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Can time be a discrete dynamical variable?
Volume 122B, number 3,4
PHYSICS LETTERS
10 March 1983
CAN TIME BE A DISCRETE DYNAMICAL VARIABLE? ~ T.D. LEE Columbza Umverszty, ?few York, N Y 10027, USA Received 19 November 1982
The posslblhty that time can be regarded as a discrete dynamical variable is examined through all phases of mechanics from classical mechanics to nonrelativlstlc quantum mechanics, and to relativistic quantum field theorms
1. Introduction. Throughout the development of physics, "time" always appears as a continuous parameter. Take the example of a nonrelativlstic particle in quantum mechanics. In Feynman's path integranon formulation, the probablhty amphtude fm the particle to be at the positron r 0 at an inltml tame t O and at rf at a final time tf is given by the amplitude sum over all paths r(t) connecting r(to) = r 0 and r(tf) = r t- Apart from a normalization constant, it is equal to
f l-I d3r(t) exp(iAc) ,
(1)
t
where the action A c 1s a functional of r(t), related to the lagrangian L by
determined by the extremity of the action Ac, which as a functional ofr(t). While r is the dynamical variable, t appears only as a continuous parameter. By setting the variational derivative 8Ac/Sr(t ) = 0, we obtain the usual Lagrange equation of motion, whose solution gives the classical path r(t). In the relativistic quantum field theory, space r and time t have to be treated symmetrically due to Lorentz invariance. Our usual approach is to regard r and t all as parameters; the operators are now the field variables, say the scalar field O(r,t). Expression (1) is then replaced by f]-[ r
]'-I d~(r,t) exp(iAc) , t
(3)
tf
a c = f L(r(t), i ( t ) ) dr, to
(2)
with the subscript c denoting the familiar case that t is continuous We note that the position of the particle r is not treated on the same basis as the time t At a given tnne the integration d3r(t) in eq (1) can be viewed as that over the whole range of eagenvalues of the operator rop(t ) This then underlies the familiar difference between r as an operator and t as a parameter Actually this asymmetry can be traced back to classical mechanics The classical trajectory of a particle is ¢~This research was supported m part by the US Department of Energy.
0 031-9163/83/0000-0000/$ 03 00 © 1983 North-Holland
where the action A c is related to the lagranglan density /2 by A c = f z ? (qS,V ~b,~) d3r dt
(4)
Again, the integration over d4~(r,t) can be viewed as that over the eigenvalues of the operator q~op(r,t). In the following we wish to explore some alternative possabthties. In place of treating time as a continuous parameter, we may ask. (1) Can time be a discrete parameter (discrete time formulation)9 (2) Can time be discrete and treated as a bona fide dynamical variable (discrete mechanics)9 As we shall see, both possibilities can be realized in all stages of mechanics. In a relatlvastac field theory, the discrete tune formulation becomes the discrete space217
Volume 122B, number 3,4
PHYSICS LETTERS
time formulation. In the corresponding discrete mechanics both space and time, in addition to being discrete, are treated as genuine dynamxcal variables. Because this approach involves a fundamental change in our concept of s p a c e - t i m e , we will develop our ideas gradually first in classical mechanics, then in nonrelatlvistlc quantum mechanics and finally in relativistic quantum field theory. The result is that in this new formalism our usual idea of continuous time or continuous s p a c e time structure will appear only as an approximation. 2 Classical m e c h a n i c s . Consider the simple example o f a nonrelativlstic particle of unit mass moving in a potential V(r). The usual lagrangian is (regarding time as continuous) L = ~ r1
.2
(5)
__ V ( r ) ,
The action is then given by the integral (2) from the initial position r ( t o ) = r 0 to the final r(tf) = rf. In the discrete time formalism we replace the continuous function r ( t ) by a sequence of discrete values: ( t o , t o ) , (r 1 , t l ) ..... (rN+ 1 , t w + l ) , where (rN+ 1 , t N + l ) = (rf, tf). The action (2) is then replaced by
N+I A= ~ (1 (rn-rn-1)n= 1
-
2
½% -
9
tn -- t n - 1
tn_l)[V(rn)+
)
V(rn_l) ] ,
(6)
with t n > t n _ l . Newton's equation of motion is derived by setting the derivative 3 A / a r n = 0.
