Classical sample space and event algebra Probability theory is based upon the concept of a sample space of mutually exclusive possibilities, one and only one of which actually occurs, or is true, in any given situation. The elements of the sample space are sometimes called points or elements or events . In classical and quantum mechanics the sample space usually consists of various possible states or properties of some physical system. For example, if a coin is tossed, there are two possible outcomes: H (heads) or T (tails), and the sample space S is { H, T }. If a die is rolled, the sample space S consists of six possible outcomes: s = 1, 2, 3, 4, 5, 6. If two individuals A and B share an office, the occupancy sample space consists of four possibilities: an empty office, A present, B present, or both A and B present. Associated with a sample space S is an event algebra B consisting of subsets of elements of the sample space. In the case of a die, “ s is even” is an event in the event algebra. So are “ s is odd”, “ s is less than 4”, and “ s is equal to 2.” It is sometimes useful to distinguish events which are elements of the sample space, such as s = 2 in the previous example, and those which correspond to more than one element of the sample space, such as “ s is even”. We shall refer to the former as elementary events and to the latter as compound events.
Classical sample space and event algebra Probability theory is based upon the concept of a sample space of mutually exclusive possibilities, one and only one of which actually occurs, or is true, in any given situation. The elements of the sample space are sometimes called points or elements or events . In classical and quantum mechanics the sample space usually consists of various possible states or properties of some physical system. For example, if a coin is tossed, there are two possible outcomes: H (heads) or T (tails), and the sample space S is { H, T }. If a die is rolled, the sample space S consists of six possible outcomes: s = 1, 2, 3, 4, 5, 6. If two individuals A and B share an office, the occupancy sample space consists of four possibilities: an empty office, A present, B present, or both A and B present. Associated with a sample space S is an event algebra B consisting of subsets of elements of the sample space. In the case of a die, “ s is even” is an event in the event algebra. So are “ s is odd”, “ s is less than 4”, and “ s is equal to 2.” It is sometimes useful to distinguish events which are elements of the sample space, such as s = 2 in the previous example, and those which correspond to more than one element of the sample space, such as “ s is even”. We shall refer to the former as elementary events and to the latter as compound events.
Definitions The concept of probability is of considerable importance in optics, as in any situation in which the outcome of a given trial or measurement is uncertain. Under these conditions it is desirable to be able to associate a measure with the likelihood of the outcome or the event in question; such a measure is called the probability of the event. Several different definitions of probability have been adopted at various times in the past. The classical definition is based on an exhaustive enumeration of the possible outcomes of an experiment or trial. If the trial has N distinguishable, mutually exclusive outcomes, which are equally likely to occur, and if n out of these N possible outcomes have an attribute or characteristic that we call ‘success’, then the probability of success in any one trial is given by the ratio n/N . For example, if we roll a die, and if each of the six digits is equally likely to be on top when the die comes to rest, there are N = 6 distinguishable outcomes. If we identify success with an even number, for example, then since there are three different ways in which success can be achieved, it follows that the probability of success when the die is rolled is given by 3/6 = 1/2. Unfortunately, an exhaustive enumeration of all possibilities is not always feasible. Another common definition of probability is based on the notion of relative frequency of success.
Quantum probability and quantum mechanics The formalism of quantum probability theory Discussions of quantum mechanics are often confused by a lack of clarity about what exactly constitutes ‘quantum mechanics’. It is therefore useful to try at the start to isolate a consistent mathematical core of quantum mechanics, and consider anything that goes beyond this core to be ‘interpretation’. For us, this core is quantum probability theory. Quantum probability is a generalization of classical probability, and therefore I begin with a brief review of the latter. I assume that the reader has some familiarity with the ideas of probability theory. What follows is just to provide a quick review, and to establish some notation and terminology. In modern classical probability theory, probabilities are defined over algebras of events. The motivation is straightforward: we begin with a set of ‘primitive’, or ‘simple’, events (the ‘sample space’), and form an algebra of events by taking all logical combinations of the simple events. For example, let us take the simple events to be the possible results of rolling a six-sided die one time, so that the sample space is the set {1,2,3,4,5,6}. We then form an algebra of events from the sample space by taking all possible logical combinations of the simple events. Logical combinations include, for example, ‘either 3 or 5’ and ‘not 3 and not 2’.
