Finite Quantum Measure Spaces

Transcription

1 Finite Quantum Measure Spaces Denise Schmitz 4 June 2012 Contents 1 Introduction 2 2 Preliminaries Finite Measure Spaces Quantum Systems Quantum Measures Grade-2 Additivity Decoherence Functions Properties of Quantum Measures Interference and Compatibility Interference Functions Compatibility Restriction to Z µ Probability Quantum Covers k-set Covers Generalization and Further Questions Antichains Super-Quantum Measures Conclusion 14 8 References 15 1

2 1 Introduction Measure theory is the branch of mathematics concerned with assigning a notion of size to sets. First developed in the late 19th and early 20th centuries, the theory has widespread applications to other areas of mathematics. One important application of measure theory is in probability, in which each measurable set is interpreted as an event and its measure as the probability that the event will occur. Naturally, as probability is such a central notion to the study of quantum mechanics, one might wish to apply techniques of measure theory to study quantum phenomena. Unfortunately, as we shall see, one of the foundational axioms of measure theory fails in its intuitive application to quantum mechanics. This paper will discuss a modification of traditional measure theory as discussed in [3] that allows us to accomodate these quirky features of quantum systems. We will define an extended notion of a measure and discuss its applications to the study of interference, probability, and spacetime histories in quantum mechanics. 2 Preliminaries 2.1 Finite Measure Spaces Measure theory allows the consideration of infinite sets; however, for simplicity we will consider only the finite case. In classical measure theory, a finite measure space is a pair of objects denoted (X, µ): a set X and a function µ : P(X) R +, where P(X) denotes the power set of X. These objects must satisfy the following properties: X is finite and nonempty, that is, X = {x 1,..., x n } for some n > 0. µ( ) = 0. µ satisfies a condition known as additivity: for any collection of mutually disjoint sets {A 1,..., A m } P(X), ( m ) m µ A i = µ(a i ). (1) A function µ satisfying these properties is known as a measure on X. In some situations it is useful to allow measures to attain negative or complex values, and we shall consider such signed measures and complex valued measures later. A finite probability space is a finite measure space (X, µ) such that µ(x) = 1. The set X is interpreted as the sample space of outcomes and P(X) is the set of events, i.e., combinations of outcomes. The measure µ(a) for any subset of X represents the probability that some trial will result in the event A. Under this interpretation, it is clear that the union operator on sets corresponds to logical disjunction of events, the intersection to the logical conjunction, and the complementation to the logical negation. 2

3 2.2 Quantum Systems Suppose X = {x 1,..., x n } is a set and let the x i represent quantum objects or quantum events. There are many situations in which it would be useful to have an interpretation of a measure on X, but unfortunately the additivity condition (1) fails in some such situations. Example Suppose (X, µ) is a finite measure space in which the x i represent particles and the measure µ represents mass. Although mass is additive in the macroscopic world, this is not the case on a quantum scale due to the effects of annihilation and binding energy. If, for instance, x 1 and x 2 represent an electron and a positron respectively, then µ(x 1 ) = µ(x 2 ) = kg whereas µ(x 1 x 2 ) = 0. At the heart of quantum mechanics is a phenomenon known as wave-particle duality. This principle states that every fermion (matter particle) and boson (force-carrying particle) is described by a wavefunction a time-varying function giving the particle s probability density at each point in space. Often these wavefunctions behave like classical waves, exhibiting properties such as diffraction and interference. A famous experiment known as the two-slit experiment demonstrated that a beam of electrons shot through two narrow slits produces an interference pattern identical to the interference patterns produced by electromagnetic (light) waves. Thus, in situations involving particles, additivity of measures will clearly fail when interference occurs. 3 Quantum Measures 3.1 Grade-2 Additivity Fortunately, it is possible to modify equation (1) to obtain a weaker, yet still useful, constraint on the additivity of measures. Suppose X = {x 1,..., x n } and µ : P(X) R +. We shall say that µ is a grade-2 measure if for all disjoint sets A, B, C P(X), µ(a B C) = µ(a B) + µ(b C) + µ(a C) µ(a) µ(b) µ(c). (2) Note that equation (2) follows trivially from (1), but the converse fails. We shall refer to a proper measure that is, a measure satisfying condition (1) as a grade-1 measure. There are two additional properties which do not follow from grade-2 additivity but are nonetheless useful. Let us say that a measure µ is regular if it satisfies the following: If A and B are disjoint and µ(a) = 0, then µ(a B) = µ(b). (3) 3