(7)
Keeping the initial and final configurations fixed, we have altogether N such equations:
1
Vn + l - vn = --~ (t n + l - tn - 1 ) V V ( r n )
(8)
where v n = (r n - r n _ l )/(tn - t n - I ) is the velocity. For any given time dxstrlbutlon t l , . . , t N , the positions, r 1 .... , t N can then be determmed. An ad hoc discrete time distribution t l , t2, .., t N is clearly not satisfactory for a closed mechanical system. Thus, in discrete mechanics we require that the time distribution should also be determined by the same action (6). This can be achieved by treating t n as dynalnlcal variables, on the same basis as r n . By setting 3 A / 3 t n = 0,
we derive N additional equations:
E n E 1~v n2 + ~1[ V ( r n ) +
V(r_l) l =L)~+~
(10)
Thus, the entire set ( r l , t 1),...,(rN,t N ) can be determined from (8) and (10). In the continuum case, the energy conservation law is a consequence of Newton's equation of motion In the discrete case, these two are Independent. By treating t n and r n both as dynamical variables, we manage to employ the same action principle for their determination. In our formulation of discrete mechanics, there is a f u n d a m e n t a l l e n g t h or time l (in natural units). Given any time interval T = tf - t 0, the total number N of discrete points that define the trajectory is given by the integer nearest T/l. As illustrations, we may consider some special examples (1) V = 0 In this case (8) gives u n = constant, which also satisfies (10) The trajectory of the particle IS always a straight line, independent of the time distribution. (11) VV = constant. It can be readily verified that (8) and (10) yield equal spacings for the tnne intervals. The trajectory consists of N discrete points (r 1 , t l ) , , (rN, t N ) all lying on a parabola, similar to the continuum case. (For more comphcated potentials, the time intervals are in general of varying lengths.) 3 N o n r e l a t i v l s t i c q u a n t u m m e c h a n i c s . As above, let the initial and final configurations be (r O, t o ) and (rf, tf). In the discrete quantum mechanics, we assume that within the given time interval T = tf - t 0, there can be maximally N measurelnents of the particle's configurations possible, where the ratio N I T = l is the fundamental constant mentioned before. Each measurement yields a set of values ( r n , t n ) where n = 1,2 ..... N. [The initial (r 0` to) can be regarded as the result of a previous experiment, and the final (rl, tf) = (rN+ 1, tN+ l ) as that of the (N + 1)th measurement.] Without any loss of generahty, we may arrange t I , .,t N into an ordered sequence t O < t 1 < .. < t N < tf. The probablhty amphtude of observing such a set (r I , t 1 ) . . . . (r N , t N ) is proportional to exp(1A) with A given by eq. (6) Summing over all possible (r I , t 1 ) . . . . (r N , t N ) we obtain the overall quantum mechanical amplitude ( f l G I 0 ) leading from (r0,t0) to (rf, tf)
N
(9) (flGI0)
218
10 March 1983
=fJ exp(iA) n=l [-I
dt n d3rn ,
(11)
Volume 122B, number 3,4
PHYSICS LETTERS
where J is a jacobian, whmh can be either a simple normalization constant independent o f t n and t n , or some other slowly varying functmns. The classical tralectory is determined by the stationary phase condition 6A = 0 In the path integration (11), the integrations d3rn and dt n may be viewed as those over the eigenvalues of the operators (rn)op and (tn)op. The discreteness in this new mechanics enters through the finite number of measurements allowed in any gwen time interval. The results of each measurement can be any of the continuous eigenvalues of (r n)op and (t n)op. In the usual contlnuuin case, as the time-parameter t vanes continuously there are an infinite number of operators rop(t ). Now, by replacing these with a finite number of operators we have elevated "time" to the status of an operator, the same as "space". As an illustration, let us consider the special case that the jacoblan is
