Classical Mechanics — Observables, States

This post begins a series on quantum mechanics. Rather than beginning with the Schrödinger equation, wavefunctions, or operators, I want to begin with a more basic question:

What is a physical theory actually trying to describe?

At a first level, physics tries to describe regular relations between events. We observe that certain things happen together, that some events reliably follow others, that some quantities remain unchanged while others vary, and that the same experimental preparation produces similar patterns again and again. A physical theory gives us a framework in which these regularities can be stated precisely. It introduces certain concepts in advance—space, time, position, motion, mass, charge, field, energy—and then uses them to organize what we observe.

This does not mean that these concepts are arbitrary inventions. Rather, they are part of the language through which a theory makes contact with experience. In Newtonian mechanics, for example, we assume that events occur in a pre-existing space and time. We can say where an object is, how its position changes, how fast it moves, and how forces alter that motion. The notions of “being somewhere” and “moving from one place to another” are therefore built into the framework from the beginning.

In this post I will be discussing classical mechanics, where space and time are treated as given background structures. There are approaches to quantum gravity in which such a fixed background spacetime is not assumed from the start. In those approaches one tries to formulate physics more relationally, perhaps in terms of correlations between physical quantities rather than in terms of objects moving through a predetermined space. Then words such as “here,” “there,” or even “at this time” may no longer have their ordinary fundamental meaning. That is a much deeper subject, and I will return to it later. For the moment, classical mechanics gives us a clean setting in which to ask a more modest question: Once space and time have been assumed, what is a physical state?

In the usual formulation of classical mechanics, one says that the state of a system is described by canonical variables The variables are generalized positions and the variables are their conjugate momenta. Together they describe a point in a phase space manifold. For a particle moving on a line, one may take to be its position and to be its momentum. For a pendulum, may be its angle and its angular momentum. For a system of many particles, the vector records all their positions and records all their momenta. One then assumes that every observable quantity is a function of these variables. The position is the observable , the momentum is the observable , and the energy is some function , called the Hamiltonian. For a particle of mass moving in a potential , the Hamiltonian is often The first term is kinetic energy and the second is potential energy. Hamilton’s equations of motion are

or the particle just mentioned, these become Thus the usual Newtonian equation is recovered. If the potential is , one obtains the harmonic oscillator. If , one obtains the classical Kepler problem. In this standard picture, a state is a point in phase space, and the equations of motion tell us how that point moves with time.

My main interest here is not yet the detailed form of the equations of motion. Instead, I want to step back and ask whether the usual order of ideas is really the most fundamental one. The standard formulation begins with a space of states and then says that observables are functions on that space. We first imagine that the system is at a point of phase space, and then an observable tells us a number . But from a physical point of view, this is not quite how we encounter a system in the laboratory. What do we actually have access to? We have preparations and measurements. We prepare a system in some reproducible way; we perform an experiment; we record an outcome; we repeat the procedure many times. The notions of “state” and “observable” are inferred from the pattern of results. An observable is not first given to us as a function on an invisible space. It is given to us as a measurement procedure: a way of coupling an apparatus to a physical system and obtaining a number. Likewise, a state is not initially presented as a point in phase space. Operationally, a state is a preparation procedure that determines the statistical behavior of all measurements. If two preparations give exactly the same expectation value for every observable, then no experiment available within the theory can distinguish them. It is therefore natural to identify them as the same physical state.

This point of view will become much more important in quantum mechanics. In classical mechanics, it eventually leads us back to ordinary phase space. In quantum mechanics, it will not. That difference is one of the central reasons quantum theory is not merely classical mechanics with some small correction added.

So let us reverse the usual order of construction. Rather than beginning with a state space and then defining observables as functions on it, let us begin with observables and ask whether the states can be recovered from them.

Those with a mathematical background may recognize the general pattern. If one is given a ring or algebra, one may ask whether it can be interpreted as an algebra of functions on some space. In algebraic geometry, for instance, one often begins with an algebra and reconstructs a geometric object from it. The same philosophical move appears here: instead of beginning with geometry and then forming functions, we begin with functions and ask whether geometry can be recovered.

