An Isometric Dynamics for a Causal Set Approach to Discrete Quantum Gravity

Size: px

Start display at page:

Download "An Isometric Dynamics for a Causal Set Approach to Discrete Quantum Gravity"

Stella Thornton
5 years ago
Views:

University of Denver Digital Commons @ DU Mathematics Preprint Series Department of Mathematics 214 An Isometric Dynamics for a Causal Set Approach to Discrete Quantum Gravity S.

1 University of Denver Digital DU Mathematics Preprint Series Department of Mathematics 214 An Isometric Dynamics for a Causal Set Approach to Discrete Quantum Gravity S. Gudder Follow this and additional works at: Part of the Mathematics Commons Recommended Citation Gudder, S. (214). An Isometric Dynamics for a Causal Set Approach to Discrete Quantum Gravity. Mathematics Preprint Series. Retrieved from This Article is brought to you for free and open access by the Department of Mathematics at Digital DU. It has been accepted for inclusion in Mathematics Preprint Series by an authorized administrator of Digital DU. For more information, please contact jennifer.cox@du.edu,dig-commons@du.edu.

2 AN ISOMETRIC DYNAMICS FOR A CAUSAL SET APPROACH TO DISCRETE QUANTUM GRAVITY S. Gudder Department of Mathematics University of Denver Denver, Colorado 828, U.S.A. sgudder@du.edu Abstract We consider a covariant causal set approach to discrete quantum gravity. We first review the microscopic picture of this approach. In this picture a universe grows one element at a time and its geometry is determined by a sequence of integers called the shell sequence. We next present the macroscopic picture which is described by a sequential growth process. We introduce a model in which the dynamics is governed by a quantum transition amplitude. The amplitude satisfies a stochastic and unitary condition and the resulting dynamics becomes isometric. We show that the dynamics preserves stochastic states. By doubling down on the dynamics we obtain a unitary group representation and a natural energy operator. These unitary operators are employed to define canonical position and momentum operators. 1 Microscopic Picture We call a finite poset (x, <) a causet and interpret a < b in x to mean that b is in the causal future of a. If x and y are causets with cardinality y = x + 1, then x produces y (denoted x y) if y is obtained from x by adjoining a 1

3 single maximal element to x. If x y we call y an offspring of x. A labeling for a causet x is a bijection l: x {1, 2,..., x } such that a, b x with a < b implies l(a) < l(b). We then call x = (x, l) a labeled causet. A labeling of x corresponds to a birth order for the elements of x. Two labeled causets x, y are isomorphic if there is a bijection φ: x y such that a < b in x if and only if φ(a) < φ(b) in y and l [φ(a)] = l(a) for all a x. A causet is covariant if it has a unique labeling (up to isomorphisms) and we call a covariant causet a c-causet. Covariance corresponds to the properties of a manifold being independent of the coordinate system used to describe it. Denote the set of c-causets with cardinality n by P n and the set of all c-causets by P. It is shown in [3] that any x P with x has a unique producer in P and precisely two offspring in P. It follows that P n = 2 n 1, n = 1, 2,.... For more background concerning the causet approach to discrete quantum gravity we refer the reader to [4, 5, 7]. For more information about c-causets the reader can refer to [1, 2, 3]. Two elements a, b x are comparable if a < b or b < a. We say that a is a parent of b and b is a child of a if a < b and there is no c x with a < c < b. A path from a to b in x is a sequence a 1 = a, a 2,..., a n 1, a n = b where a i is a parent of a i+1, i = 1,..., n 1. The height h(a) of a x is the cardinality minus one of a longest path in x that ends with a. If there is no such path, we set h(a) =. It is shown in [3] that a causet x is covariant if and only if a, b x are comparable whenever h(a) h(b). If x P, we call the sets S j (x) = {a x: h(a) = j}, j =, 1, 2,... shells and the sequence of integers s j (x) = S j (x), j =, 1, 2,... is the shell sequence for x [1]. A c-causet is uniquely determined by its shell sequence and we think of {s j (x)} as describing the shape or geometry of x. The tree (P, ) can be thought of as a growth model and an x P n is a possible universe at step (time) n. An instantaneous universe x P n grows one element at a time in one of two ways. If x P n has shell sequence (s (x), s 1 (x),..., s m (x)), then x x or x x 1 where x, x 1 have shell sequence (s (x), s 1 (x),..., s m (x) + 1) and (s (x), s 1 (x),..., s m (x), 1), respectively. In this way, we recursively order the c-causets in P using the notation 2