10 March 1983
There is quite a latitude in the choice of the jacoblan. For example, if instead of (12), J is simply a normahzanon constant independent of t n and rn, then to O(l 0) the effective hamiltonian remains the same H0; the O(/2) correction differs from (16) only by a simple numerical factor. Different J correspond to different discrete mechamcal systems that have the same continuum liniit
4. Analytical continuation. As in the continuum case, it is often useful to introduce r = it and consider the analytic continuation of the region of real t to that of real r. For a nonrelatlvistic particle, the action A in (6) as replaced by N+I (1 (rn - - r n - l ) 2
~--iA=,,=~ 2 r.-rn_I+ ½(rn - rn - 1 ) [ V(rn ) + V(rn - 1) ] ) '
N+I 1 ]3/2 j = N! l-] ( • \ T N n=l 27re,~]
(12)
where en =- t n - tn_ 1 > O, G is then related to the operator U(e) ==-e x p ( l ~.1c V ) e x p 0.1~ e g 2 ) exp(1½eV) (13)
(17)
The mlnnnum of ~ determines the classical path (r 1 , r 1) ..... (rN, rN). As is well known, the resulting trajectory as one in whmh the particle is mowng as i f m a "potential" - V. The corresponding Green's function G in quantum mechamcs is now given by, instead of (11), N
by
= f J e x p ( - . ~ ) N!
G=T -~fU(eN+I)U(eN)U(el)
~1 den,
(14)
where V 2 is the laplaclan operator and T = z N + I en Thus, for example, for a free particle (V = 0), G = exp(i½ TV2), ldentmal to the usual continuum case. In a subsequent paper with R. Friedberg, we shall prove that for arbitrary V when T becomes very large G e x p ( i T H ) (after averaging over an appropriate grand canonical ensemble), where H is a hermitian whose power series expansion in l can be written as
H = H 0 + 12H1 + 0(13),
(15)
in winch H 0 = - ~1~79- + Vis the usual continuum hamiltonlan and H 1 =-~[g 2 -
V,[V 2 ' V ] ]
1--I drnd3rn .
(18)
1
?l
(16)
gives the first correction, We may call H an effectwe hamfltonian. Its hermitIclty insures the unitarity of the S-matrix.
In the following extension of the &screte mechanics to a relativistic theory, we shall stay in the region of real z The usual Lorentz transformation becomes a rotation an the four-dimensional euclidean space. For convenience, we shall now refer to r as "time".
5. Relativistic field theory. The relativistic generahzation is identical to the annealed version of the random lattice field theory [ 1 - 3 ]. Consider the example of a scalar field O(r,r). For any given finite four-dimensional volume g2, we assume that maximally there can only be N measurements, each yielding a set o f values (ri,rl,(~i) , where i = 1,..,N. The ratio gZ/N = l 4 is a fixed constant. The probability amplitude of observing such a sequence (rl, r l , ¢1 ) .... (rN,rN,¢N) is proportional to exp(--s~) where ~ is the action; its integral N
J exp(-~)i_[I1 d¢~ dr~ d3r i
(19) 219
Volume 122B, number 3,4
PHYSICS LETTERS
gives the overall quantum mechanical amplitude where J is the jacoblan which again can be taken as a simple normalization constant, independent o f r t and "ci. By integrating over the entire domain d'ctd3ri , we see that rotational lnvariance is maintained, provided sd is invariant. To construct M , let us consider an arbitrary distribution o f N sites o f coordinates (r 1, ~'l),''',(rN, rN) in ~ . To each site we assign a value q5i for the scalar field. The sites are hnked according to the algorithm given in ref. [ 1 ]. The action N is (20)
lzl where the first sum extends over all hnks ltj and the second over all sites i. The weights X~I and coz are functions of the sites, given explicitly in ref. [3]. This discrete formalism can, of course, be apphed to fields o f higher spins. In the usual continuum relativistic quantum field theory, the s p a c e - t i m e coordinates are not regarded as dynamical variables; e.g., in eq. (3) the integration is over HrIlrd4)(r,r), which corresponds to an infinite number o f operators 4~op(r,r), with r and r as parameters. Here, there is a finite number of operators (rz)op and (rz)op, each of which can have a continuous range of eigenvalues that can be observed. The discreteness appears, as before, only in the maximal number N of measurements allowed in any finite s p a c e - t i m e volume ~2. Each measurement determines the physical
