AN EINSTEIN EQUATION FOR DISCRETE QUANTUM GRAVITY

Transcription

1 AN EINSTEIN EQUATION FOR DISCRETE QUANTUM GRAVITY S. Gudder Department of Mathematics University of Denver Denver, Colorado 80208, U.S.A. Abstract The basic framework for this article is the causal set approach to discrete quantum gravity (DQG). Let Q n be the collection of causal sets with cardinality not greater than n and let K n be the standard Hilbert space of complex-valued functions on Q n. The formalism of DQG presents us with a decoherence matrix D n (x, y), x, y Q n. There is a growth order in Q n and a path in Q n is a maximal chain relative to this order. We denote the set of paths in Q n by Ω n. For ω, ω Ω n we define a bidifference operator n ω,ω on K n K n that is covariant in the sense that n ω,ω leaves D n stationary. We then define the curvature operator R n ω,ω = n ω,ω n ω,ω. It turns out that R n ω,ω naturally decomposes into two parts R n ω,ω = D n ω,ω +T n ω,ω where Dω,ω n is closely associated with D n and is called the metric operator while Tω,ω n is called the mass-energy operator. This decomposition is a discrete analogue of Einstein s equation of general relativity. Our analogue may be useful in determining whether general relativity theory is a close approximation to DQG. 1 Causet Approach to DQG A causal set (causet) is a finite partially ordered set x. Thus, x is endowed with an irreflexive, transitive relation < [1, 8, 10]. That is, a a for all a x 1

2 and a < b, b < c imply that a < c for a, b, c x. The relation a < b indicates that b is in the causal future of a. Let P n be the collection of all causets with cardinality n, n = 1, 2,..., and let P = P n. For x P, an element a x is maximal if there is no b x with a < b. If x P n, y P n+1, then x produces y if y is obtained from x by adjoining a single new element a to x that is maximal in y. In this way, there is no element of y in the causal future of a. If x produces y, we say that y is an offspring of x and write x y. A path in P is a string (sequence) x 1 x 2 where x i P i and x i x i+1, i = 1, 2,.... An n-path in P is a finite string x 1 x 2 x n where again x i P i and x i x i+1. We denote the set of paths by Ω and the set of n-paths by Ω n. We think of ω Ω as a possible universe (or universe history). The set of paths whose initial n-path is ω 0 Ω n is called an elementary cylinder set and is denoted by cyl(ω 0 ). Thus, if ω 0 = x 1 x 2 x n, then cyl(ω 0 ) = {ω Ω: ω = x 1 x 2 x n y n+1 y n+2 } The cylinder set generated by A Ω n is defined by cyl(a) = ω A cyl(ω) The collection A n = {cyl(a): A Ω n } forms an increasing sequence of algebras A 1 A 2 on Ω and hence C(Ω) = A n is an algebra of subsets of Ω. We denote the σ-algebra generated by C(Ω) as A. It is shown [6, 11] that a classical sequential growth process (CSGP) on P that satisfies natural causality and covariance conditions is determined by a sequence of nonnegative numbers c = (c 0, c 1,...) called coupling constants. The coupling constants specify a unique probability measure ν c on A making (Ω, A, ν c ) a probability space. The path Hilbert space is given by H = L 2 (Ω, A, ν c ). If νc n = ν c A n is the restriction of ν c to A n, then H n = L 2 (Ω, A n, νc n ) is an increasing sequence of closed subspaces of H. A bounded operator T on H n will also be considered as a bounded operator on H by defining T f = 0 for all f Hn. We denote the characteristic function of a set A A by χ A and use the notation χ Ω = 1. A q-probability operator is a bounded positive operator ρ n on H n that satisfies ρ n 1, 1 = 1. Denote the set of q-probability operators on H n by Q(H n ). For ρ n Q(H n ) we define the n-decoherence functional [3, 5, 8] D n : A n A n C by D n (A, B) = ρ n χ B, χ A 2