Let us first see what the algebra of observables, functions on the phase space, looks like in ordinary classical mechanics. Suppose the phase space is a compact space . The compactness assumption is mainly for simplicity. It ensures that continuous functions are bounded, so their maximum size is finite. The observables are continuous functions For a physical observable such as position, momentum, or energy, one usually wants real-valued functions. But it is mathematically useful to allow complex-valued functions as well, because complex numbers make the algebraic structure cleaner. The real observables will then be precisely those functions satisfying . If and are observables, then so are their sum , their product , and any scalar multiple . The product has a simple classical meaning: For example, if is the position of a particle and is its momentum, then is the observable “position times momentum.” More generally, if one can simultaneously assign values to two classical quantities, then one can multiply those values. This is why the observable algebra in classical mechanics is commutative: The order does not matter because both sides evaluate to the same number at every phase-space point. There is also a natural involution. For a complex-valued observable , define If is real-valued, then . Thus the physical observables, in the ordinary sense, are represented by self-adjoint elements of the algebra. Finally, define the supremum norm

This is the largest possible magnitude of the observable anywhere in phase space. Because is compact and is continuous, this supremum is finite and is actually attained at some point of . The continuous functions on , equipped with addition, multiplication, complex conjugation, and the supremum norm, form a commutative unital -algebra, usually denoted by The word “unital” refers to the constant function , defined by for every point . This function acts as the identity for multiplication: The -condition is the identity For an ordinary function this is almost obvious, since . The point is that this relation survives in the abstract definition of a -algebra, even when one is no longer told that its elements are functions on a space.

Consider a system whose relevant classical state space is a circle. For example, it could be a phase variable, an angle variable, or the configuration of an idealized rotor. Strictly speaking, the full phase space of a particle moving around a circle is , not just . Here the circle is being used as a compact example in which the algebraic reconstruction can be seen completely and explicitly. Usually one starts by saying that the state space is the circle . Rather than using an angle , which has the awkward feature that and represent the same point, one may use the functions These observables satisfy the relation The equation of the circle has become an algebraic relation between observables. Thus the geometry of the circle is encoded in the way its observables are related. The algebra does not merely contain isolated measurement functions; it contains the structure that tells us how those functions fit together.

We can make this much more concrete. Let be the universal commutative unital -algebra generated by one element satisfying Such an element is called a unitary. At this stage is just an abstract element of an algebra. We have not yet declared that it is a function on a circle. Physically, it will eventually be the complex combination of two real observables. Because , every polynomial expression in and can be simplified into a finite Laurent polynomial For example, is an element of the algebra. The multiplication rule is and the involution is The relation says, in an algebraic form, that the magnitude of should be one. We now ask what the points of the space described by this algebra must be.

A point of a classical space should assign a definite value to every observable. Algebraically, this is represented by a nonzero multiplicative linear map called a character. If represents a point, set Because respects products, the involution, and the identity, we have

Therefore Every character sends the abstract generator to a complex number of unit magnitude. But the complex numbers of unit magnitude are exactly the points of the unit circle: Writing , every character gives an angle modulo . Conversely, every defines a character. On a Laurent polynomial, define This is exactly what one gets by replacing the abstract symbol by the complex number . For instance, Since , the relation is preserved. Hence every point of the circle gives a character, and every character comes from a point of the circle. The Gelfand spectrum of this algebra is therefore

This is the central calculation: The circle has been reconstructed from the algebra. Its points are the multiplicative ways of assigning values to all observables. We did not begin by declaring that a system lives on a circle. We began with an algebra generated by one unitary element. The circle appeared because a multiplicative assignment of values to a unitary must send it to a complex number of modulus one.