10 March 1983
field as well as the s p a c e - t i m e location of the measurement itself. In this sense, this new approach seems to be nearer to our actual experimental experience. Our usual concept that all physical fields should be embedded in a continuous s p a c e - t i m e manifold may well be only an approximation, far from fundamental. In this new formulation, any finite number of observations may be made very close to each other m space and in time. However, within an infinitesimal s p a c e time volume it is not possible to perform an infinite number of measurements. In the usual continuum quantum field theory, such unphysical possibilities are in principle allowed, and that leads to the well-known complications in the ultraviolet region. If one wishes, one may also regard the fundamental length l of the discrete mechanics as a covarlant regularizatlon parameter for the elimination of ultraviolet divergence. I wish to thank R. Frledberg for discussions. I am grateful for the very kind hospitality of the theoretical group at CERN, especially during the summer of 1982 when most o f the present Idea was formed.
References
(~i)op,
220
[1] N H Christ, R. Frledberg and T D. Lee, Nucl. Phys B202 (1982) 89 [2] N.H. Christ, R Frmdberg and T.D. Lee, Nucl Phys. B210 (1982) 310 [3] N H. Christ, R. Fnedberg and T D Lee, Nucl. Phys. B210 (1982) 337
PHYSICS LETTERS
10 March 1983
CAN TIME BE A DISCRETE DYNAMICAL VARIABLE? ~ T.D. LEE Columbza Umverszty, ?few York, N Y 10027, USA Received 19 November 1982
The posslblhty that time can be regarded as a discrete dynamical variable is examined through all phases of mechanics from classical mechanics to nonrelativlstlc quantum mechanics, and to relativistic quantum field theorms
1. Introduction. Throughout the development of physics, "time" always appears as a continuous parameter. Take the example of a nonrelativlstic particle in quantum mechanics. In Feynman's path integranon formulation, the probablhty amphtude fm the particle to be at the positron r 0 at an inltml tame t O and at rf at a final time tf is given by the amplitude sum over all paths r(t) connecting r(to) = r 0 and r(tf) = r t- Apart from a normalization constant, it is equal to
f l-I d3r(t) exp(iAc) ,
(1)
t
where the action A c 1s a functional of r(t), related to the lagrangian L by
determined by the extremity of the action Ac, which as a functional ofr(t). While r is the dynamical variable, t appears only as a continuous parameter. By setting the variational derivative 8Ac/Sr(t ) = 0, we obtain the usual Lagrange equation of motion, whose solution gives the classical path r(t). In the relativistic quantum field theory, space r and time t have to be treated symmetrically due to Lorentz invariance. Our usual approach is to regard r and t all as parameters; the operators are now the field variables, say the scalar field O(r,t). Expression (1) is then replaced by f]-[ r
]'-I d~(r,t) exp(iAc) , t
(3)
tf
a c = f L(r(t), i ( t ) ) dr, to
(2)
with the subscript c denoting the familiar case that t is continuous We note that the position of the particle r is not treated on the same basis as the time t At a given tnne the integration d3r(t) in eq (1) can be viewed as that over the whole range of eagenvalues of the operator rop(t ) This then underlies the familiar difference between r as an operator and t as a parameter Actually this asymmetry can be traced back to classical mechanics The classical trajectory of a particle is ¢~This research was supported m part by the US Department of Energy.