3 The functional D n (A, B) gives a measure of the interference between the events A and B when the system is described by ρ n. It is clear that D n (Ω n, Ω n ) = 1, D n (A, B) = D n (B, A) and A D n (A, B) is a complex measure for all B A n. It is also well known that if A 1,..., A n A n, then the matrix with entries D n (A j, A k ) is positive semidefinite. In particular the positive semidefinite matrix with entries D n (ω i, ω j ) = D n (cyl(ω i ), cyl(ω j )), ω i, ω j Ω n is called the n-decoherence matrix. We define the map µ n : A n R + given by µ n (A) = D n (A, A) = ρ n χ A, χ A Notice that µ n (Ω) = 1. Although µ n is not additive in general, it satisfies the grade-2 additivity condition [2, 3, 5, 7, 9]: if A, B, C A n are mutually disjoint then µ n (A B C) = µ n (A B)+µ n (A C)+µ n (B C) µ n (A) µ n (B) µ n (C) We call µ n the q-measure corresponding to ρ n and interpret µ n (A) as the quantum propensity for the occurrence of the event A A n. A simple example is to let ρ n = I, n = 1, 2,.... Then D n (A, B) = χ B, χ A = ν n c (A B) and µ n (A) = ν n c (A), the classical measure. A more interesting example is to let ρ n = 1 1, n = 1, 2,.... Then D n (A, B) = 1 1 χ B, χ A = ν n c (A)ν n c (B) and µ n (A) = [ν n c (A)] 2, the classical measure square. We call a sequence ρ n Q(H n ), n = 1, 2,..., consistent if D n+1 (A, B) = D n (A, B) for all A, B A n. A quantum sequential growth process (QSGP) is a consistent sequence ρ n Q(H n ) [3, 4]. We consider a QSGP as a model for discrete quantum gravity. It is hoped that additional theoretical principles or experimental data will help determine the coupling constants and hence ν c, which is the classical part of the process, and also the ρ n Q(H n ), which is 3

4 the quantum part. Moreover, it is believed that general relativity will eventually be shown to be a close approximation to this discrete model. Until now it has not been clear how this can be accomplished. However, in Section 3 we shall derive a discrete Einstein equation which might be useful in performing these tasks. 2 Difference Operators Let Q n = n P j be the collection of causets with cardinality not greater than j=1 n and let K n be the (finite-dimensional) Hilbert space C Qn with the standard inner product f, g = x Q n f(x)g(x) Let L n = K n K n which we identify with C Qn Qn having the standard inner product. We shall also have need to consider the Hilbert space { } K = f C P : x P f(x) 2 < with the standard inner product and we define L = K K. Notice that K 1 K 2 K form an increasing sequence of subspaces of K that generate K in the natural way. Let ρ n Q(H n ) be a QSGP with corresponding decoherence matrix D n (ω, ω ), ω, ω Ω n. If ω = ω 1 ω 2 ω n Ω n and ω j = x for some j, we say that ω goes through x. For x, y Q n we define D n (x, y) = {D n (ω, ω ): ω goes through x, ω goes through y} Due to the consistency of ρ n, D n (x, y) is independent of n. That is, D n (x, y)= D m (x, y) if x, y Q n Q m. Moreover, D n (x, y) are the components of a positive semidefinite matrix. We view Q n as the analogue of a differentiable manifold and D n (x, y) as the analogue of a metric tensor. One might think that the elements of causets should be analogous to points of a differential manifold and not the causets themselves. However, if x Q n, then x is intimately related to its producers, each of which determines a unique a x. 4

