Finite Quantum Measure Spaces

Finite Quantum Measure Spaces Denise Schmitz 4 June 2012

Contents 1 Introduction

2

2 Preliminaries 2.1 Finite Measure Spaces . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Quantum Systems . . . . . . . . . . . . . . . . . . . . . . . . . .

2 2 3

3 Quantum Measures 3.1 Grade-2 Additivity . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Decoherence Functions . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Properties of Quantum Measures . . . . . . . . . . . . . . . . . .

3 3 4 6

4 Interference and Compatibility 4.1 Interference Functions . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Compatibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Restriction to Zµ . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 7 8 8

5 Probability 10 5.1 Quantum Covers . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 5.2 k-Set Covers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 6 Generalization and Further Questions 12 6.1 Antichains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 6.2 Super-Quantum Measures . . . . . . . . . . . . . . . . . . . . . . 12 7 Conclusion

14

8 References

15

1

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 = {x1 , . . . , xn } for some n > 0. • µ(∅) = 0. • µ satisfies a condition known as additivity: for any collection of mutually disjoint sets {A1 , . . . , Am } ∈ P(X), ! m m [ X µ Ai = µ(Ai ). (1) i=1

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

2.2

Quantum Systems

Suppose X = {x1 , . . . , xn } is a set and let the xi 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 xi 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, x1 and x2 represent an electron and a positron respectively, then µ(x1 ) = µ(x2 ) = 9.11 × 10−31 kg whereas µ(x1 ∪ x2 ) = 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 = {x1 , . . . , xn } 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)

• 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: D(A, B) = D(B, A)

(5)

D(A, A) ≥ 0

(6)

2

|D(A, B)| ≤ D(A, A)D(B, B)

(7)

and if A and B are disjoint, 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

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

possible (or decoherent) outcome. So decoherence is a precise formulation of the basic principle underlying the Schr¨odinger’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 {A1 , . . . , Am } are mutually disjoint subsets of X. Then ! m m m [ X X µ Ai = µ(Ai ∪ Aj ) − (m − 2) µ(Ai ). (9) i=1

i