4 x n,j, n = 1, 2,..., j =, 1, 2,..., 2 n 1 1, where n = x n,j. For example, in terms of their shell sequences we have: x 1, = (1), x 2, = (2), x 2,1 = (1, 1), x 3, = (3), x 3,1 = (2, 1), x 3,2 = (1, 2), x 3,3 = (1, 1, 1) x 4, = (4), x 4,1 = (3, 1), x 4,2 = (2, 2), x 4,3 = (2, 1, 1), x 4,4 = (1, 3), x 4,5 = (1, 2, 1) x 4,6 = (1, 1, 2), x 4,7 = (1, 1, 1, 1) In the microscopic picture, we view a c-causet as a framework or scaffolding for a possible universe. The vertices of x represent small cells that can be empty or occupied by a particle. The shell sequence for x gives the geometry of the framework. In [1] we have shown how to construct a metric or distance function on x. This metric has simple and useful properties. However, the present paper is mainly devoted to the macroscopic picture and the quantum dynamics that can be developed in that picture. Figure 1 illustrates the first four steps of the sequential growth process (P, ). Notice that this is a multiverse model in which infinite paths represent the histories of completed universes [4]. 2 Macroscopic Picture We now study the macroscopic picture which describes the evolution of a universe as a quantum sequential growth process. In such a process, the probabilities and propensities of competing geometries are determined by quantum amplitudes. These amplitudes provide interferences that are characteristic of quantum systems. A transition amplitude is a map ã: P P C satisfying ã(x, y) = if x y and y P ã(x, y) = 1 for every x P. Since x n,j only has the offspring x n+1,2j and x n+1,2j+1 we have that 1 ã(x n,j, x n+1,2j+k ) = 1 (2.1) for all n = 1, 2,..., j =, 1, 2,..., 2 n 1 1. We call ã a unitary transition amplitude (uta) if ã also satisfies y P ã(x, y) 2 = 1 or as in (2.1) we have 1 ã(x n,j, x n+1,2j+k ) 2 = 1 (2.2) 3

5 x 4, x 4,1 x 3, x 4,2 x 4,3 x 4,4 x 4,5 x 4,6 x 4,7 x 3,1 x3,2 x 3,3 x 2, x 2,1 x 1, Figure 1 (Ordering for first four steps.) 4

6 One might suspect that these restrictions on a uta are so strong that the possibilities are very limited. This would be true if ã were real valued. In this case, ã(x, y) = 1 for one y with x y and ã(x, y) =, otherwise. However, in the complex case, the next result shows that there are a continuum of possibilities. Theorem 2.1. Two complex numbers a, b satisfy a + b = a 2 + b 2 = 1 if and only if there exists a θ [, π) such that a = cos θe iθ and b = i sin θe iθ. Moreover, θ is unique. Proof. Necessity is clear. For sufficiency, suppose the conditions a + b = a 2 + b 2 = 1 hold. Then 1 = a 2 + b 2 = a a 2 = a 2 + (1 a)(1 a) = 1 2 Re a + 2 a 2 Hence, a 2 = Re a. Letting a = a e iθ we have that a 2 = a cos θ. If a =, the result holds with θ = π/2. If a, we have that a = cos θ and Re a = a cos θ. Hence, a = cos θe iθ and b = 1 cos θe iθ = 1 cos 2 θ i cos θ sin θ = sin θ(sin θ i cos θ) = i sin θe iθ Uniqueness follows from the fact that cos θ is injective on [, π). If ã: P P C is a uta, we call c k n,j = ã(x n,j, x n+1,2j+k ), k =, 1 the coupling constants for ã. It follows from Theorem 2.1 that there exist θ n,j [, π) such that c n,j = cos θ n,j e iθ n,j, c 1 n,j = i sin θ n,j e iθ n,j It follows that c n,j + c 1 n,j = c 2 n,j + c 1 2 n,j = 1 for all n = 1, 2,..., j =, 1, 2..., 2 n 1 1. Let H n be the Hilbert space with the standard inner product H n = L 2 (P n ) = {f : P n C} f, g = x P n f(x)g(x) 5

