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Quantum Logic

In mathematics, a logic is a formal language (from logos, meaning "word" in greek). In a logic, statements, or propositions have a particular form and are given truth values. In classical, Boolean, or two-valued logic, truth values are either 0 or 1, for statements which are FALSE or TRUE respectively. Quantum logic is a many valued logic. In many valued logics, truth values between 0 and 1 are allowed for statements which are neither true nor false, but have some kind of intuitive level of truth somewhere in between. Fuzzy logic is an example of a many valued logic, much used in control engineering. Probability theory is also a many valued logic, described as The Logic of Science, in the classic book by E. T. Jaynes, and also called Bayesian reasoning.

Quantum logic (due to Birkhoff and von Neumann) is a language which tells us about possible results of measurements. It does not tell us, in any direct way, what happens between measurements (later, when we are fluent in the language, we can draw some conclusions). Quantum logic is usually treated in a very abstract, and somewhat obscure, way. This treatment aims keep it as concrete as possible and to show how to translate the mathematical symbols of standard quantum theory into statements in the English language.

Feynman is famously quoted as having said “I think it is safe to say that no one understands quantum mechanics”. The plethora of “interpretations” seems to bear this out. Yet I claim quantum mechanics can be understood. In essence, what I will describe is the “orthodox”, or Dirac-Von Neumann interpretation. If I have contributed anything, it is to make the interpretation easier to understand, not to alter it in a fundamental way. When I read von Neumann on interpretation, I do not think that what he says is really different from what I say. However, von Neumann was one of the most brilliant mathematicians in history. He had a way of talking over one’s head, which is difficult to follow. Mathematics is a language, but the natural ability to treat it like an ordinary language is not really human. This goes beyond the more common ability to manipulate equations according to rules, which does not require understanding. One way to learn language is to start by translating simple phrases and sentences into English from a primer. This section is a primer in the language of quantum logic, as well as an introduction to the mathematical structure of quantum theory.

Quantum logic is often rejected as an interpretation of quantum mechanics on the ground that it consists of obscure truth values for simple propositions. In my view this is wrong. It is better described as intuitive truth values for sophisticated propositions. In order to understand it we need to understand what the propositions of quantum mechanics are, and how to translate them into ordinary language. We will understand that these propositions refer to hypothetical measurements, and are not, in general, strictly true or false, but have levels of truth somewhere in between. In this section I describe the language for measurements of position of a single interacting particle. Later sections will treat states of more than one particle, interactions between particles, and introduce the mysterious property of spin.

Truth Values

Classical logic applies to sets of statements about the real world which are definitely true or definitely false. For example, when we make a statement,


we tend to assume that it is definitely true or definitely false. Such statements are said to be sharp, meaning that they have truth values from the set {0,1}. When it is the case that P(x) is definitely either true or false then classical logic and classical mechanics apply.

We cannot say that a statement about the future is strictly true or false.


We may, however, assign a probability to such a statement. If we consider probabilities as truth values, then probability theory is a many valued logic, applying to sentences in the future tense.

In quantum mechanics we also talk about situations in which there is not going to be a measurement. Hypothetical measurement results can be described using statements in the subjunctive mood:


R(x) is intuitively sensible, even when no measurement is done, but it is neither strictly true nor false, and cannot sensibly be given a crisp truth value. Its truth is distinguished from a probability because, when no measurements are to be done, we cannot sensibly discuss the potential frequency of individual measurement results. In quantum theory we are not always going to do a measurement, but we still want to talk about what would happen if we were to do a measurement, i.e. we need to be able to make statements about hypothetical measurement results. Quantum logic provides a way of discussing levels of truth for statements about hypothetical measurement, like R(x), in the subjunctive mood.

Sentences Describing Hypothetical Measurement Results

Statements in the subjunctive consist of two clauses, the conditional clause “If a measurement of position were done, …”, and the consequent clause “…, then the result would be x”. Quantum mechanics is based on statements composed of a conditional clause and a consequent clause. To formally describe physics using mathematics, we need to be more precise. The conditional clause must also contain whatever information is known before measurement. This information comes from a prior measurement. We therefore discuss two measurements, the first to determine the condition and the second to determine the outcome, or consequence. We represent the results of these measurements symbolically.

The conditional clause, referring to the first measurement, is represented by a ket. It is described as a formal conditional clause to indicate that only clauses formally described in the rules are allowed as part of the logic, or formal language. The basic conditional clauses, on which the language is built, refer to measurements of position:


Measured positions are always discrete values, determined by the range and resolution of a measurement apparatus. We are discussing hypothetical measurement, and in practice, rather than use the values of x generated by a particular apparatus to generate the basic propositions (represented by blue dots), it is simpler to use an equally spaced lattice, containing N positions given by decimals terminating at some value beyond the best available resolution of any existing appearatus (cyan grid), where N is a very large, but finite, number. The use of a discrete lattice has implications to the mathematical statement of covariance in the quantum domain. This will be considered in Quantum Covariance. For the immediate development, it will be assumed that Lorentz covariance holds in an approximation valid at the limit of experimental accuracy.

