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Discrete-time general relativity and hyperspace

R. F. BORDLEY

Operating Sciences Department, General Motors Research Labs Warren, MI 48090-9055, USA

(ricevuto il 13 Maggio 1996; approvato il 30 Agosto 1996)

Summary. — Two popular, but distinctly different, approaches to a unified field theory include: i) general relativity using ten(or more) dimensions; ii) discrete-time Lagrangians. But as we note, discrete-time Lagrangians can be approximated by Lagrangians dependent on position, velocity and s 21 higher derivatives of velocity. Furthermore, Lagrangians dependent on s 21 higher derivatives of velocity can be reformulated as conventional Lagrangians dependent on only the first derivative of velocity in a ( 3 s 11)-dimensional space. Hence discrete-time Lagrangians and hyperspace models are formally interrelated.

PACS 04.20 – Classical general relativity.

1. – Introduction

Quantum theory in its present formulation assumes that the arena of space and time is continuous and any discreteness in the spectrum of observables is a result of boundary conditions on the wave functions or from non-trivial topological effects. If, however, configurations of matter and geometry become very compressed, space and time themselves might take on a grainy-like discrete structure.

Both Wheeler [1] and Finkelstein [2] have suggested this on fundamental grounds;

furthermore, the very existence of a quantum space (Planck length B10233_{cm )}

suggests that below a certain scale of length, it does not make sense to measure continuous lengths. In order to facilitate calculations in both QED and quantum gravity, Snyder [3] and ’t Hooft [4] have suggested a lattice structure of space-time and Lee [5] has discussed the discretization of time in order to make the measure in path integrals less ambiguous.

Indeed Caldirola suggested a discrete-time version of quantum theory driven by the belief that there is an uncertain response of the wave function in time that results in the application of the Hamiltonian at time t [6, 7]. Wolf [8] investigated this idea and indicated the experimental limits on the discrete-time interval. He also showed that modifications to such phenomena as electron spin resonance [9], electron spin polarization precession [10], and the spectral shift in hydrogen [11] arise because of discrete-time effects. He also discussed how discrete-time quantum effects could be

used to probe for the composite structure of leptons and gauge bosons [12, 13]. Montvay and Munster’s recent book [14] provides an overview of this exciting new subfield.

The discrete-time approach replaces the derivatives x1

(t) 4ˇx(t)Oˇt in the convention-al Lagrangian, L

(

x(t), x1_(t)

₎

by the discrete-time operator, Dx4

(

x(t1T )2x(t)

)

OT. The next section examines the consequences of making this substitution in the conventional non-relativistic classical mechanics. As we show, we get a formalism which seems to be consistent with quantum mechanics. The next step is to explore the implications of substituting discrete-time derivatives in relativistic mechanics.

Since Dx 4

!

k 41 Q _Tk 21 k! x k_{(t) ,}

replacing the standard derivative in the Lagrangian with Dx makes the Lagrangian a function of all the higher derivatives of velocity. As Riahi noted, a Lagrangian in 3-dimensional space which depends on velocity and s 21 higher derivatives of velocity is equivalent to a conventional Lagrangian, dependent only on position and velocity, in a 3s-dimensional space. As a result, replacing velocity in the Lagrangian by the

discrete-time operator is equivalent to modelling the Lagrangian on an

infinite-dimensional space (where the higher dimensions of that space are scaled by powers of T ).

Hence a discrete-time of relativity theory is equivalent to relativity theory in an infinite-dimensional space—where the higher dimensions of the space are extremely small. This suggests that there may be some connections between discrete-time general relativity and superstring theory. As a step toward making such a comparison, the closing section of the paper will sketch out this discrete-time formulation of general relativity.

While our results will be far from definitive, we hope they encourage further examination of the discrete-time formulation of general relativity.

2. – Discrete-time classical mechanics and quantum mechanics

2._{1. The discrete calculus. – While the conventional derivative is defined by dxOdt4}

lim

d K0

(

x(t 1d)2x(t)

)

Od , we define the discrete derivative by

Dx(t) 4 lim d KT x(t 1d)2x(t) d 4 x(t 1T)2x(t) T ,

where T is the smallest interval of time. Since the finite-difference operator, d, of the calculus of finite differences [15] is defined by dx(t) 4x(t1T )2x(t), our discrete derivative is trivially related to the finite-difference operator by Dx(t) 4dx(t)OT. Based on the properties of dx from the finite-difference calculus, we can derive the following properties for the discrete derivative:

D[ f (x) 1g(x) ] 4Df(x)1Dg(x) ,

D[ f (x) g(x) ] 4f(x) Dg(x)1g(x) Df(x)1TDf(x) Dg(x) , D

y

f (x)

g(x)

z

4

g (x) Df (x) 2f(x) Dg(x) g(x)

(

g(x) 1TDg(x)

)

.