4 If µ(a B) = 0, then µ(a) = µ(b). (4) The reason for equation (3) is immediately clear. To understand the importance of (4), consider a situation involving destructive interference. In order for two waves to produce complete destructive interference, thereby cancelling out each other, their original amplitudes must have been equal. A grade-2 measure µ is a quantum measure, or q-measure, if it is regular. 3.2 Decoherence Functions Since interference plays such a prominenent role in quantum mechanics and its mathematical formulation, it can be useful to define functions capturing this notion that can be used to define q-measures. Such functions will be called decoherence functions, and they behave rather like inner products. A function D : P(X) P(X) C is a decoherence function if it satisfies the following: and if A and B are disjoint, D(A, B) = D(B, A) (5) D(A, A) 0 (6) D(A, B) 2 D(A, A)D(B, B) (7) D(A B, C) = D(A, C) + D(B, C). (8) Note that condition (5) implies that D(A, A) is real, so conditions (6) and (7) are well posed. Now, for two sets A, B X representing quantum objects, Re[D(A, B)] can be interpreted as the interference between A and B. As one might expect, this allows for a convenient way to define a q-measure on X. Proposition 1 Let D : P(X) P(X) C be a decoherence function. Then the function µ : P(X) R + defined by µ(a) = D(A, A) is a q-measure. Proof We shall show that µ is grade-2 additive and leave the proof of regularity to the reader. Suppose A, B, C X are disjoint. Then from the definition of a decoherence function, we have µ(a B) + µ(b C) + µ(a C) µ(a) µ(b) µ(c) = D(A B, A B) + D(B C, B C) + D(A C, A C) µ(a) µ(b) µ(c) = 2D(A, A) + 2D(B, B) + 2D(C, C) + 2Re[D(A, B)] + 2Re[D(A, C)] + 2Re[D(B, C)] µ(a) µ(b) µ(c). 4

5 But since we have defined µ(a) = D(A, A), the above expression is equal to D(A, A) + D(B, B) + D(C, C) + 2Re[D(A, B)] + 2Re[D(A, C)] + 2Re[D(B, C)] = D(A, A) + D(B, B) + D(C, C) + D(A, B) + D(A, C) + D(B, C) + D(B, A) + D(C, A) + D(C, B) = D(A B, A B) + D(C, C) + D(A, C) + D(B, C) + D(C, A) + D(C, B) = D(A B C, A B C). Thus µ is grade-2 additive. We will now give an example of the use of a decoherence function for describing quantum systems and defining a q-measure. Example Suppose ν is a complex-valued grade-1 measure on P(X) (often interpreted as a quantum amplitude). Then we can define a decoherence function as follows (verification that this is a decoherence function is left to the reader): D(A, B) = ν(a)ν(b). The corresponding quantum measure is therefore µ(a) = D(A, A) = ν(a) 2. If A, B X are disjoint, then computing the measure of A B will show that µ is not grade-1 additive: µ(a B) = ν(a B) 2 = ν(a) + ν(b) 2 = ν(a) 2 + ν(b) 2 + 2Re[ν(A)ν(B)] = µ(a) + µ(b) + 2Re[D(A, B)]. This measure µ satisfies grade-1 additivity for the union of disjoint A and B if and only if the real part of the decoherence function of A and B is zero. This lends some meaning to the earlier statement that the real part of a decoherence function represents interference. In fact, we can discuss the importance of a decoherence function in more detail. In quantum mechanics, decoherence occurs when a wavefunction becomes coupled to its environment (that is, when the objects involved interact with the surroundings) and refers to the assignment of a particular outcome to the system. This phenomenon is sometimes referred to in more casual terms as wavefunction collapse, and it is of key importance for allowing the classical limit to emerge on the macroscopic scale from a collection of quantum events. Once decoherence has occurred, the components of the system can no longer interfere and it becomes possible to assign a well-defined probability to each 5