Classically, probability is considered in terms of the number of possible outcomes. For example, a die toss has six possible outcomes and a coin toss, two. Probability calculations make use of probabilities and hence of the total number of possible outcomes. Thus, they are not deterministic calculations which follow “what happens” as in Newtonian mechanics. We suggest that if free particle quantum mechanics is probabilistic, it must also involve the notion of possible outcomes in an experiment and not represent a deterministic approach. We suggest, however, that something interesting applies to quantum mechanics because we have suggested that free particle quantum mechanics arises from the Lorentz invariant A = -Et+px. We note that there exist x,t trajectory points which follow x=vt. We also note, however, that one may introduce the idea of intervals dx, dt (because Newton’s dx→0, dt→0 is an idealization) and that A is unchanged for x,t and x+hbar/p , t+hbar/E. In other words, we suggest that there are intervals with probability about each x,t trajectory point. A trajectory point is associated with deterministic physics, but probability about this point is not. In other words, one has a kind of hybrid probability scheme. Thus, exp(ipx) and exp(-iEt) really represent fluctuation probabilities about a trajectory point x and t. These ranges of possible x and t fluctuations about a trajectory point x,t allow for further probability to occur in certain interactions. We consider two examples. The first is 2-slit interference which involves a deterministic particle moving along a straight line towards a 2-slit mechanism perpendicular to its path. At the 2-slit apparatus, one considers a probabilistic calculation as the uncertainty in x, i.e. dx=hbar/p allows for possible interaction with both slits. This involves considering all possible outcomes (i.e. motion through one slit or the other which is a classical situation), but exp(i p dot r1) and exp(i p dot r2) “sit on top” of these trajectories at the 2-slit apparatus and one has interference of the probabilities linked with each classical possible outcome. Thus, there are two sets of probabilities which are intertwined, the probability range hbar/p for each path and the two possible paths which have equal probability. One must consider the possible outcomes for both sets together. We suggest that a similar analysis applies to the case of 1-D reflection-refraction for x=0 at an n1-n2 index of refraction junction. Thus, we suggest that a deterministic free particle with x=vt may encounter probabilistic interaction situations, such as a two-slit apparatus or an n1-n2 index of refraction junction. These are probabilistic only because there is a probability fluctuation scheme in x associated with each deterministic x point. It is this probability which allows for a second probability to arise due to interaction with the slits or at the 1-D n1-n2 junction. In such a case, one must consider all possible outcomes which means that one does not only consider the exp(ipx) type probabilities, but also those linked with the physical interaction, i.e. two possible paths, one from each slit, or an incident, reflected and refracted photon at x=0 (n1-n2 junction). It is a combination of all possible paths which must be used in calculations. We thus suggest that even though there is a sense of determinism, i..e free particles move as x=vt and these exist in the 2-slit case and the 1-D reflection-refraction case, in a probabilistic calculation one must consider all probabilistic outcomes consistent with these. Thus, quantum mechanical calculations are true probabilistic calculations in the sense that they involve all possible outcomes in an analysis.
We restrict ourselves to finite level systems. Elementary classical probability theory is woven around the notions of a finite sample space or an alphabet A, events which are subsets of A, random variables which are real or complex-valued functions on A and probability distributions which are, once again, functions on A with values in the unit interval adding to unity. An event E ⊂ A can be equivalently replaced by the two-valued indicator function 1 E which is one on the set E and zero on the complement E′ of E in A. Thus all the basic objects, namely, events, random variables and distributions are functions on A. Complex-valued functions on A constitute a commutative algebra A A with complex conjugation as an involution. Thus an event can be identified as an element f ∈ A A satisfying f = f2 = f̄ where the bar indicates complex conjugation. A real-valued random variable is an element f ∈ A A satisfying f = f̄ whereas a probability distribution is an element p ∈ A A satisfying p(x) ≥ 0 ∀ x ∈ A and ∑ x∈A p( x )=1 . x∈A∑p(x)=1. . Note that every nonnegative function on A has the form f f̄ for some f ∈ A A .
This introductory chapter begins with a review of uncorrelated classical probabilities and then extends these concepts to correlated quantum systems. This is done both to establish notation and to provide a basis for those who are not experts to understand material in later chapters. Probability of a transition Seldom does one know with certainty what is going to happen on the atomic scale. What can be determined is the probability P that a particular outcome (i.e. atomic transition) will result when many atoms interact with photons, electrons or protons. A transition occurs in an atom when one or more electrons jump from their initial state to a different final state in the atom. The outcome of such an atomic transition is specified by the final state of the atom after the interaction occurs. Since there are usually many atoms in most systems of practical interest, we can usually determine with statistical reliability the rate at which various outcomes (or final states) occur. Thus, although one is unable to predict what will happen to any one atom, one may determine what happens to a large number of atoms. 1 Single particle probability A simple basic analogy is tossing a coin or dice. Tossing of a coin is analogous to interacting with an atom. In the case of a simple coin there are two outcomes: after the toss one side of the coin (‘heads’) will either occur or it will not occur.
The probability [Formula: see text] of the occurrence of an event pertaining to a physical system which is observed in different states [Formula: see text] determines a function [Formula: see text] from the set [Formula: see text] of states of the system to [Formula: see text]. Function [Formula: see text] is called a numerical event or more precisely an [Formula: see text]-probability. When appropriately structured, sets [Formula: see text] of numerical events form so-called algebras of [Formula: see text]-probabilities, which are orthomodular posets that can serve as quantum logics. If one deals with a classical physical system, then [Formula: see text] will be a Boolean algebra. Starting with a supposed quantum logic or logics obtained by individual measurements, newly added numerical events may turn out to be crucial for the assumption of classicality or nonclassicality of the physical system. We will call those numerical events critical and will study their impacts among various classes of algebras of numerical events. Moreover, we will consider the situation of enlarging [Formula: see text] by new states and will describe how elements of an algebra of [Formula: see text]-probabilities contribute to the character of a logic when there exists a certain relation between them and the element that is newly added.