Letting T go to zero gives the standard rules of the differential calculus. To define a discrete integral I—which is the inverse of the D-operator—let

DF(x) 4f(x)

i.e. f (x) 4

(

F(x 1T )2F(x)

)

OT . If C(x) is a periodic function with C(x 1 T ) 4 C(x), then D[F(x) 1C(x) ] 4f(x), i.e. we can only infer F(x) up to an arbitrary periodic function, C(x).

Then the discrete integral I is defined by

I[ f (x) ] 4F(x)1C(x) 4T[ f(x2T)1f(x22T)1R1f(0) ]1C(x) .

Now the summation operator, S * , of the finite-difference calculus is given by

S * f 4f(x2T)1f(x22T)1R1f(0)1C(x) ,

so that I[ f (x) ] 4TS*[ f(x) ]. We can use the known properties of the summation operator to infer the following properties for the discrete integral I:

.

/

´

I[ f (x) 1g(x) ] 4If(x)1Ig(x) , I[af (x) ] 4aI[ f(x) ] , I[ f (x) Dg(x) ] 4f(x) g(x)2I[Df(x)

(

g(x) 1TDg(x)

)

] , (1)

where we suppress the arbitrary period function, C(x). These formulas reduce to the standard formulas when T goes to zero. We also have

lim

T K0

[

I[ f (x) ] 1C(x)

]

4



_{f (x) dx 1c .}

This paper will only focus on discrete integrals with specified limits of integration so that we can drop the arbitrary function C(x).

2.2. The discrete Euler-Lagrange equations.

Theorem 1. Let X 4x(t) be the curve joining the points (t1, x1), (t2, x2) which makes

It2

t1L(t , x , Dx) an extremum. Let T be the smallest meaningful interval and let Lx, LDx

denote ˇLOˇx, ˇLOˇDx, respectively. Then x(t) satisfies the following finite-difference generalization of the Euler equation:

Lx4 D[LDx2 TLx] .

Proof. See appendix A.

Thus our finite-difference generalization of the calculus of variations modifies the Euler-Lagrange equations to include a discrete-time correction factor.

One of the difficulties implicit in the discrete calculus is that the usual chain rule dV(x , t) dt 4 Vx dx dt 1 Vt dt dt

does not apply. But when T is small, we can derive the following result:

Theorem 2. DV(x , t) BVxDx 1VtDt 1 T 2 DVxDx 1 T 2 DVtDt .

Proof. See appendix A.

Hence we can replace the standard chain rule with an approximate chain rule. Defining Vav

(t) 4

(

V(t) 1V(t1T )

)

O2 4 V(t) 1 (TO2 ) DVt gives

DV(x , t) 4Vav

x Dx 1VavDt .

In other words, evaluating our partial derivatives at the center of the differencing interval—and not at the start of that interval—gives the conventional chain rule.

2.3. Discrete-time classical mechanics and the Schrödinger equation. – Defining

the discrete-time Hamiltonian by

H 4

k

LDx2

T

2 Lx

l

Dx 2L gives:

Lemma. DH 42[Lt1 (TO2 ) DLt] Dt .

Proof. See appendix A.

If we define S 4I[L] to be the action, then integrating DH gives

H 42

k

St1 T 2 DSt

l

4 H 4 2St2 T 2

y

St(t 1T)2St(t) T

z

¨H 1 1 2[St(t) 1St(t 1T) ] 40 . Let Sav

4

(

S(t) 1S(t1T )

)

_{O2 be the average action so that}

H 1Sav t 4 0 . Note that ˇSav ˇx 4 S av x 4 ˇ ˇx

k

S 1 T 2 L

l

4

k

LDx2 TLx1 T 2 Lx

l

4 LDx2 T 2 Lx. Let L 4 m 2 (Dx) 2 2 V(x , t) .

Then H 4LDxDx 2 T 2 LxDx 2L4 Dx 2 [LDx2 TLx] 1V . But Sav x 4 LDx2 T 2 Lx¨Dx 4 Sav x 1 (TO2 ) Lx m . Substituting gives H 4 1 2 m

k

S av x 1 T 2 Lx

lk

S av x 2 T 2 Lx

l

1 V 4 1 2 m

k

(S av x )22 T2 4 L 2 x

l

1 V .

Hence, our discrete-time Hamilton-Jacobi equation becomes (Sav x )2 2 m 2 T2 8 mL 2 x1 V 1 Stav4 0 .