Every element now becomes a function on the reconstructed circle by the Gelfand transform For the generator , this gives Thus the abstract generator becomes the ordinary circle-coordinate function More generally, the Laurent polynomial becomes the trigonometric polynomial For example, becomes the continuous function The norm is Completing the Laurent polynomials in this norm gives all continuous functions on the circle: To recover the ordinary real coordinate functions, define Both and are self-adjoint: Thus they represent real observables. Their relation is computed directly from :

At the character , these observables take the values and Thus the abstract algebraic identity becomes the ordinary geometric equation The geometry of the circle is therefore encoded in the algebraic relations satisfied by its observables.

This circle computation is the concrete model for the general theorem. The commutative part of the Gelfand–Naimark theorem says that every commutative unital -algebra can be realized as an algebra of continuous complex-valued functions on some compact Hausdorff space .

More precisely, if is an abstract commutative unital -algebra, one defines its Gelfand spectrum to be the set of all nonzero multiplicative linear maps Such a map is called a character. If , every point gives a character by evaluation: The theorem says that, conversely, every character arises this way. Thus the points of the space can be recovered from the algebra itself. An abstract observable becomes a continuous function on the spectrum by the rule In this way one obtains an isomorphism

So we can forget the original space and begin only with the algebra of observables. If the algebra is commutative, then the space can be reconstructed from the algebra. The geometry has not disappeared; it has been encoded algebraically.

There is one small qualification worth keeping in mind. The commutative -algebra determines the compact topological space and its continuous observables. But Hamiltonian mechanics requires more structure: one must also know a Poisson bracket, or equivalently a symplectic structure in favorable situations, together with a Hamiltonian. The algebra tells us what can be measured and how observables combine. The Poisson bracket and Hamiltonian tell us how those observables change with time.

States as expectation-value functionals

Let us now return to the operational meaning of a state. Suppose a system has been prepared in some reproducible way. We measure an observable many times on identically prepared systems. The outcomes may fluctuate from trial to trial. What the preparation determines is not necessarily one definite number, but a probability distribution of outcomes. Provided the expectation exists, the average outcome is denoted by Thus a state gives an expectation-value rule This rule should be linear. If and are observables, then the expectation value of their sum should be the sum of their expectation values: Similarly, It should also be positive. The observable is nonnegative at every classical phase-space point, since Therefore its expectation value must satisfy A linear functional with this property is called positive. The constant observable always takes the value . Hence a normalized physical state should satisfy

A positive normalized linear functional on the observable algebra is called a state. One can see why normalization is natural. Positivity gives a Cauchy–Schwarz inequality: Taking , one gets Therefore, if , then for every . In other words, a positive functional with is the zero functional. Any nonzero positive functional can therefore be normalized by dividing by .

Suppose a particle may be found in different regions of phase space with probability density . Then the expectation value of an observable is provided the integral makes sense. A thermal state is a familiar example. At inverse temperature , one often has a probability density proportional to The system is not assumed to occupy one known phase-space point. Instead, the preparation gives a probability distribution over many possible points, weighted by their energies.

Consider again a particle on a circle. A completely uniform preparation is described by the normalized angular measure For the observables and , one gets This does not mean that the particle is at the origin; the origin is not even on the circle. It means that the preparation is rotationally symmetric, so positive and negative values cancel in the average. This illustrates an important point. A state should not be identified merely with the expectation values of a few chosen observables. Different probability distributions may have the same average position or the same average energy. A state is the full rule for all observables.

The general mathematical statement is the Riesz–Markov representation theorem. If is compact and is a positive linear functional on , then there exists a unique regular Borel measure on such that

If , then so is a probability measure. Thus the operational definition of a state as a positive normalized linear functional is equivalent, in the classical commutative case, to the familiar notion of a probability distribution on phase space.

Let us now return to the circle algebra . A state is a linear functional such that and for every . The first condition says that the constant observable has expectation value one. The second says that squares have nonnegative expectation values. The state gives the expected value of every observable. In particular, is the expected value of the periodic observable , while and are the expected horizontal and vertical coordinates. Because is positive, one has

Define the Fourier moments of the state by Then and Also, since is unitary, But not every sequence of complex numbers with these properties comes from a state. Positivity imposes a stronger condition. Let Then

Since , we obtain

This says that the Toeplitz matrix

is positive.