6 possible (or decoherent) outcome. So decoherence is a precise formulation of the basic principle underlying the Schrödinger s Cat thought experiment the outcome of a quantum event is undetermined until the system interacts with its environment. The decoherence function is thus used to define the probabilities of all decoherent outcomes for a particular event by quantifying the amount of interference betweent the various components of the system. 3.3 Properties of Quantum Measures We next obtain a result regarding the q-measure of the union of more than three mutually disjoint sets. Proposition 2 Suppose µ : P(X) R + is a grade-2 measure, m 3, and {A 1,..., A m } are mutually disjoint subsets of X. Then ( m ) m m µ A i = µ(a i A j ) (m 2) µ(a i ). (9) i<j=1 Proof By induction on m. The base case, m = 3, is simply the hypothesis, so assume the result holds for m 1 3. Then ( m ) µ A i = µ (A 1 A 2... (A m 1 A m )) = i<j=1 (m 3) = i<j=1 µ(a i A j ) + µ (A i (A m 1 A m )) ( µ(a i ) + µ(a m 1 A m ) µ(a i A j ) + µ(a i A m 1 ) + µ(a i A m ) ) + (m 2)µ(A m 1 A m ) µ(a i ) (m 2)µ(A m 1 ) (m 2)µ(A m ) (m 3) = i<j=1 ( µ(a i ) + µ(a m 1 A m ) µ(a i A j ) + µ(a i A m 1 ) + µ(a i A m ) + µ(a m 1 A m ) (m 2) m µ(a i ) ) 6

7 = = m 1 i<j=1 m i<j=1 m 1 µ(a i A j ) + µ(a i A m ) (m 2) µ(a i A j ) (m 2) m µ(a i ). m µ(a i ) Thus the result holds by induction. Note that the result also holds for signed or complex valued measures. 4 Interference and Compatibility 4.1 Interference Functions An important consequence of (9) is that any grade-2 measure on X is uniquely determined by its values on single elements and pairs of elements in X. This result suggests that it is possible to define a function describing the amount of interference between pairs of elements and express the measure µ as the sum of this function and a classical grade-1 measure. Let µ be a q-measure on X and define the classical part of µ as a measure ν µ : P(X) R + such that ν µ is grade-1 additive and ν µ (x i ) = µ(x i ) for each x i X. The condition of grade-1 additivity ensures that ν µ is unique. Then the quantum interference function of µ, I µ : X X R is defined as I µ (x i, x j ) = µ({x i, x j }) µ(x i ) µ(x j ). (10) This function represents the difference between the values of the q-measure µ and the values of the grade-1 measure ν µ. It allows us to define the interference part of µ, which is a function λ µ : P(X) R such that λ µ (A) = I µ (x i, x j ). (11) (x i,x j) A Lemma 1 The function δ : P(X) R defined by δ(a) = λ µ (A A) is a grade-2 signed measure on X. As we shall discuss later, Lemma 1 can be extended in order to generalize the notion of a grade-2 measure. It will also be important later to note that λ µ is symmetric, that is λ µ (A B) = λ µ (B A) for all A, B X. For now, we shall conclude this section with a nice result relating µ to ν µ and λ µ ; the proof is omitted here, but it folllows readily from Lemma 1. Theorem 1 Any q-measure can be decomposed into its classical and interference parts. That is, if µ is a q-measure on X, then for any A X, µ(a) = ν µ (A) λ µ(a). (12) 7

8 4.2 Compatibility Although some quantum objects interfere with each other, some do not. It will be useful to have a precise definition of such objects, i.e., those sets for which a q-measure µ behaves like a grade-1 measure. Let A, B X; then A and B are µ-compatible (denoted AµB) if µ(a B) = µ(a) + µ(b) µ(a B). (13) A set that is µ-compatible with every set in P(X) is called a macroscopic set. It is not too difficult to justify this name a macroscopic set does not interfere with any set and thus behaves in the manner of a non-quantum object in the macroscopic world. The set Z µ of all macroscopic sets in P(X) is called the µ-center of X. We shall now state a number of trivial properties of µ-compatibility. Proposition 3 If A B, then AµB. If AµB, then õ B. If A Z µ, then à Z µ., X Z µ. 4.3 Restriction to Z µ As one might expect, we can obtain a grade-1 measure by restricting a q-measure µ to its µ-center. To make this precise, let us say that a set A X is µ-splitting if for all B X, µ(b) = µ(b A) + µ(b Ã). (14) Lemma 2 A set A is µ-splitting if and only if A Z µ. Proof ( ) Suppose A is µ-splitting and let B P(X). Then and thus A Z µ. µ(a B) = µ((a B) A) + µ((a B) Ã) = µ(a) + µ(b Ã) = µ(a) + µ(b) µ(a B) 8