7 A path in P is a sequence ω = ω 1 ω 2 where ω i P i and ω i ω i+1 Similarly, an n-path has the form ω = ω 1 ω 2 ω n where again ω i P i and ω i ω i+1. We denote the set of paths by Ω and the set of n-paths by Ω n. Since every x P n has a unique n-path terminating at x, we can identify P n with Ω n and we write P Ω n. Similarly, we identify H n with L 2 (Ω n ). If ã is a uta and ω = ω 1 ω 2 ω n Ω n, we define the amplitude of ω to be a(ω) = ã(ω 1, ω 2 )ã(ω 2, ω 3 ) ã(ω n 1, ω n ) Moreover, we define the amplitude of x P n to be a(ω) where ω Ω n terminates at x. Let x n, be the unit vector in H n given by the characteristic function χ xn,j. Then clearly, { x n,j : j =, 1,..., 2 n 1 1} forms an orthonormal basis for H n. Define the operators U n : H n H n+1 by U n x n,j = and extend U n to H n by linearity. 1 c k n,j x n+1,2j+k Theorem 2.2. (i) The adjoint of U n is given by U n : H n+1 H n where U n x n+1,2j+k = c k n,j x n,j, k =, 1 (2.3) (ii) U n is a partial isometry with UnU n = I n and 2 n 1 1 U n Un 1 1 = c k n,j x n+1,2j+k c k n,j x n+1,2j+k (2.4) j= Proof. (i) To show that (2.3) holds, we have Un x n+1,2j +k, x n,j = x n+1,2j +k, U n x n,j 1 = x n+1,2j +k, c k n,k x n+1,2j+k = c k n,jδ jj δ kk = (ii) To show that U nu n = I n we have by (i) that c k n,j x n,j x n,j U nu n x n,j = 1 c k n,jun x n+1,2j+k = 6 1 c k n,j 2 x n,j = x n,j

8 Since { x n,j : j =, 1,..., 2 n 1 1} forms an orthonormal basis for H n, the result follows. Equation (2.4) holds because it is well-known that U n Un is the projection onto the range of R(U n ). We can also show this directly as follows 2 n c k n,j x n+1,2j+k c k n,j x n+1,2j+k x n+1,2j +k j= = 2 n 1 1 j= 1 c k n,j x n+1,2j+k c k n,j δ jj = ck n,j = c k n,j U n x n,j = U n U n x n+1,2j +k 1 c k n,j x n+1,2j +k It follows from Theorem 2.2 that the dynamics U n : H n H n+1 for a uta ã is an isometric operator. As usual a state on H n is a positive operator ρ on H n with tr(ρ) = 1. A stochastic state on H n is a state ρ that satisfies ρ1 n, 1 n = 1 where 1 n = χ Pn ; that is, 1 n (x) = 1 for every x P n. Notice that U n1 n+1 = 1 n. Lemma 2.3. (i) If ρ is a state on H n, then U n ρu n is a state on H n+1. (i) If ρ is a stochastic state on H n, then U n ρu n is a stochastic state on H n+1. Proof. (i) To show that U n ρ n U n is positive, we have U n ρu nφ, φ = ρu nφ, U nφ for all φ H n+1. Moreover, by Theorem 2.2(ii) we have (ii) Since U n1 n+1 = 1 n we have tr(u n ρu n) = tr(u nu n ρ) = tr(ρ) = 1 U n ρu n1 n+1, 1 n+1 = ρu n1 n+1, U n1 n+1 = ρ1 n 1 n = 1 Denoting the time evolution of states by ρ n ρ n+1, Lemma 2.3 shows that ρ U n ρu n gives a quantum dynamics for states. We now show this explicitly for the transition amplitude. Since x n+1,2j+k, U n x n,j = c k n,j = ã(x n,j, x n+1,2j+k ) we have for every ω = ω 1 ω 2 ω n Ω n that a(ω) = ω 2, U 1 ω 1 ω 3, U 2 ω 2 ω n, U n 1 ω n 1 Define the operator ρ n on H n by ω, ρ n ω = a(ω)a(ω ) where ω = χ {ω} H n for any ω Ω n. 7