5 Moreover, if y x we view (y, x) as a tangent vector at x. In this way, there are as many tangent vectors at x as there are producers of x. Finally, the elements of Ω n are analogues of curves and the elements of K n are analogues of smooth functions on a manifold. For x Q n, x denotes the cardinality of x. Notice if ω = ω 1 ω 2 ω n Ω n and ω j = x, then j = x and ω goes through x if and only if ω x = x. We see that a path ω through x determines a tangent vector (ω x 1, ω x ) at x (assuming that x 2). For ω Ω n we define the difference operator n ω on K n by n ωf(x) = [ f(x) f(ω x 1 ) ] δ x,ω x for all f K n, where δ x,ω x is the Kronecker delta. Thus, n ωf(x) gives the change of f along the tangent vector (ω x 1, ω x ) if ω goes through x. It is clear that n ω is a linear operator on K n. We now show that n ω satisfies a discrete form of Leibnitz s rule. For f, g K n we have n ωfg(x) = [ f(x)g(x) f(ω x 1 )g(ω x 1 ) ] δ x,ω x = {[ f(x)g(x) f(x)g(ω x 1 ) ] + [ f(x)g(ω x 1 ) f(ω x 1 )g(ω x 1 ) ]} δ x,ω x = f(x) n ω g(x) + g(ω x 1 ) n ω f(x) (2.1) Of course, it also follows that n ωfg(x) = n ωgf(x) = g(x) n ω f(x) + f(ω x 1 ) n ω g(x) (2.2) Given a function of two variables f C Qn Qn = L n we have a function f K n of one variable where f(x) = f(x, x) and given a function f K n we have the functions of two variables f 1, f 2 L n where f 1 (x, y) = f(x) and f 2 (x, y) = f(y) for all x, y Q n. For ω, ω Ω n we want an operator n ω,ω : L n L n that extends n ω and satisfies a discrete Leibnitz s rule. That is, and n ω,ω f 1(x, y) = n ωf(x)δ y,ω x, n ω,ω f 2(x, y) = n ω f(y)δ x,ω x (2.3) n ω,ω fg(x, y) = f(x, y) ω,ω g(x, y) + g(ω x 1, ω x 1) n ω,ω f(x, y) (2.4) 5

6 Theorem 2.1. A linear operator n ω,ω : L n L n satisfies (2.3) and (2.4) if and only if n ω,ω has the form n ω,ω f(x, y) = [ f(x, y) f(ω x 1, ω 1) ] δ x,ω x δ y,ω (2.5) Proof. If n ω,ω is defined by (2.5), then for f K n we have n ω,ω f 1(x, y) = [ f 1 (x, y) f 1 (ω x 1, ω 1) ] δ x,ω x δ y,ω = [ f(x) f(ω x 1 ) ] δ x,ω x δ y,ω In a similar way, n ω,ω have = n ωf(x)δ y,ω satisfies the second equation in (2.3). Moreover, we n ω,ω f(x, y) = [ f(x, y)g(x, y) f(ω x 1 ω 1)g(ω x 1 ω 1) ] δ x,ω x δ y,ω = [ f(x, y)g(x, y) f(x, y)g(ω x 1, ω 1) ] δ x,ω x δ y,ω + [ f(x, y)g(ω x 1, ω 1) f(ω x 1 ω 1)g(ω x 1 ω 1) ] δ x,ω x δ y,ω = f(x, y) n ω,ω g(x, y) + g(ω x 1, ω 1) n ω,ω f(x, y) Conversely, suppose the linear operator n ω,ω : L n L n satisfies (2.3) and (2.4). If f L n has the form f(x, y) = g(x)h(y), then n ω,ω f(x, y) = n ω,ω gh(x, y) = n ω,ω g 1h 2 (x, y) = g 1 (x, y) n ω,ω h 2(x, y) + h 2 (ω x 1 ω 1) n ω,ω g 1(x, y) = g(x) n ω h(y)δ x,ω x + h(ω 1) n ω g(x)δ y,ω = { g(x) [ h(y) h(ω 1) ] + h(ω 1) [ g(x) g(ω 1 ) ]} δ x,ω x δ y,ω = [ g(x)h(y) g(ω x 1 )h(ω 1) ] δ x,ω x δ y,ω = [ f(x, y) f(ω x 1 ω 1) ] δ x,ω x δ y,ω Since n ω,ω, is linear and every element of L n product functions, the result follows. is a linear combination of 6