For the circle, the Riesz–Markov theorem says that every state has a unique probability measure on such that

For the Fourier observables , this gives

Writing , we obtain

Thus the numbers are exactly the Fourier coefficients of the probability distribution. For a trigonometric polynomial we have

Since trigonometric polynomials are dense in , knowing all the numbers determines the expectation value of every continuous observable. Hence it determines the state completely. It also determines the probability measure uniquely. If two probability measures have the same Fourier coefficients, then they agree on every trigonometric polynomial. By density, they agree on every continuous function. Therefore they are the same measure.

So, on the circle, a state is exactly the same thing as a complete list of consistent Fourier moments satisfying the positivity condition above. One can make this reconstruction more explicit. Given a state , define

This is an element of the algebra for each value of . Now define

Because is of the form , positivity gives

Expanding the expression gives

Moreover, Thus is an honest probability distribution on the circle. These are the Fejér approximations to the state. They are smooth probability densities constructed directly from the expectation values . For every fixed integer ,

once . Therefore, as , the Fourier coefficients of converge to the Fourier coefficients of the state. The measures converge weakly to the unique measure representing the state. So this is not merely an abstract theorem. Starting from the expectation values of the observables , one can explicitly construct better and better probability distributions whose limit is the state.

Take a point . The associated state is evaluation at that point: In particular, Thus its Fourier moments are The associated measure is the Dirac measure This means

The point state is multiplicative: This multiplicativity is the algebraic signature of a classical point. The Fejér approximation of this point state is

where The function is nonnegative and sharply peaked at . As increases, the density becomes more concentrated around . In the limit it becomes the point mass . Thus even a perfectly definite classical point can be approximated operationally by increasingly narrow ordinary probability densities.

The usual textbook state—a point of phase space—appears as a special case. It corresponds to the Dirac measure concentrated at , denoted by . This measure satisfies

Hence the associated state is Equivalently, and the measure gives zero weight to every region not containing . This is the mathematical form of a perfectly sharp classical state. For such a point state, every observable has zero uncertainty. Indeed, By contrast, a general probability distribution usually gives a nonzero variance. If a classical particle is known only to lie somewhere in an interval, then its position observable has an uncertainty. If a gas is in thermal equilibrium, then its energy fluctuates. If a die is rolled, then its outcome is uncertain before the roll, even though each individual outcome is definite. The point states are often called pure classical states. General probability measures are mixtures of point states.

We have therefore recovered the usual classical picture from the operational one. Beginning with the algebra of observables, we defined a state as a positive normalized linear functional. The Riesz–Markov theorem then showed that such states are probability measures on a compact phase space. The pure states are the Dirac measures, hence the ordinary points of phase space. The usual formulation of classical mechanics says: a state is a point, and observables are functions on the set of points. The operational formulation says: a state is a consistent assignment of expectation values to observables.

In the classical commutative case these two pictures fit together perfectly. The algebra of observables can be viewed as for a compact Hausdorff space , and the pure states are precisely the points of . To summarize, one may begin with an algebra of observables rather than with a geometric phase space. If the algebra is commutative, then the Gelfand–Naimark theorem reconstructs a compact space from it. States are positive normalized linear functionals, and the Riesz–Markov theorem identifies them with probability measures on that space. The familiar phase-space points arise as the sharp, pure states.

Even time evolution can be formulated algebraically. If is the classical flow carrying a phase-space point forward by time , then observables evolve by The family is a one-parameter group of automorphisms of the algebra: Thus the entire classical theory can be described in terms of observables, states, and their algebraic evolution. To recover Hamiltonian mechanics specifically, one adds the Poisson bracket and the Hamiltonian. In canonical coordinates, the infinitesimal evolution of an observable is

The important point for the next stage of the story is this: in classical mechanics the algebra of observables is commutative, and this is why it can be understood as an algebra of functions on an ordinary phase space. In quantum mechanics the observable algebra will generally be noncommutative. Then the same operational language of states and observables still survives, but the ordinary picture of a system as occupying one point of a classical phase space no longer does.