9 Theorem 2.4. The operator ρ n is a stochastic state on H n. Proof. To show that ρ n is positive we have f, ρ n f = γi, f γ i, ρ n γj, f γ j = γ i, f γ j, f γ i, ρ n γ j = γ i, f γ j, f a(γ i )a(γ i ) = a(γ i ) γ i, f 2 To show that ρ n is a state on H n we have that tr(ρ n ) = γ i, ρ n γ i = a(γ i )a(γ i ) = a(γ i ) 2 = ω 2, U 1 ω 1 2 ω 3, U 2 ω 2 2 ω n, U n 1 ω n 1 2 ω 2 ω 3 ω n = ω 2, U 1 ω 1 2 ω 2 ω 3 ω n 1 ω n 1, U n 2 ω n 2 2 ω n ω n, U n 1 ω n 1 2 = ω 2, U 1 ω 1 2 ω n 1, U n 2 ω n 2 2 ω 2 ω 3 ω n 1. = ω 2 ω 2, U 1 ω 1 2 = 1 Finally, ρ n is stochastic on H n because 1 n, ρ n 1 n = γi, ρ γ j = γ i, ρ n γ j i,j = a(γ i )a(γ j ) = a(γ i ) 2 i,j 8

10 As before, we obtain a(ω) = ω 2, U 1 ω 1 ω 3, U 2 ω 2 ω n, U n 1 ω n 1 ω Ω n ω 2 ω 3 ω n = ω 2, U 1 ω 1 ω 3, U 2, ω 2 ω n 1 U n 2 ω n 2 ω 2 ω 3 ω n 1. = ω 2, U 1 ω 1 = 1 ω 2 If ω = ω 1 ω 2 ω n Ω n, we have seen that ω n produces two offspring ω n,, ω n,1 P n+1. We call the set (ω ) = {ω 1 ω 2 ω n ω n,, ω 1 ω 2 ω n ω n,1 } Ω n+1 the one-step causal future of ω. We say that the sequence ρ n is consistent if (ω ), ρ n+1 (ω ) = ω, ρ n ω for every ω, ω Ω n where (ω ) = χ (ω ). Consistency is important because it follows that the probabilities and propensities given by the dynamics ρ n are conserved in time [2, 3]. Theorem 2.5. The sequence ρ n is consistent. Proof. Let ω = ω 1 ω 2 ω n, ω = ω 1, ω 2 ω n Ω n and suppose that ω n ω n,, ω n,1 and ω n ω n,, ω n,1. We then have (ω ), ρ n+1 (ω ) = (ωω n, ) + (ωω n,1 ), ρ n+1 [ (ω ω n,) + (ω ω n,1) ] = (ωω n, ), ρ n+1 (ω ω n,) + (ωω n, ), ρ n+1 (ω ω n,1) + (ωω n,1 ), ρ n+1 (ω ω n,) (ωω n,1 ), ρ n+1 (ω ω n,1) = a(ω)ã(ω n ω n, )a(ω ) [ ã(ω n, ω n,) + ã(ω n, ω n,1) ] + a(ω)ã(ω n, ω n,1 )a(ω ) [ ã(ω n, ω n,) + ã(ω n, ω n,1) ] = a(ω)a(ω ) = ω, ρ n ω 9