For some function W with Wx4 Lx, defining R 4exp[2WOa] gives

Rxx4 1 a2(Lx) 2 R 1 1 aLxxR , so that (Sav x )2 2 m 2 a2_T2 8 m Rxx R 1 aT2 8 m Lxx1 V 1 S av t 4 0 . (2)

Now Bohm [16] has shown that substituting the wave function

f 4R exp [iSOh]

into the Schrödinger equation leads to two equations

s2 x 2 m 2 h2 4 m Rxx R 1 V 1 st4 0 , (3) RRt1 ˇ ˇx

k

R 2 sx m

l

4 0 . (4)

If we let a 421 O2hOT and assume that a2T2caT2—so that the second term in (2) is negligible—eq. (2) becomes identical to (1_{) (3) with s}

t4 Sav. So the Schrödinger

equation is consistent with the discrete-time Hamilton-Jacobi equation.

(1_{) Substituting r 4R}2 _{and noting that s}

x4 LDx2 (TO2 ) LxB mx 8 2 (TO2 ) mx 9 transforms the

second equation into

rt1

ˇ

ˇx

k

r

k

x 82 T

2x 9

ll

4 0 which closely approximates the familiar continuity equation.

3. – Discrete-time general relativity

Relativity often defines the free-particle Lagrangian to be

L(t) 4 m

2

!

ij

gijx 8i x 8j

(

where, hereafter, we supress

!

in following the Einstein summation convention

)

.

Then x(t), x8(t) extremizes m 2



0 T [ gijx 8i x 8j ] dt .

We now discretize the Lagrangian by replacing x 8i by Dxi. This gives

L(t) 4 M 2 gijDxiDxj. We now define x(k) 4 dkxOdtk _{with x}( 0 ) 4 x . Then Dx 4

!

k (Tk 21_{Ok! ) x}k . Letting a 4

!

k F2 (Tk 22_{Ok! ) x}k gives Dx 4x1

1 Ta . Substituting into our discretized action gives



M 2 gij(x 1 i 1 Tai)(xj11 Taj) dt 4 M 2



_gij[x1 i xj11 2 Txi1aj1 T2aiaj] dt .

3.1. Generalized mechanics. – For a generalized Lagrangian of the form

l(xi( 0 ), xi( 1 ), R , xi(s)Ni 4 1 , 2 , 3 ) ,

we define li , k4 ˇlOˇxi(k) and li , k(r)4 ( drOdtr) li , k with li , k( 0 )4 li , k. As generalized

mechanics [17, 18] prescribes, the Euler-Lagrange equations become

li , 01

!

r 41 l(r) i , r(21)r4

!

r 40 l(r) i , r(21)r4 0 . (5)

Now Riahi [19] noted that any Lagrangian (in three-dimensional space) dependent on the first s higher derivatives of position could be reformulated as a Lagrangian in 3s-dimensional space dependent on only position and velocity. To show this, define generalized momenta by pi(k 21) 4

!

r 40 Q l(r) i , k 1r(21)r, k 41, R, s , with p 8i( 0 ) 4

!

r 40 Q l_{i , r 11}(r 11)(21)r 4

!

r 41 Q li , r(r)(21)r 214 li0

so that the conventional form of Newton’s law is consistent with the generalized Euler-Lagrange equations.

Note that the Hamiltonian becomes

H 4

!

k 41 s

pi(k 21) xi(k)2 L , k 41, 2, R , s .

Given the definition of Hamiltonian and momenta, we have ˇH ˇt 4 2 ˇL ˇt , ˇH ˇpi(k) 4 xi(k 11), ˇH ˇx(k) i 4 2p 8i (k) .