7 Of course, Theorem 2.1 is not surprising because (2.5) is the natural extension of n ω to L n. Also n ω,ω extends n ω in the sense that for any f L n we have n ω,ωf(x, x) = [ f(x, y) f(ω x 1, ω x 1 ) ] δ x,ω x [ = f(x) f(ω x 1 )] δ x,ω x = n f(x) ω As before, n ω,ω satisfies n ω,ω fg(x, y) = n ω,ω gf(x, y) = g(x, y) n ω,ω f(x, y) + f(ω x 1, ω 1) n ω,ω g(x, y) The next result characterizes n ω and n ω,ω up to a multiplicative constant. Theorem 2.2. (a) An operator T ω : K n K n satisfies (2.1) and T ω f(x) = 0 if ω x x if and only if there exists a function β ω K n such that T ω = β ω n ω. (b) An operator T ω,ω : L n L n satisfies (2.4) and T ω,ω f(x, y) = 0 if ω x x or ω y if and only if there exists a function β ω,ω L n such that T ω,ω = β ω,ω n ω,ω. Proof. If T ω satisfies (2.1), it follows from (2.2) that f(x)t ω g(x) + g(ω x 1 )T ω f(x) = g(x)t ω f(ω) + f(ω x 1 )T ω g(x) Hence, [ g(x) g(ω x 1) ] T ω f(x) = T ω g(x) [ f(x) f(ω x 1) ] Therefore, if g(x) g(ω x 1 ) 0, we have Letting T ω f(x) = T ω g(x) [ f(x) f(ω x 1 ) ] g(x) g(ω x 1 ) β ω (x) = T ω g(x) g(x) g(ω x 1 ) gives the result. The converse is straightforward. The proof of (b) is similar. 7

8 It is clear that µ n (x) = D n (x, x) is not stationary; that is n ωµ n (x) 0 for all x Q n in general. Is there a function α ω K n such that ( n ω+α ω )µ n (x) = 0 for all x Q n? If α ω exists, we obtain [ µn (x) µ n (ω x 1 ) ] δ x,ω x + α ω(x)µ n (x) = 0 If ω x = x and µ n (x) = 0, this would imply that µ n (ω x 1 ) = 0. Continuing this process would give µ n (ω x 2 ) = µ(ω x 3 ) = = 0 which leads to a contradiction. It is entirely possible for µ n (x) to be zero for some x Q n so we abandon this attempt. How about functions α ω, β ω K n such that (β ω n ω +α ω )µ n (x) = 0 for all x Q n? We then obtain β ω (x) [ µ n (x) µ n (ω x 1 ) ] δ x,ω x + α ω (x)µ n (x) = 0 (2.6) If ω x = x and µ n (x) = 0 but β ω (x) 0 we obtain the same contradiction as before. We conclude that β ω (x) = 0 whenever µ n (x) = 0. The simplest choice of such a β ω is β ω (x) = µ n (x). This choice also has the advantage of being independent of ω. Equation (2.6) becomes µ n (x) [ µ n (x) µ n (ω x 1 ) ] δ x,ω x + α ω (x)µ n (x) = 0 (2.7) If µ n (x) = 0, then (2.7) holds. If µ n (x) 0, then we obtain α ω (x) = [ µ n (ω x 1 ) µ n (x) ] δ x,ω x = n ω µ n (x) The numbers α ω (x) are an analogue of the Christoffel symbols. We call n ω = µ n n ω +α ω the covariant difference operator. The operator n ω is not a difference operator in the usual sense because n ω1 0. Instead, we have n ω1 = α ω. Following the previous steps for n ω,ω we define the covariant bidifference operator n ω,ω = D n n ω,ω +α ω,ω where α ω,ω (x, y) = [ D n (ω x 1, ω 1) D n (x, y) ] δ x,ω x δ y,ω and again, α ω,ω (x, y) are analogous to Christoffel symbols. Notice that n ω,ω D n(x, y) = 0 for all x, y Q n and n ω,ωf(x, x) = n f(x). ω Complete expressions for n ω and n ω,ω are n ωf(x) = [ µ n (ω x 1 )f(x) µ n (x)f(ω x 1 ) ] δ x,ω x (2.8) 8