11 The n-decoherence functional is the map D n : 2 Ωn 2 Ωn C defined by [4, 5, 7] D n (A, B) = χ B, ρ n χ A The functional D n (A, B) gives a measure of the interference between A and B when the system is in the state ρ n. Clearly D n (Ω n, Ω n ) = 1, D n (A, B) = D n (A, B) and A D n (A, B) is a complex measure for every B 2 Ωn. It is also well-known that if A 1,..., A n 2 Ωn, then the matrix with entries D n (A j, A k ) is positive semidefinite [5]. Notice that D n ({ω}, {ω }) = a(ω)a(ω ) for every ω, ω Ω n and D(A, B) = } {a(ω)a(ω ): ω A, ω B Since ρ n is consistent, we have that D n+1 ((A ), (B )) = D n (A, B) for every A, B 2 Ωn where (A ) = {(ω ): ω A}. The corresponding q-measure [2, 5, 6] is the map µ n : 2 Ω R + defined by µ n (A) = D n (A, A) = χ A, ρ n χ A It follows that µ n (Ω n ) = 1 and µ n+1 ((A )) = µ n (A) for all A 2 Ωn. Although µ n is not additive, it satisfies the grade 2-additive condition: if A, B, C 2 Ωn are mutually disjoint then [4, 5, 6, 7] 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) Since µ n is not a measure we do call it a probability but we interpret µ n (A) as the quantum propensity for the occurrence of A. We have discussed in [2, 3] ways of extending the µ n s to a q-measure µ on suitable subsets of Ω. A uta is completely stationary (cs) with parameter θ [, π) if θ n,j = θ for all n, j. For example, let ã be cs with parameter. Then the path x 1, x 2, x 3, has q-measure 1 and all other paths have q-measure. Now consider a general cs uta ã with parameter θ (, π). When a path turns left ã has the value cos θe iθ and when it turns right ã has the value i sin θe iθ. Hence if ω Ω n turns left l times and right r times we have a(ω) = (cos θ) l ( i) r (sin θ) r e inθ 1

12 We then have µ n ({ω}) = a(ω) 2 = cos θ 2l sin θ 2r Hence, lim n µ n ({ω}) = and is is natural to define µ ({ω}) =. A vector v = 2 n 1 1 j= v j x n,j = (v o, v 1,..., v 2 n 1 1) H n is called a stochastic state vector if v = 1 and v, 1 n = 1. we call the vector â n = (a(x n,o ), a(x n,1 ),..., a(x n,2 n 1 1)) H n an amplitude vector. Of course, â n is a stochastic state vector. Theorem 2.6. (i) If v H n is a stochastic state vector, then U n v H n+1 is also. (ii) U n â n = â n+1. (iii) U nâ n+1 = â n. Proof. (i) This follows from the fact that U n is isometric and U n1 n+1 = 1 n. (ii) This holds because 2n n 1 1 [ ] U n â n = U n a(x n,j ) x n,j = a(xn,j )c n,j x n+1,2j + a(x n,j )c 1 n,j x n+1,2j+1 = 2 n 1 j+ j= j= a(x n+1,j ) x n+1,j = â n+1 (iii) This is obtained from U nâ n+1 = [a(x n+1,2j ) x n+1,2j + a(x n+1,2j+1 ) x n+1,2j+1 ] = [ a(xn+1,2j )c n,j x n,j + a(x n+1,2j+1 )c 1 n,j x n,j ] = [ c n,j a(x n,j )c n,j + c 1 n,ja(x n,j )c 1 n,j] xn,j = a(x n,j ) x n,j = â n Actually, (iii) follows from (ii) in Theorem 2.6 because â n+1 = U n â n R(U n ) so U nâ n+1 = U nu n â n = â n. Interference in P n or Ω n can be described by the nonadditivity of the q-measure µ n. We say that x, y P n do not interfere if µ n ({x, y}) = µ n ({x}) + µ n ({y}) The next result gives an application of this concept. 11