Letting y(i 13k) 4xi(k) and p

(

i 13(k21)

)

4 pi(k) gives

ˇH ˇt 4 2 ˇL ˇt , ˇH ˇp(k) 4 y 8 (k) , ˇH ˇy(k) 4 2p 8 (k) ,

so that the dynamics of a system whose Lagrangian involves s higher derivatives of velocity is equivalent to the dynamics of a system in 3s dimensions. Since

p 8i (k 21) 4li , k 211 pi(k 22) ,

the system’s motion in the i13(k21)-th dimension is likewise subjected to a force.

3.2. Application to our Lagrangian. – For our Lagrangian,

li04 M 2 gjk , i[x 1 j xk11 2 Txj1ak1 T2ajak] B M 2 gjk , i[x 1 j xk11 Txj1xk2] ,

if we neglect terms of order T2_{. Then}

li14 Mgij[xj11 Taj] BMgij

k

xj11 T 2 x 2 j

l

, lir4 Mgij

k

Tr 21 r! x 1 j1 Tr 21 r! Taj

l

4 Tr 21 r! li1.

Substituting into the Euler-Lagrange equations gives

k

!

s 41 Q (21)s 21 Ts 21 s! l s i1

l

4 M 2 gjk , i[x 1 j xk11 Txj1xk2] .

Since we neglect terms of order T2,

!

s 41 Q (21)s 21 Ts 21 s! l s i1B l 1 i12 T 2 l 2 i1B Mgij , kx 1 k

k

x 1 j 1 T 2 x 2 j

l

1 Mgij

k

x 2 j 1 T 2 x 3 j

l

2 2TM 2 g 1 ij , kxk1xj12 TM 2 gij , k[x 2 kxj11 xk1xj2] 2 TM 2 gij , kx 1 kxj22 TM 2 gijx 3 j4 4 Mgij , k

k

xk1xj11 T 2 x 1 kxj22 T 2 x 2 kxj12 T 2 x 1 kxj22 T 2 x 1 kxj2

l

1 1Mgij

k

xj21 T 2 x 3 j 2 T 2 x 3 j

l

2 TM 2 g 1 ij , kxk1xj14 4 Mgij , k

k

xk1xj12 T 2 [x 2 kxj11 xk1xj2]

l

1 Mgijxj22 TM 2 x 1 kxj1gij , k1 .

Substituting back into the Euler-Lagrange equation (and cancelling the common factor of M) gives gij , k

k

xk1xj12 T 2 [x 2 kxj11 xk1xj2]

l

1 gijxj22 T 2 x 1 kxj1gij , k1 4 1 2gjk , i[x 1 j xk11 Txj1xk2] . Rearranging gives gijxj21 gij , kxk1xj12 1 2 gjk , ix 1 j xk14 T 2

[

gjk , ix 1 j xk21 gij , k[xk2xj11 xk1xj2]

]

1 T 2 x 1 kxj1gij , k1 , or gijxj21 gij , kxk1xj12 1 2gjk , ix 1 j xk14 T 2 x 1 j [gjk , i1 gij , k1 gik , j] xk21 T 2 x 1 kxj1gij , k1 .

Defining the Christoffel symbol by

Gz jk4 1 2g zi_(g ij , k1 gik , j2 gjk , i)

and substituting gives

x2 z1 xj1Gzjkxk14 T 2 g zi_x1 j[gij , k1 gik , j1 gjk , i] xk21 gij , k1 xk1] ,

where T 40 gives us the standard equation for a geodesic. Thus discretizing has induced a force on the system. If we view our system as operating in a

higher-dimensional space, then this motion is a function of x2

k—reflecting the

coordinates of the system in this higher-dimensional space and the second derivative of

gi j, derivatives of g. (For a derivation of the analogous equations defined in terms of

finite-difference operators, see the appendix.) Defining generalized momenta gives us

pi(k 21) 4

!

r 40 Q l(r) i , k 1r(21)r4

!

r 40 Q _Tk 1r21 k 1r! l r i1(21)r4 Tk 21 k! r 40

!

Q (2T)r k! (k 1r)!l r i14 4 T k 21 k!

y

pi( 0 ) 1r 41

!

Q (2T)r_lr i1

y

k! (k 1r)! 2 1 (r 11)!

zz

4 4 T k 21 k!

y

pi( 0 ) 2Tr 40

!