9 ( ) Suppose A Z µ and let B P(X). Note that A B = A (B Ã), where A and B à are disjoint. Thus and so Thus A is µ-splitting. µ(a B) = µ(a (B Ã)) = µ(a) + µ(b Ã) µ(b) = µ(a B) µ(a) + µ(a B) = µ(b A) + µ(b Ã). The notion of an algebra of sets is significant in measure theory because it places conditions on which sets are measurable. One important type of algebra is a Boolean algebra. A collection A of sets in P(X) is a Boolean subalgebra of P(X) if X A and whenever A, B A, the sets à and A B are in A. Theorem 2 Z µ is a Boolean subalgebra of P(X). The restriction of µ to Z µ is a grade-1 measure which satisfies the following condition: if {A 1,..., A m } are mutually disjoint sets in Z µ, then for any B X ( m ) m µ (B A i ) = µ(b A i ). (15) Proof Proposition 3 states that X Z µ and if A Z µ, then à Z µ. Thus, to prove Z µ is a Boolean subalgebra it suffices to show that if A, B Z µ, (A B) Z µ. Consider sets A, B Z µ and C P(X). Then µ(c (A B)) = µ((c A) (A B)) + µ((c Ã) (A B)) since A is µ-splitting, and thus = µ(c A) + µ(c à B) µ(c) = µ(c A) + µ(c Ã) = µ(c A) + µ(c à B) + µ(c à B) = µ(c (A B)) + µ(c (A B)) since B is µ-splitting. Thus A B is µ-splitting, and so by Lemma 2 it is in Z µ. Thus Z µ is a Boolean subalgebra of P(X). Next, note that it is clear from the definition of Z µ that the restriction of µ to Z µ is grade-1 additive. 9

10 Finally, we will prove the last statement by induction on m. The claim is vacuously true for m = 1, so suppose the result holds for some m 1. Let A i,..., A m be mutually disjoint sets in Z µ and let S m = m A i. Then S m is in Z µ by the first part of this theorem, so S m is µ-splitting and therefore µ(b S m+1 ) = µ(b S m+1 S m ) + µ(b S m+1 S m ) = µ(b S m ) + µ(b A m+1 ) m = µ(b A i ) + µ(b A m+1 = m+1 µ(b A i ). Thus the result holds by induction. 5 Probability 5.1 Quantum Covers We now wish to interpret q-measures probabilistically. Recall that a measure µ is a probability measure on X if µ(x) = 1. To turn an arbitrary measure into a probability measure requires a process of normalization that is, defining a new measure µ P such that µ P (A) = µ(a)/µ(x), as long as µ(x) is nonzero. However, if X is very large, verifying that µ(x) 0 can be quite complicated. One way to verify whether X has measure zero is to check whether it can be covered by sets of measure zero (the reader is referred to Proposition 3.2 in [4] for a proof that these conditions are equivalent for grade-1 measures). However, the preceding statement fails for grade-2 measures, because interference allows the union of sets of measure zero to have nonzero measure. The solution, as was the case with additivity, is to modify our definition of a cover. Recall that a cover of X is a collection of sets whose union is X. We shall say that a collection {A 1,..., A m } is a quantum cover of X if for every q-measure µ on X, the condition µ(a i ) = 0 for all i implies that µ(x) = 0. Lemma 3 Suppose A = {A 1,..., A m } is a cover of X and the A i are mutually disjoint. Then A is a quantum cover. 10