13 Theorem 2.7. If x, y P have the same producer, then x and y do not interfere. Proof. Suppose x = x n+1,2y, y = x n+1,2j+1 so x, y have the same producer x n,j. Then µ n+1 ({x, y}) = a(x) + a(y) 2 = a(x n,j )c n,j + a(x n,j )c 1 2 n,j = a(x n,j ) 2 c n,j + c 1 2 n,j = a(x n,j ) 2 [ = a(x n,j ) 2 c n,j 2 ] + c 1 n,j 2 = a(x n,j )c 2 n,j + a(x n,j )c 1 2 n,j = a(x n+1,2j ) 2 + a(x n+1,2j+1 ) 2 = µ n+1 ({x}) + µ n+1 ({y}) Hence, x and y do not interfere. In general, the noninterference result in Theorem 2.7 does not hold if x and y have different producers. This is shown in the next two examples. Example 1. For simplicity, suppose the uta is cs so we have just two coupling constants c, c 1. We have seen in Theorem 2.7 that x 3, and x 3,1 do not interfere. In a similar way, we see that x 3, and x 3,2 do not interfere. We also have that x 3,1 does not interfere with x 3,j, j =, 1, 2 and x 3,2 does not interfere with x 3,j, j =, 1, 3. Let us now consider x 3, and x 3,3. We have that On the other hand µ 3 ({x 3,, x 3,3 }) = a(x 3, ) + a(x 3,3 ) 2 = (c ) 2 + (c 1 ) 2 2 = cos 2 θ sin 2 θ = cos 2 2θ µ 3 ({x 3, }) + µ 2 ({x 3,3 }) = a(x 3, ) 2 + a(x 3,3 2 so x 3, and x 3,3 interfere, in general. = cos 4 θ + sin 4 θ = 1 2 (1 + cos2 2θ) Example 2. If the uta is not cs, the situation is more complicated and we incur more interference. In the cs case, we saw in Example 1 that x 3, and x 3,2 do not interfere. However, in this more general case we have µ 3 ({x 3,, x 3,2 }) = a(x 3, ) + a(x 3,2 ) = c 1, c 2, + c 1 1,c 2 2,1 12

14 On the other hand µ 3 ({x 3, }) + µ 3 ({x 3,2 })= a(x 3, ) 2 + a(x 3,2 ) 2 = c 1, 2 c 2, 2 + c 1 1, 2 c 2,1 2 But these two quantities do not agree unless Re(c 1,c 2,c 1 1,c 2,1) = so x 3, and x 3,3 interfere, in general. 3 Double-Down To Unitary We have seen that corresponding to a uta with coupling constants c k n,j there are isometries U n : H n H n+1 that describe the dynamics for a quantum sequential growth process on (P, ). The operators U n cannot be unitary because H n and H n+1 are different dimensional Hilbert spaces. However, we can double-down the U n to form operators V n+1 : H n+1 H n+1 by V n+1 x n+1,2j = c n,j x n+1,2j + c 1 n,j x n+1,2j+1 V n+1 x n+1,2j+1 = c 1 n,j x n+1,2j + c n,j x n+1,2j+1 Theorem 3.1. The operators V n+1 are unitary and V n+1 1 n+1 = 1 n+1, n = 1, 2,.... Proof. Since V n+1 x n+1,2j = V n+1 x n+1,2j+1 = 1 and V n+1 x n+1,2j, V n+1 x n+1,2j+1 = c n,jc 1 n,j + c 1 n,jc n,j = we conclude that V n+1 sends an orthonormal basis to an orthonormal basis. Hence, V n+1 is unitary. To show that V n+1 1 n+1 = 1 n+1 we have V n+1 1 n+1 = j = j (V n+1 x n+1,2j + V n+1 x n+1,2j+1 ) [ (c n,j + c 1 n,j) x n+1,2j + (c 1 n,j + c n,j) x n+1,2j+1 ] = j ( x n+1,2j + x n+1,2j+1 ) = 1 n+1 13