Q (2T)r_lr 11 i1

y

k! (k 1r11)! 2 1 (r 12)!

zz

B B T k 21 k!

k

pi( 0 ) 2T

g

1 k 11 2 1 2

h

l 1 i

l

B Tk 21 k!

y

pi( 0 ) 2T k 21 2(k 11)p 8i( 0 )

z

. In other words, all the hyperspace momenta are approximately proportional to the normal space momenta minus some fraction of the normal space force. Since

conclude that the induced hyperspace force is proportional to the normal space force minus some fraction of the time derivative of that normal space force.

4. – Conclusions

This paper showed that a discrete-time version of classical mechanics seems consistent with the Schrödinger equation. We then explored the extension of discrete-time classical mechanics to a discrete-time general relativity. We found that this extension was consistent with standard relativity on an infinite-dimensional space. Although preliminary, these findings will hopefully stimulate further investigation of the relationship between discrete-time general relativity and hyperspace models.

AP P E N D I X A

A.1. Proof of theorem 1

Proof. _{Let X(e) 4X(t)1eh(t) for fixed functions x(t) and h(t) represent a}

neighbouring curve joining the points t1and t2. We require h(t1) 40. The value of the

discrete integral along this curve is

V(e) 4It2

t1L

(

t , x(t) 1eh(t), Dx(t)1eDh(t)

)

.

Let

L(e) 4L

(

t , x(t) 1eh(t), Dx(t)1eDh(t)

)

. Since e 40 is an extremum, we must have dVOde40 at e40. Then

dV de 4 I

y

ˇL(e)

ˇx h(t) 1

ˇL(e)

ˇDx Dh(t)

z

4 I[Lx(e) h(t) 1LDx(e) Dh(t) ] 40 , e 40 .

But (1) gives

I[LDxDh(t) ] 4LDxh(t) NT02 I[DLDxh(t) ] 2TI[DLDxDh(t) ] 4

4 LDxh(t) NT02 I[DLDxh(t) ] 2TDLDxh(t) NT01 1TI[D2LDxh(t) ] 1T2I[D2LDxDh(t) ] 4 4 R 4

k

!

k 40 Q (2T)k_Dk

g

ˇL ˇDx

h

h(t)

l

2 I

k

k 40

!

Q (21)k_Tk_Dk 11

k

ˇL ˇDx

ll

h(t) . Hence Ve4 I

kk

Lx(e) 2

!

k 40 Q (21)k_Tk_Dk 11_[L Dx]

l

h(t)

l

1

k

!

k 40 Q (2T)k_Dk_(L Dx) h(t)

l

.

of integration, we conclude that the curve x(t) must satisfy Lx(e) 4

!

k 40 Q (21)k_Tk_Dk 11_[L Dx] . Since 2TD[Lx] 4

!

k 41 Q (21)k_Tk_Dk 11_[L Dx] , we conclude Lx1 TD[Lx] 4D[LDx] . (A.1)

A.2. Proof of theorem 2

Let Vjk4 (ˇj 1kV ) O(ˇxjˇtk). Let Vjk04 Vjk evaluated at x 40, t40. Then

V 4V0 001 V100x 1V200 x2 2 1 V 0 11xt 1V010t 1V020 t2 2 1j 1kD2

!

V0 jk xj_tk j! k! .

Taking differences gives

DV 4V0 10Dx 1V010Dt 1V110[xDt 1tDx1TDxDt]1 1V200

k

xDx 1 T 2 (Dx) 2

l

1 V020

k

tDt 1 T 2 (Dt) 2

l

1

!

j 1kD2 Vjk0 D[xj_tk_] j! k! . Since V104 V1001 V110t 1V200x 1

!

j 1kD2 Vjk0 xj 21_tk ( j 21)! k! , we get V0 104 V102 V110t 2V200x 2

!

j 1kD2 V0 jk xj 21_tk ( j 21)! k! . We also have V014 V0101 V110x 1V020t 1

!

j 1kD2 Vjk0 xj_tk 21 j!(k 21)! , so that V0 014 V012 V110x 1V020t 1

!

j 1kD2 V0 jk xjtk 21 j!(k 21)! ,

Substituting V100 and V010 into our expression for DV gives DV 4V10Dx 1V01Dt 1TV110DxDt 1 TV200 2 (Dx) 2 1 TV02 2 (Dt) 2 1 1

!

j 1kD2 Vjk

y

D[xjtk] j! k! 2 xj 21tkDx ( j 21)! k! 2 xjtk 21Dt j!(k 21)!