9 and n ω,ω f(x, y) = [ D n (ω x 1, ω 1)f(x, y) D n (x, y)f(ω x 1, ω 1) ] δ x,ω x δ y,ω (2.9) are weighted differ- The form of (2.8) and (2.9) shows that n ω and n ω,ω ence operators. 3 Curvature Operators The linear operator R n ω,ω : L n L n defined by R n ω,ω = n ω,ω n ω,ω is called the curvature operator. Applying (2.9) we have R n ω,ω f(x, y) = D n (x, y) [f(ω x 1, ω 1 )δ x,ω δ y,ω f(ω x x 1, ω 1)δ ] x,ω x δ y,ω + [D n (ω x 1, ω 1)δ x,ω x δ y,ω D n (ω x 1, ] ω 1 )δ x,ω δ y,ω f(x, y) x If x = y, then (3.1) reduces to (3.1) R n ω,ω f(x, x) = µ n(x) [ f(ω x 1, ω x 1 ) f(ω x 1, ω x 1) ] δ x,ω x + 2i ImD n (ω x 1, ω x 1)f(x, x)δ x,ω x We call the operator D n ω,ω : L n L n given by D n ω,ω f(x, y) = D n (x, y) [f(ω x 1, ω 1 )δ x,ω δ y,ω f(ω x x 1, ω 1)δ ] x,ω x δ y,ω the metric operator and the operator T n ω,ω : L n L n given by Tω,ω n = f(x, y) [D n (ω x 1, ω 1)δ x,ω x δ y,ω D n (ω x 1, ] ω 1 )δ x,ω x δ y,ω f(x, y) 9

10 the mass-energy operator. Then (3.1) gives R n ω,ω = Dn ω,ω + T n ω,ω (3.2) Equation (3.2) is a discrete analogue of Einstein s equation [12]. In this sense, Einstein s equation always holds in the present framework no matter what we have for the quantum dynamics ρ n. One might argue that we obtained this discrete analogue of Einstein s equation just by definition. However, R n ω,ω is a reasonable counterpart of the curvature tensor in general relativity [12] and Dω,ω n is certainly a counterpart of the metric tensor. Equation (3.2) does not give direct information about D n (x, y) and D n (ω, ω ) (which are, after all, what we want to find), but it may give useful indirect information. If we can find D n (ω, ω ) such that the classical Einstein equation is an approximation to (3.2), then this would give information about D n (ω, ω ). Moreover, an important problem in discrete quantum gravity theory is how to test whether general relativity is a close approximation to the theory. Whether Einstein s equation is an approximation to (3.2) would provide such a test. Another variant of a discrete Einstein equation can be obtained by defining the operator R n x,y for x, y Q n by R n x,y = { R n ω,ω : ω x = x, ω = y } With similar definitions for D n x,y and T n x,y we obtain R n x,y = D n x,y + T n x,y In order to consider approximations by Einstein s equation, it may be necessary to let n in (3.2). However, the convergence of the operators depends on D n and will be left to a later paper. In a similar vein, it may be possible that limit operators R ω,ω, D ω,ω and T ω,ω can be defined as (possibly unbounded) operators directly on the Hilbert space L. 4 Matrix Elements We have introduced several operators on K n and L n in Sections 2 and 3. In order to understand such operators more directly, it is frequently useful to write them in terms of their matrix elements. First we denote the standard basis on K n by e n x, x Q n. The matrix that is zero except for a one in 10

