SPACE-TIME : ARENA OR ILLUSION - Stanford University

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