z

. We also have DV104 V110Dt 1V200Dx 1

!

j 1kD2 V0 jkD

y

xj 21tk ( j 21)! k!

z

, DV014 V110Dx 1V020Dt 1

!

j 1kD2 V0 jkD

y

xjtk 21 j!(k 21)!

z

, so that V0 20 T(Dx)2 2 4 T 2 Dx

y

DV102 V 0 11Dt 2

!

j 1kD2 V0 jkD

y

xj 21_tk ( j 21)! k!

zz

, V0 02 T(Dt)2 2 4 TDt 2

y

DV012 V 0 11Dx 2

!

j 1kD2 V0 jkD

y

xj_tk 21 j!(k 21)!

zz

. Substituting V0

20Dx and V020Dx into our expression for DV gives

DV 4V10Dx 1V01Dt 1 T 2 DV10Dx 1 T 2 DV10Dt 1 1

!

j1kD2 Vjk

y

D[xj_tk_] j! k! 2 xj 21_Dx ( j21)! k! 2 xj_tk21_Dt j! (k21)! 2 TD[xj21_tk_{] Dx} 2( j 21)! k! 2 TD[xj_tk21_{] Dt} 2 j!(k 21)!

z

. Define Rjk4 D[xjtk] 2jxj 21tkDx 2kxjtk 21Dt 2 j 2TD[x j 21_tk ] Dx 2 k 2TD[x j tk 21] Dt , so that Rjk4

!

r 40, s40

u

j r

vu

k s

v

x j 2r_(TDx)r tk 2s(T)s 2

u

j 0

vu

k 0

v

x j tk₂

u

j 1

vu

k 0

v

x j 21_tk 2 2

u

j 0

vu

k 1

v

x j_tk 21₂ j 2

y

r 40, s40

!

u

j 21 r

vu

k s

v

x j 212r_(TDx)r_tk 2s_Ts 2 xj 21tk

z

2 2k 2

y

r 40, s40

!

u

j r

vu

k 21 s

v

x j 2r_(TDx)r tk 212sTs_{2 x}jtk 21

z

.

We can write Rjk4 1 T[A 2x j_tk ] 2jxj 21_tk Dx 2kxj_tk 21_{Dt 2} 2 j 2TDx[B 2x j 21_tk ] 2 k 2TDt[C 2x j tk 21] , where A 4 (x1TDx)j(t 1TDt)k4 xjtk_{1 T [ jx}j 21tk_{Dx 1kx}jtk 21_{Dt] 1} 1T 2 2 [ j( j 21) x j 22_tk_(Dx)2 1 2 jkDxDtxj 21tk 21_{Dt 1k(k21)(Dt)}2_xj_tk 22_{] 1} 1T 3 6 [ j( j 21)( j22) x j 23_tk_(Dx)3 1 j( j 2 1 ) kxj 22tk 21_(Dx)2 Dt 1 1jxj 21k(k 21) tk 22(Dx)(Dt)21 xjk(k 21)(k22) tk 23(Dt)3] , B 4 (x1TDx)j 21_{(t 1TDt)}k 4 xj 21tk 1 T[ ( j 2 1 ) xj 22tk Dx 1kxj 21_tk 21_{Dt] 1} 1T 2 2 [ ( j 21)( j22) x j 23_tk_(Dx)2 1 ( j 2 1 ) xj 22ktk 21_{DxDt 1} 1k(k 2 1 ) xj 21tk 22(Dt)2 ] 1 T 2 2 b , C 4 (x1TDx)j(t 1TDt)k 21 4 xjtk 21_{1 T [ jx}j 21tk 21_{Dx 1 (k21) x}jtk 22_{Dt] 1} 1T 2 2 [ j( j 21) x j 22_tk 21_(Dx)2 1 jkxj 21tk 22_{DxDt 1k(k21) x}j 21_tk 23_(Dt)2_{] .} Substituting gives Rjk4 T2(a 1b1c) .