11 5.2 k-set Covers Let us define the k-set cover of X to be the collection of subsets of X with k elements. Theorem 3 Suppose X = {x 1,..., x n }. For any k n, the k-set cover of X is a quantum cover. Proof By induction on k. The k = 1 case is trivial by Lemma 3, so suppose the result holds for k 1 and suppose each element of the k-set cover of X has measure zero. By Proposition 2, we have for distinct values of i 1,..., i k n, k k 0 = µ({x i1,..., x ik }) = (2 k) µ(x ij ) + µ({x ij, x il }). j=1 j<l=1 From combinatorics, we have that each x ij appears in ( n 1 k 1) k-sets, by the following reasoning: if we choose x ij out of our n choices for one of the k spots in a k-set, there remain n 1 choices left to fill k 1 spots. By a similar argument, {x ij, x il } is a subset of ( n 2 k 2) k-sets. Thus, ( ) n 1 n ( ) k n 2 0 = µ({x i1,..., x ik }) = (2 k) µ(x j ) + µ({x j, x l }) k 1 k 2 j=1 j<l=1 and so Now, and thus k j<l=1 (k 2) ( ) n 1 k 1 ) = ( n 2 k 2 k j<l=1 Finally, Proposition 2 gives µ({x j, x l }) = (k 2)( ) n 1 k 1 ) = µ({x j, x l }) = µ(x) = µ({x 1,..., x n }) = ( n 2 k 2 n µ(x j ). j=1 (k 2)(n 1)! (k 2)!(n k)! (k 1)!(n k)! (n 2)! (k 2)(n 1) k 1 (k 2)(n 1) k 1 n j<l=1 n µ(x j ). j=1 µ({x j, x l }) (n 2) n µ(x j ) j=1 ( ) (k 2)(n 1) n = (n 2) µ(x j ) k 1 = k n k 1 n µ(x j ) j=1 j=1 11

12 which is less than or equal to 0 since k n and µ is a positive function. Thus µ(x) = 0, and so this cover is a quantum cover. 6 Generalization and Further Questions 6.1 Antichains As a chain is a sequence of sets such that each set is a subset of the next, an antichain is a collection of sets such that no set in the antichain is a subset of another. A maximal antichain of X is thus an antichain {A 1,..., A n } in X such that for all B X, either B A i or A i B for some i n. Clearly, any maximal antichain in X covers X. The converse is obviously false, but admits an intriguing refinement. Proposition 4 The k-set cover of X is a maximal antichain. The proof is rather trivial, so we leave it for the reader along with the following conjecture (Conjecture 1 in [7]). Conjecture Every maximal antichain in X is a quantum cover of X. We will now briefly investigate the significance of this conjecture. Quantum measures can be interpreted as probability measures over possible histories, that is, spacetime configurations of a quantum system. The issue of reconciling quantum mechanics with relativity is an extremely important one in physics today, and so it seems advantageous to find a way to define a space of quantum histories in which space and time are unified as required by relativity. Such a space of histories would be an ideal space over which to define a q-measure representing the probabilities of various histories. The conjecture above (if it is true) will allow for new ways to define quantum covers, which, as discussed earlier, are useful for interpreting q-measures probabilistically. 6.2 Super-Quantum Measures Just as ordinary grade-1 measures can be generalized to grade-2 measures, grade-2 measures can be generalized even further through the use of conditions called grade-m additivity. We shall say that a measure µ is grade-m additive if for any mutually disjoint collection of sets A 1,..., A m, µ(a 1,..., A m ) = m i 1<...<i m=1 m ( 1) m+1 µ(a i1,..., A im ) µ(a i ). m+1 i 1<...<i m 1=1 µ(a i1,..., A im 1 ) 12