11 the xy entry is denoted by e n x e n y and we call this the xy matrix element, x, y Q n. Of course, in Dirac notation, e n x e n y can be considered directly as a linear operator without referring to a matrix. In any case, every linear operator T on K n can be represented uniquely as T = t x,y e n x e n y x,y Q n for t x,y C. In a similar way, e n x e n y, x, y Q n form an orthonormal basis for L n = K n K n and every linear operator T on L n has a unique representation T = { tx,y;x,y e n x e n y Theorem 4.1. If ω = ω 1 ω 2 ω n Ω n, then and n ω = n ( ) e n ω j e n ω j e n ω j 1 j=1 e n x e n y : x, y, x, y Q n } n ω = n ] e n ω j [µ n (ω j 1 ) e n ω j µn (ω j ) e nωj 1 j=1 where we use the conventions µ n (ω 0 ) = e n ω 0 = e n ω n+1 = 0. Proof. We first observe that n ( e ) e n n ω j ωn e n ω j 1 e n ω k = e n ω k e n ω k+1 j=1 On the other hand n ωe n ω k (x) = [ e n ω k (x) e n ω k (ω x 1 ) ] δ x,ω x (4.1) Now the right side of (4.1) is zero if ω x x, 1 if ω k = x and 1 if ω k+1 = x. The first result now follows. The second result is similar. The proof of the next theorem is similar to that of Theorem

12 Theorem 4.2. If ω = ω 1 ω 2 ω n, ω = ω 1ω 2 ω n Ω n, then n ω,ω = and n [ ] e n ω j e n ω e n k ω j e n ω k e n ω j 1 e n ω k 1 j,k=1 n ω,ω = n [ ] e n ω j e n ω D k n (ω j 1, ω k 1) e n ω j e n ω D k n (ω j, ω k) e n ω j 1 e n ω k 1 j,k=1 It follows from Theorem 4.2 that R n ω,ω = n ω,ω n ω,ω n [ = D n (ω j 1, ω k 1) e n ω j e n ω e n k 1 ω j e n ω k j,k=1 ] D(ω j 1, ω k 1 ) e n ω j en ω k e n ω j en ω k n [ + D n (ω k, ω j ) e n ω j en ω k e n ω j 1 en ω k 1 j,k=1 ] D(ω j, ω k) e n ω j e n ω e n k ω j 1 e n ω k 1 (4.2) The matrix element representations of Dω,ω n from (4.2) and T n ω,ω can now be obtained References [1] L. Bombelli, J. Lee, D. Meyer and R. Sorkin, Spacetime as a casual set, Phys. Rev. Lett. 59 (1987), [2] F Dowker, S. Johnston and S. Surya, On extending the quantum measure, arxiv: quant-ph (2010). [3] S. Gudder, Discrete quantum gravity, arxiv: gr-gc (2011). [4] S. Gudder, Models for discrete quantum gravity, arxiv: gr-gc (2011). 12

13 [5] J. Henson, Quantum histories and quantum gravity, arxiv: gr-gc (2009). [6] D. Rideout and R. Sorkin, A classical sequential growth dynamics for causal sets, Phys. Rev. D 61 (2000), [7] R. Sorkin, Quantum mechanics as quantum measure theory, Mod. Phys. Letts. A 9 (1994), [8] R. Sorkin, Causal sets: discrete gravity, arxiv: gr-qc (2003). [9] R. Sorkin, Toward a fundamental theorem of quantal measure theory, arxiv: hep-th (2011) and Math. Struct. Comp. Sci. (to appear) [10] S. Surya, Directions in causal set quantum gravity, arxiv: gr-qc (2011). [11] M. Varadarajan and D. Rideout, A general solution for classical sequential growth dynamics of causal sets, Phys. Rev. D 73 (2006), [12] R. Wald, General Relativity, University of Chicago Press, Chicago