0 031-9163/83/0000-0000/$ 03 00 © 1983 North-Holland
where the action A c is related to the lagranglan density /2 by A c = f z ? (qS,V ~b,~) d3r dt
(4)
Again, the integration over d4~(r,t) can be viewed as that over the eigenvalues of the operator q~op(r,t). In the following we wish to explore some alternative possabthties. In place of treating time as a continuous parameter, we may ask. (1) Can time be a discrete parameter (discrete time formulation)9 (2) Can time be discrete and treated as a bona fide dynamical variable (discrete mechanics)9 As we shall see, both possibilities can be realized in all stages of mechanics. In a relatlvastac field theory, the discrete tune formulation becomes the discrete space217
Volume 122B, number 3,4
PHYSICS LETTERS
time formulation. In the corresponding discrete mechanics both space and time, in addition to being discrete, are treated as genuine dynamxcal variables. Because this approach involves a fundamental change in our concept of s p a c e - t i m e , we will develop our ideas gradually first in classical mechanics, then in nonrelatlvistlc quantum mechanics and finally in relativistic quantum field theory. The result is that in this new formalism our usual idea of continuous time or continuous s p a c e time structure will appear only as an approximation. 2 Classical m e c h a n i c s . Consider the simple example o f a nonrelativlstic particle of unit mass moving in a potential V(r). The usual lagrangian is (regarding time as continuous) L = ~ r1
.2
(5)
__ V ( r ) ,
The action is then given by the integral (2) from the initial position r ( t o ) = r 0 to the final r(tf) = rf. In the discrete time formalism we replace the continuous function r ( t ) by a sequence of discrete values: ( t o , t o ) , (r 1 , t l ) ..... (rN+ 1 , t w + l ) , where (rN+ 1 , t N + l ) = (rf, tf). The action (2) is then replaced by
N+I A= ~ (1 (rn-rn-1)n= 1
-
2
½% -
9
tn -- t n - 1
tn_l)[V(rn)+
)
V(rn_l) ] ,
(6)
with t n > t n _ l . Newton's equation of motion is derived by setting the derivative 3 A / a r n = 0.
(7)
Keeping the initial and final configurations fixed, we have altogether N such equations:
1
Vn + l - vn = --~ (t n + l - tn - 1 ) V V ( r n )
(8)
where v n = (r n - r n _ l )/(tn - t n - I ) is the velocity. For any given time dxstrlbutlon t l , . . , t N , the positions, r 1 .... , t N can then be determmed. An ad hoc discrete time distribution t l , t2, .., t N is clearly not satisfactory for a closed mechanical system. Thus, in discrete mechanics we require that the time distribution should also be determined by the same action (6). This can be achieved by treating t n as dynalnlcal variables, on the same basis as r n . By setting 3 A / 3 t n = 0,
we derive N additional equations:
E n E 1~v n2 + ~1[ V ( r n ) +
V(r_l) l =L)~+~
(10)
Thus, the entire set ( r l , t 1),...,(rN,t N ) can be determined from (8) and (10). In the continuum case, the energy conservation law is a consequence of Newton's equation of motion In the discrete case, these two are Independent. By treating t n and r n both as dynamical variables, we manage to employ the same action principle for their determination. In our formulation of discrete mechanics, there is a f u n d a m e n t a l l e n g t h or time l (in natural units). Given any time interval T = tf - t 0, the total number N of discrete points that define the trajectory is given by the integer nearest T/l. As illustrations, we may consider some special examples (1) V = 0 In this case (8) gives u n = constant, which also satisfies (10) The trajectory of the particle IS always a straight line, independent of the time distribution. (11) VV = constant. It can be readily verified that (8) and (10) yield equal spacings for the tnne intervals. The trajectory consists of N discrete points (r 1 , t l ) , , (rN, t N ) all lying on a parabola, similar to the continuum case. (For more comphcated potentials, the time intervals are in general of varying lengths.) 