13 It can be shown inductively that grade-m additivity implies grade-(m+1) additivity, but not the converse. (Since the proof requires additional algebraic machinery, we refer the reader to Lemma 2 in [5] for the proof). It is now possible to generalize the interference functions of Section 4.1 to show that certain measures on Cartesian products of X can be used to define grade-m measures on X. Let X m denote the m-fold Cartesian product of X with itself, and let λ : P(X m ) R be a signed grade-1 measure on X m. We shall say that λ is symmetric if for all A 1,..., A m P(X), λ(a 1,..., A m ) = λ(a π(1),..., A π(m) ) (16) for every permutation π of 1,..., m. The following lemma is a straightforward extension of Lemma 1. Lemma 4 Let λ be a symmetric signed measure on P(X m ) and define µ : P(X) R by µ(a) = λ(a m ). Then µ is a grade-m signed measure. We omit the proof, but we shall use the result to generalize Theorem 1. Recall from Section 4.1 that any grade-2 measure µ can be expressed in terms of a classical part ν µ and an interference part λ µ derived from an interference function I µ. For the sake of notational convenience, we shall denote the classical part ν µ and the interference part λ µ as λ 1 µ and λ 2 µ respectively, and the interference funtion I µ as I 2 µ, renaming it the two-point interference function. We shall now define a signed three-point interference function analogously: I 3 µ(x i, x j, x k ) = µ({x i, x j, x k }) µ({x i, x j }) µ({x i, x k }) µ({x j, x k }) + µ(x i ) + µ(x j ) + µ(x k ) when i j k i, and otherwise Iµ 3 = 0. Finally, we shall define a signed measure on P(X 3 ) representing a third interference term. λ 3 µ(a) = Iµ(x 3 i, x j, x k ). (17) x i,x j,x k A As with λ 2 µ, λ 3 µ is symmetric, and thus can be used to characterize a grade-3 signed measure in the spirit of Theorem 1. Theorem 4 Suppose µ is a grade-3 signed measure on X. Then for any A P(X), µ(a) = λ 1 µ(a) + 1 2! λ2 µ(a 2 ) + 1 3! λ3 µ(a 3 ). (18) Indeed, if µ is a grade-m signed measure, then m 1 µ(a) = i! λi µ(a i ), (19) 13

14 where the higher-order versions of λ are defined in an analogous manner. Proof By Lemma 4, the right-hand side of the equation is a grade-m signed measure, so to prove the result it suffices to show that the two sides agree for k-sets with k = 1,..., m. The case k = 1 is obvious and k = 2 follows from Theorem 1. Proceeding inductively, all cases k = i for i < m follow from the inductive hypothesis. The case k = m can be proven in a rather tedious and straightforward manner, so we leave it to the reader to check. We call these grade-m measures super-quantum measures because they do not represent quantum mechanics as we know it but may have applications to more general theories. These measures are explored in more detail in [5], but thus far it appears that physical interpretations for such functions have not yet been thoroughly investigated. It is suggested in [3] that the concept may be applicable to quantum field theory; this is certainly an intriguing possibility for future research. 7 Conclusion Although classical measure theory imposes strict additivity conditions on measures, it is nonetheless possible to obtain a rich theory of nonadditive measures; we have shown that the less restrictive grade-2 additivity embodies many properties of quantum systems and that grade-2 measures can be generated by decoherence functions. These concepts have deep connections to the history approach to quantum mechanics, an interpretation which seeks to understand the nature of possible spacetime configurations of a quantum system rather than individual quantum events. Those configurations which are physically meaningful, i.e. which conform to certain consistency conditions, are known as decoherent histories [2] for the system. Under such an interpretation, a quantum system is not thought of as having certain probabilities for possessing various properties, but rather as having certain probabilities for following various histories, each of which represents a definite set of properties. As discussed in Section 6.1, one advantage of the history approach is that it may be useful for reconciling quantum mechanics with relativity. Similarly, histories may be used to transition from quantum behavior to the macroscopic classical limit. The decoherence function as discussed in Section 3.2 can be used to define the necessary consistency conditions for histories. In this way, the notion of decoherent histories may be considered without requiring the phenomenon of decoherence or wavefunction collapse, so that classical physics may be regarded as simply quantum physics on a larger domain (for which the µ-center of section 4.2 and 4.3 would likely be quite useful). But of course, to study such concepts requires a variant of probability theory capable of accomodating interference effects a role that is even now being filled by quantum measure theory. 14

15 8 References [1] D. Braun, Dissipative Quantum Chaos and Decoherence. Springer, Berlin, [2] H. F. Dowker and J. J. Halliwell, Quantum Mechanics of History: The Decoherence Functional in Quantum Mechanics, Physical Review D, 46(1992), [3] Stan Gudder, Finite Quantum Measure Spaces, American Mathematical Monthly, 117(2012), [4] H. L. Royden, Real Analysis, 2nd Edition. The Macmillan Company, Toronto, [5] R. Salgado, Some Identities for the Quantum Measure and its Generalizations, Modern Physics Letters A, 9(1994), [6] Robert Scherrer, Quantum Mechanics: An Accessible Introduction. Pearson Addison Wesley, San Francisco, [7] S. Surya and P. Wallden, Quantum Covers in Quantum Measure Theory, Foundations of Physics, 40(2010),