
first, that this
values, since 7 tends
constant but c vanishes in the limit of large n. For this
reason we will refer to this case as being fluctuation
dominated.
We can rescue all of our previous results if we extend our basic Hamiltonian to include what we will call generalized plaquette
terms, ;.e, if we add to (3.6) a
term of the form vAangle ,
which is independent
=
-
C k