3 N o n r e l a t i v l s t i c q u a n t u m m e c h a n i c s . As above, let the initial and final configurations be (r O, t o ) and (rf, tf). In the discrete quantum mechanics, we assume that within the given time interval T = tf - t 0, there can be maximally N measurelnents of the particle's configurations possible, where the ratio N I T = l is the fundamental constant mentioned before. Each measurement yields a set of values ( r n , t n ) where n = 1,2 ..... N. [The initial (r 0` to) can be regarded as the result of a previous experiment, and the final (rl, tf) = (rN+ 1, tN+ l ) as that of the (N + 1)th measurement.] Without any loss of generahty, we may arrange t I , .,t N into an ordered sequence t O < t 1 < .. < t N < tf. The probablhty amphtude of observing such a set (r I , t 1 ) . . . . (r N , t N ) is proportional to exp(1A) with A given by eq. (6) Summing over all possible (r I , t 1 ) . . . . (r N , t N ) we obtain the overall quantum mechanical amplitude ( f l G I 0 ) leading from (r0,t0) to (rf, tf)
N
(9) (flGI0)
218
10 March 1983
=fJ exp(iA) n=l [-I
dt n d3rn ,
(11)
Volume 122B, number 3,4
PHYSICS LETTERS
where J is a jacobian, whmh can be either a simple normalization constant independent o f t n and t n , or some other slowly varying functmns. The classical tralectory is determined by the stationary phase condition 6A = 0 In the path integration (11), the integrations d3rn and dt n may be viewed as those over the eigenvalues of the operators (rn)op and (tn)op. The discreteness in this new mechanics enters through the finite number of measurements allowed in any gwen time interval. The results of each measurement can be any of the continuous eigenvalues of (r n)op and (t n)op. In the usual contlnuuin case, as the time-parameter t vanes continuously there are an infinite number of operators rop(t ). Now, by replacing these with a finite number of operators we have elevated "time" to the status of an operator, the same as "space". As an illustration, let us consider the special case that the jacoblan is
10 March 1983
There is quite a latitude in the choice of the jacoblan. For example, if instead of (12), J is simply a normahzanon constant independent of t n and rn, then to O(l 0) the effective hamiltonian remains the same H0; the O(/2) correction differs from (16) only by a simple numerical factor. Different J correspond to different discrete mechamcal systems that have the same continuum liniit
4. Analytical continuation. As in the continuum case, it is often useful to introduce r = it and consider the analytic continuation of the region of real t to that of real r. For a nonrelatlvistic particle, the action A in (6) as replaced by N+I (1 (rn - - r n - l ) 2
~--iA=,,=~ 2 r.-rn_I+ ½(rn - rn - 1 ) [ V(rn ) + V(rn - 1) ] ) '
N+I 1 ]3/2 j = N! l-] ( • \ T N n=l 27re,~]
(12)
where en =- t n - tn_ 1 > O, G is then related to the operator U(e) ==-e x p ( l ~.1c V ) e x p 0.1~ e g 2 ) exp(1½eV) (13)
(17)
The mlnnnum of ~ determines the classical path (r 1 , r 1) ..... (rN, rN). As is well known, the resulting trajectory as one in whmh the particle is mowng as i f m a "potential" - V. The corresponding Green's function G in quantum mechamcs is now given by, instead of (11), N
by
G=T -~fU(eN+I)U(eN)U(el)
~1 den,
(14)
where V 2 is the laplaclan operator and T = z N + I en Thus, for example, for a free particle (V = 0), G = exp(i½ TV2), ldentmal to the usual continuum case. In a subsequent paper with R. Friedberg, we shall prove that for arbitrary V when T becomes very large G e x p ( i T H ) (after averaging over an appropriate grand canonical ensemble), where H is a hermitian whose power series expansion in l can be written as
H = H 0 + 12H1 + 0(13),
(15)
in winch H 0 = - ~1~79- + Vis the usual continuum hamiltonlan and H 1 =-~[g 2 -
V,[V 2 ' V ] ]
1--I drnd3rn .