Dropping all terms of order T2 _gives

DV 4V10Dx 1V01Dt 1

T

2 DV10Dx 1

T

2 DV01Dt , which is our theorem.

A.3. Proof of lemma

If L 4L(Dx, x, t) then DL 4

k

LDx1 T 2 DLDx

l

D 2 x 1

k

Lx1 T 2 DLx

l

Dx 1

k

Lt1 T 2 DLt

l

Dt 4 4 D[LDxDx] 2TD[LDx] D2x 1 T 2 DLxDx 2 T 2 DLxDx 1

k

Lt1 T 2 DLt

l

Dt 4

4 D[LDxDx] 2 T 2 [DLDxD 2 x 1DLxDx] 1

k

Lt1 T 2 DLt

l

Dt 4 4 D[LDxDx] 2 T 2 [LxD 2 x 1TDLDxD2x 1DLxDx] 1

k

Lt1 T 2 DLt

l

Dt 4 4 D[LDxDx] 2 T 2 D[LxDx] 1

k

Lt1 T 2 DLt

l

Dt 4 4 D

k

g

LDx2 T 2 Lx

h

Dx

l

1

k

Lt1 T 2 DLt

l

Dt . Hence E 4

k

LDx2 T 2 Lx

l

Dx 2L

will be constant whenever the Lagrangian is not an explicit function of time.

AP P E N D I X B

General relativity with discrete-time operators

If we work through our equations using only discrete-time operators, then the Euler-Lagrange conditions optimizing



_{L(t) 4} M 2



_gijDxiDxj are Lxi4 M 2 Gjk , iDxkDxj4 D[LDxi2 TLxi] 4 4 D

k

MgijDxj2 T M 2 gjk , iDxjDxk

l

4 MD

kk

gij2 T 2 Gjk , iDxk

l

Dxj

l

. Theorem 2 indicates that Dgi j4 gi j , kDxk1 (TO2 ) Dgi j , kDxk. Making this substitution

and neglecting all terms of order T2 gives 1 2gjk , iDxjDxj4 gij , kDxk1 T 2 Dgij , kDxk2 T 2 Dgjk , iDxk2 2T 2 gjk , iD 2_x k[Dxj1 TD2xj] 1

k

gij2 T 2 gjk , iDxk

l

D 2_x j.

We can rewrite this as 1 2gjk , iDxjDxj2 gij , kDxjDxk2 gijD 2 xj4 4 T 2 Dxj(D[gij , k2 gjk , i] Dxk[ 2 gik , j2 gjk , i2 gkj , i] D 2_x k)

which can, again, be written as

D2_x z2 DxjGzjkDxk4 T 2 Dxjg zi_(D[g jk , i2 Gij , k] Dxkgzi[gjk , i1 gkj , i2 2 gik , j] D2xk)

so that the induced force once again depends on D2_x

k.

R E F E R E N C E S

[1] WHEELERJ., Superspace and nature of quantum geometrodynamics, in Battilee Recontres:

1967 Lectures in Mathematics and Physics, edited by B. DEWITT and J. WHEELER (Benjamin) 1968.

[2] FINKELSTEIND., Int. J. Theor. Phys., 27 (1985) 473. [3] SNYDER H., Phys. Rev., 71 (1987) 38.

[4] ’THOOFT G., Phys. Rep., 104 (1984) 133. [5] LEE T., Phys. Lett. B, 122 (1983) 217.

[6] CALDIROLAP., Suppl. Nuovo Cimento, 3 (1956) 297. [7] CALDIROLAP., Lett. Nuovo Cimento, 16 (1976) 151.

[8] WOLF C., Proceedings of the IV Workshop on Hadronic Mechanics and Nonpotential

Interactions, Skopje, Yugoslavia, 1988 (Nova Scientific Publ., New York) 1990.

[9] WOLF C., Phys. Lett. A, 123 (1987) 208. [10] WOLF C., Nuovo Cimento B, 100 (1987) 431. [11] WOLF C., Eur. J. Phys., 10 (1989) 197. [12] WOLF C., Hadronic J., 13 (1990) 2.

[13] WOLF C., Ann. Fond. L. de Broglie, 18 (1993) 403.

[14] MONTVAYI. and MUNSTER G., Quantum Field on a Lattice (Cambridge University Press) 1994.

[15] LAKSHMIKANTHAM L. and TRIGANTED., Theory of Difference Equations (Academic Press) 1988.

[16] BOHM D., Phys. Rev., 85 (1952) 160. [17] BORNEAS M., Am. J. Phys., 27 (1959) 265. [18] BORNEAS M., Nuovo Cimento, 16 (1960) 5. [19] RIAHI F., Am. J. Phys., 40 (1972) 386.