15 The unitary operator V 2 corresponds to the coupling constants c 1,, c 1 1, and relative to the basis { x 2,, x 2,1 } has the form [ ] c 1, c 1 1, V 2 = c 1 1, c 1, Besides being unitary, V 2 is doubly stochastic (row and column sums are one). Of course, this is also true of V n. By Theorem 2.1, there exists a unique θ [, π) such that c 1, = cos θe iθ, c 1 1, = i sin θe iθ. To make θ explicit, we write V 2 = V 2 (θ). Lemma 3.2. The operator V 2 (θ) has eigenvalues 1, e 2iθ with corresponding unit eigenvectors 2 1/2 (1, 1), 2 1/2 (1, 1). Proof. By direct verification we have [ ] [ ] 1 c 1, + c 1 [ ] 1, 1 V 2 = = 1 c 1 1, + c 1 1, [ ] [ ] 1 c 1, c 1 [ ] 1, 1 V 2 = = (c 1 c 1 1, c 1, c 1 1,) 1 1, But c 1, c 1 1, = c 1, (1 c 1,) = 2c 1, 1 = 2 cos θe iθ 1 = 2 cos 2 θ + 2i cos θ sin θ 1 = cos 2 θ sin 2 θ + i sin 2θ = cos 2θ + i sin 2θ = e 2iθ We can write the 2 n -dimensional Hilbert space H n+1 as H n+1 = H 2 H 2 H 2 where there are 2 n 1 summands and the jth summand has the basis{ x n+1,2j, x n+1,2j+1 }. In general, V n+1 has the form V n+1 (θ 1, θ 2,..., θ 2 n 1) = V 2 (θ 1 ) V 2 (θ 2 ) V (θ 2 n 1) It follows from Lemma 3.2 that V n+1 (θ 1, θ 2,..., θ 2 n 1) has eigenvalues 1 (with multiplicity 2 n 1 ) and e 2iθ 1, e 2iθ 2,..., e 2iθ 2 n 1. The unit eigenvectors corresponding to 1 are 2 1/2 ( x n+1,2j + x n+1,2j+1 ), j =, 1,..., 2 n

16 and the unit eigenvector corresponding to e 2iθj is 2 1/2 ( x n+1,2j x n+1,2j+1 ) Let S(H n+1 ) be the set of operators on H n+1 of the form S(H n+1 ) = {V n+1 (θ 1, θ 2,..., θ 2 n 1): θ n [, π)} Now [, π) forms an abelian group with operations a b = a + b (mod π). Lemma 3.3. For θ 1, θ 2 [, π) we have V 2 (θ 1 )V 2 (θ 2 ) = V 2 (θ 1 + θ 2 ). Proof. Since V 2 (θ 1 ) and V 2 (θ 2 ) have the same eigenvectors, they commute and can be simultaneously diagonalized as [ ] [ ] 1 1 V 2 (θ 1 ) = V e 2iθ 1 2 (θ 2 ) = e 2iθ 2 Hence, if θ 1 + θ 2 < π then V 2 (θ 1 )V 2 (θ 2 ) = [ 1 e 2i(θ 1+θ 2 ) ] = V 2 (θ 1 θ 2 ) and if θ 1 + θ 2 π then θ 1 θ 2 = θ 1 + θ 2 π and we have [ ] 1 V 2 (θ 1 )V 2 (θ 2 ) = e 2i(θ 1+θ 2 = V π) 2 (θ 1 + θ 2 π) = V (θ 1 θ 2 ) We now form the product group [, π) 2n 1 = [, π) [, π) to obtain the following result. Corollary 3.4. Under operator multiplication, S(H n+1 ) is an abelian group and (θ 1,..., θ 2 n 1) V n+1 (θ 1,..., θ 2 n 1) is a unitary representation of the group [, π) 2n 1. Since V n+1 is unitary, there exists a unique self-adjoint operator K n+1 on H n+1 such that V n+1 = e ik n+1. We call K n+1 a Hamiltonian operator. For V n+1 (θ 1,..., θ 2 n 1 the eigenvalues of K n+1 are (with multiplicity 2 n 1 ) and 2θ 1,..., 2θ 2 n 1. Hence, θ j = 2 1 λ j where λ j is the jth energy value, 15