(18)
1
?l
(16)
gives the first correction, We may call H an effectwe hamfltonian. Its hermitIclty insures the unitarity of the S-matrix.
In the following extension of the &screte mechanics to a relativistic theory, we shall stay in the region of real z The usual Lorentz transformation becomes a rotation an the four-dimensional euclidean space. For convenience, we shall now refer to r as "time".
5. Relativistic field theory. The relativistic generahzation is identical to the annealed version of the random lattice field theory [ 1 - 3 ]. Consider the example of a scalar field O(r,r). For any given finite four-dimensional volume g2, we assume that maximally there can only be N measurements, each yielding a set o f values (ri,rl,(~i) , where i = 1,..,N. The ratio gZ/N = l 4 is a fixed constant. The probability amplitude of observing such a sequence (rl, r l , ¢1 ) .... (rN,rN,¢N) is proportional to exp(--s~) where ~ is the action; its integral N
J exp(-~)i_[I1 d¢~ dr~ d3r i
(19) 219
Volume 122B, number 3,4
PHYSICS LETTERS
gives the overall quantum mechanical amplitude where J is the jacoblan which again can be taken as a simple normalization constant, independent o f r t and "ci. By integrating over the entire domain d'ctd3ri , we see that rotational lnvariance is maintained, provided sd is invariant. To construct M , let us consider an arbitrary distribution o f N sites o f coordinates (r 1, ~'l),''',(rN, rN) in ~ . To each site we assign a value q5i for the scalar field. The sites are hnked according to the algorithm given in ref. [ 1 ]. The action N is (20)
lzl where the first sum extends over all hnks ltj and the second over all sites i. The weights X~I and coz are functions of the sites, given explicitly in ref. [3]. This discrete formalism can, of course, be apphed to fields o f higher spins. In the usual continuum relativistic quantum field theory, the s p a c e - t i m e coordinates are not regarded as dynamical variables; e.g., in eq. (3) the integration is over HrIlrd4)(r,r), which corresponds to an infinite number o f operators 4~op(r,r), with r and r as parameters. Here, there is a finite number of operators (rz)op and (rz)op, each of which can have a continuous range of eigenvalues that can be observed. The discreteness appears, as before, only in the maximal number N of measurements allowed in any finite s p a c e - t i m e volume ~2. Each measurement determines the physical
10 March 1983
field as well as the s p a c e - t i m e location of the measurement itself. In this sense, this new approach seems to be nearer to our actual experimental experience. Our usual concept that all physical fields should be embedded in a continuous s p a c e - t i m e manifold may well be only an approximation, far from fundamental. In this new formulation, any finite number of observations may be made very close to each other m space and in time. However, within an infinitesimal s p a c e time volume it is not possible to perform an infinite number of measurements. In the usual continuum quantum field theory, such unphysical possibilities are in principle allowed, and that leads to the well-known complications in the ultraviolet region. If one wishes, one may also regard the fundamental length l of the discrete mechanics as a covarlant regularizatlon parameter for the elimination of ultraviolet divergence. I wish to thank R. Frledberg for discussions. I am grateful for the very kind hospitality of the theoretical group at CERN, especially during the summer of 1982 when most o f the present Idea was formed.
References
(~i)op,
220
[1] N H Christ, R. Frledberg and T D. Lee, Nucl. Phys B202 (1982) 89 [2] N.H. Christ, R Frmdberg and T.D. Lee, Nucl Phys. B210 (1982) 310 [3] N H. Christ, R. Fnedberg and T D Lee, Nucl. Phys. B210 (1982) 337