An actual position found by a real apparatus is described by a set of points in the lattice (magenta set boundaries). To describe this we need to extend the language, by introducing a structure corresponding to OR. In grammar OR is a coordinating conjunction. In logic and mathematics it is logical disjunction. OR will be represented by the symbol +. A simple logical disjunction does not distinguish the likelihood of the two possibilities; “If measured position were x or y, …”. To express the idea that one possibility is more likely than the other we introduce a weighting. Thus, if the magnitude of a is greater than that of b, then means “if measured position were either x or y, but more likely x, …”. Precise values for a and b in a given situation will be determined as the language develops.

We want to be able to express many possibilities, “if the particle were found at x or y or z or …”, and we want weighting between all options. This is done recursively, by starting from the basic conditional clauses and repeating the following rule as many times as we require:


The set of formal conditional clauses, or kets, now has the mathematical structure of an N-dimensional vector space. The symbol H¹(t) is used to denote this vector space, and consists of a family of formal conditional clauses concerning the measurement of position of a single particle at time t. The basic conditional clauses, , described in rule I are a basis for H¹(t).

Kets are often known as states. They are not strictly states of a particle, but formal conditional clauses describing the likelihood of particular measurement results. That is something of a mouthful. I will use “state”, in keeping with common practice when no confusion arises.

One may ask why we are using complex numbers to construct the propositions of quantum theory. So long as we are talking about measurements of position at a particular time, real numbers would work just as well. In fact, real numbers would be in some ways more precise, because complex numbers introduce an extra level of freedom. The reason for using them lies in what happens when we want to compare a measurement at one time with a measurement at another. Before we can study the evolution of states in time we need to complete the language for talking about measurements of position at a particular time.

A conditional clause on its own has little meaning. To complete a formal sentence in quantum theory we need to put it together with a consequent clause. Consequent causes refer to a second measurement, at the same time as the first measurement. For hypothetical measurements, there is no problem with the idea that they both take place at the same time. To make statements about real measurement results we will also need to know how kets change in time.

There is no fundamental difference between one measurement and another, so the grammatical structure, weighted disjunction described in rule II, applies equally well to consequent clauses. These also form an N-dimensional vector space defined from a basis of consequent clauses in one-one correspondence with the basic conditional clauses, or kets, described by rule I. Consequent clauses are represented symbolically by bras, according to rule III:


We put the two clauses together, to make a braket, representing a statement about measurement at a given time:


From observation we know that, if, at some particular time, a particle is measured at position x, then its position is definitely x and it cannot be measured separately at some other position y at the same time. The statement is strictly true or false, depending on whether or not x = y. Its truth value is given by a Kronecker delta, and we write:
Taken together with linearity and complex conjugation, this is sufficient to define an inner product between any two kets, and in H¹(t). Thus, H¹(t) is a Hilbert space and the basic states of rule I are an orthonormal basis.

Integral Representation

In the language of quantum theory, absolute magnitude has no meaning; only relative magnitudes are important in weighted logical OR. For any complex number a, the clause means exactly the same thing as . In other words, normalisation is irrelevant. For large values of N, it is often convenient to normalise basis states so that the inner product becomes a Dirac delta function instead of a Kronecker delta,
This normalisation is natural when using the language of integrals rather than the language of sums. If we bear in mind that strictly the integrals stand for finite sums with N terms, everything remains well defined, and we be able to control the divergence problems of quantum electrodynamics. With this normalisation, the resolution of unity,
is replaced by,
The integral is just what was defined previously, but uses three dimensions instead of one. It is regarded strictly as a sum of a N terms where N is a finite number of possible positions, rather than an infinite sum.

Plane Wave States

For a 3-vector, p, at the origin, define a particular ket, , as a sum of position states, :
Taking the inner product with defines a sinusoidal wave,
This function can be seen to have planar wave crests, perpendicular to vector p and at equally spaced intervals 2π/|p|. It is a plane wave at constant time.


Using this approach to quantum theory, this is the fundamental definition of 3-momentum. The justification for the definition is that it turns out later that p is a conserved quantity which corresponds precisely to the classical notion of momentum. Momentum is here defined using natural units, in which the Dirac constant, . To convert to conventional units, substitute


Using the dot product with the laws of indices, the delta function in 3 dimensions is the product of three 1 dimensional delta functions. So,
Thus, we can precisely recover the coefficients of position if we know the momentum space wave function at any time. This is the Fourier inversion theorem. In standard quantum theory there are some subtle mathematical problems with Fourier inversion, because of the use of continuous transforms. If we remember that the integrals over p and over x (or y) really stand for finite sums with N terms, where N is large and the same in both sums, these problems never arise. There are mathematical issues to resolve concerning discreteness and Lorentz transformation. These issues do not arise for discrete transforms, and do not appear in the limit in which N goes to infinity, provided that N is kept the same in both sums. One may then recover a continuum theory without divergence problems. This willl be considered in more depth in Quantum Covariance and in Regularisation.

The Resolution of Unity in Momentum Space

With a little rearrangement,
Using the integral representation of the resolution of unity, substituting for , and using the resolution of unity again, for any kets and ,
So, we have another resolution of unity, this time in terms of momentum states:
So, any ket can be written as a sum of plane wave states,
This shows that plane wave states are a basis for Hilbert space. Mathematically, switching between representations in terms of momentum and position is simply a change of basis, analogous to a change of coordinate axes for 3-vectors.

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