17 j = 1,..., 2 n 1. This gives a physical significance for the angles θ j. The corresponding eigenvectors are the same as those given for V n+1. It is natural to define the position operator Q n+1 on H n+1 by Q n+1 f( x n+1,k ) = k. Thus, Q n+1 x n+1,2j = 2j and Q n+1 x n+1,2j+1 = 2j + 1. Since Q n+1 is diagonal, we immediately see that its eigenvalues are, 1,..., 2 n 1 with corresponding eigenvector x n+1,k. It also seems natural to define the canonical momentum operator P n+1 on the subspace generated by { x n+1,2j, x n+1,2j+1 } as P 2 (θ j ) = V 2 (θ j ) Q 2 (θ j )V 2 (θ j ) [ ] [ ] [ ] c n,j c 1 n,j 2j c n,j c 1 n,j = c 1 n,j c n,j 2j + 1 c 1 n,j c n,j = 2j + [ c 1 n,j 2 c n,jc 1 n,j 2j + sin c n,jc 1 n,j 2j + 2 i θ n,j 2 c n,j 2 = sin 2θ ] n,j i sin 2θ 2 n,j 2j + cos 2 θ n,j The eigenvalues of P 2 (θ j ) are 2j and 2j + 1 with corresponding unit eigenvectors [ ] [ ] 1 V 2 (θ j ) c = n,j c 1 n,j [ ] [ ] V 2 (θ j ) c = 1 n,j 1 c n,j The complete momentum operator P n+1 is given by P n+1 (θ 1,..., θ 2 n 1) = P 2 (θ 1 ) P 2 (θ 2 ) P 2 (θ 2n 1 ) We now compute the commutator [P 2 (θ j ), Q 2 (θ j )] = P 2 (θ j )Q 2 (θ j ) Q 2 (θ j )P 2 (θ n ) = c n,jc 1 n,j [ ] 1 [ ] 1 1 = i 2 sin 2θ j 1 16

18 The complete commutation relation is [P n+1 (θ 1,..., θ 2 n 1), Q n+1 (θ 1,..., θ 2 n 1)] = [P 2 (θ 1 ), Q 2 (θ 1 )] [P 2 (θ 2 n 1)Q 2 (θ 2 n 1)] As in the Heisenberg uncertainty relation, the number φ, [P n+1, Q n+1 ] φ gives a lower bound for the product of the variances of P n+1 and Q n+1. We now compute this number for an amplitude state â n+1. We have that â n+1, [P n+1, Q n+1 ] â n+1 = [ ] [ ] [ ] a(x n+1,2j ) 1, c a(x n+1,2j+1 ) n,jc 1 a(x n+1,2j ) n,j 1 a(x n+1,2j+1 ) j = j = j c n,jc 1 n,j [a(x n+1,2j )a(x n+1,2j+1 ) + a(x n+1,2j+1 )a(x n+1,2j )] c n,jc 1 n,j a(x n,j ) 2 [ c n,c 1 n,j + c 1 n,jc n,j] = This shows that even though P n+1 and Q n+1 do not commute, there is no lower bound for the product of their variances when the system is in an amplitude state. References [1] S. Gudder, A unified approach for discrete quantum gravity, arxiv: grqc (214). [2] S. Gudder, A covariant causal set approach to discrete quantum gravity, arxiv: gr-qc (213). [3] S. Gudder, The universe as a quantum computer, arxiv: gr-qc (214). [4] J. Henson, Quantum histories and quantum gravity, arxiv: gr-qc (29). [5] R. Sorkin, Quantum mechanics as quantum measure theory, Mod. Phys. Letts. A9 (1994),

19 [6] R. Sorkin, Causal sets: discrete gravity, arxiv: gr-qc 399 (23). [7] S. Surya, Directions in causal set quantum gravity, arxiv: gr-qc (211). 18

LABELED CAUSETS IN DISCRETE QUANTUM GRAVITY

LABELED CAUSETS IN DISCRETE QUANTUM GRAVITY S. Gudder Department of Mathematics University of Denver Denver, Colorado 80208, U.S.A. sgudder@du.edu Abstract We point out that labeled causets have a much