Classical Mechanics–Observables, states

This post will be an introduction to a series of posts that I am going to write on Quantum Mechanics. In this post I will discuss classical mechanics from a perspective which differs from traditional presentation.

I will start with a basic question of what we do in physics. We can say physics is about the description of the relationship between two events. We prescribe frameworks with some priori concepts and use them to describe these relationships. The concepts of time, space are a few such concepts which are assumed and used in the description. Then we choose to describe objects we see and their behaviour , for instance how they move in time and space. In this post I will be talking about classical mechanics in which we assume many priori notions like space and time. ( There are theories physicists are building to get a consistent theory of quantum gravity in which there is no background spacetime. Physics is described in terms of some correlation functions and there is no meaning attached to terms like ” moving” ” being somewhere” . I will try to post on this topic sometime later.)

Once we have priori notions like space and time, next comes the question how do we describe the state of a system. In classical mechanics they assume that the state of system is described by a set of canonical variable ${q,p}$ , where $q=(q_1,q_2,...,q_n), p=(p_1,p_2,...,p_n)$ describes a point in a phase space manifold. The observables of the system are assumed to be some functions of these $q,p$ And we assume that dynamics of the system is given a time evolution equations $\displaystyle \dot p = -\frac{\partial \mathcal{H}}{\partial q}$ $\displaystyle \dot q =~~\frac{\partial \mathcal{H}}{\partial p}.$

( I will be writing a post on this later. My interest in this post is not in this)

So we assumed that a system is described by some points in the phase space manifold and that every observable of the system is a function on this set of points. But, physically, it’s not the way one views states. One looks at observables and states from an operational point of view. That is, states and observables are assumed to be known from the experiments they do and quantities they measure. ( You will later appreciate this when we move to quantum mechanics). We are now going to reverse the things we did above. We start with some set of observables and try to define what a state is , instead of assuming that observables are functions on states. Those with mathematical background would have observed that we are seeking the general idea of trying to view a set of objects like rings(algebra) as a ring(algebra) of functions on some space.

Looking back, we had observables as a set of functions on the phase space. Each observable is of the form $f(p,q)$ where is f is some continuous function. These set of functions form an algebra. Assume that the phase space $X$ is compact. Then the supremum norm on the functions makes the algebra a normed algebra. So we have a normed algebra of functions to $\mathbb{C}$ say, so that we can define $f^*$ by $f^*(P)=f(P)^*$ . This makes the algebra into a $\mathbb C^*$ -algebra. Now a theorem called Gelfand Naimark representation theorem says every Abelian $\mathbb{C}^*$ – algebra ( with identity) can be seen as an algebra of complex continuous functions on some compact Hausdorff space $X.$ This space is called the Gelfand spectrum of the algebra. Thus we can forget about the space $X$ we started with and just talk about the algebra of observables we have. It contains all the information needed to describe the system.

From an operational view-point the observable $f$ at a state is measured by a series of experiments $f_1,f_2,...$ and the expected value of the results is supposed to be the value of observable at that state (It is an assumption that measured values are distributed so that expectation exists.). So the expected value of sum of two observables is the sum of the expected values and the expectation of $f^*f$ is non-negative. So this expectation operator depends on the state, acts as a linear functional on the algebra and has the properties that $\omega(f^*f)\ge0$ . We assume that this expectation operator defines the state i.e., we further have to assume that no two state gives the same expected values for all the observables. The expectation operator has to define the state.

We know that $\omega(\bf 1)\ge0$ and it is easy to show that if $\omega(\bf 1)=0$ , $\omega(f)=0$ for all $f$ (by the Cauchy -Schwarz inequality and positivity & linearity of $A$ . So we thus normalise $\omega$ and, for a non-trivial functional, get a normalised linear functional.( $\bf 1$ is the function which takes the value 1 on all points viz., the identity on the algebra.)

So we now have a compact phase space and a normed algebra of continuous functions, normed linear functionals on this algebra.If $\omega$ is such a linear functional then Riez Markov representation theorem implies that we have a unique Borel measure defined on $X$ which satisfies $\displaystyle \omega (f)= \int_{X}{f d\mu_{\omega}} , \mu(X)= \omega (\bf 1) =1$ .

Note that a state in this operational definition defines a probability distribution over the space $X$ rather than just a point. But you can see that the states that are defined usually are also states under the general definition. The probability measures correspondingly are measures concentrated completely at a single point in the phase space $X$ . So we have $\omega_{P}(f)=\int f d\mu_{P}=f(P)$ where $\mu_P(P)=1$ and $\mu(P)=0$ everywhere else. This is a point distribution at $P$ . From this we got back the normal definitions of states as points on the phase space and observables as functions on these states.
To summarize, one just needs an algebra of observables to describe the system. What we have done, in a nutshell, is to start with algebra rather than geometry. With this we can have states which are defined in a very general fashion using operational characterisation as linear functionals on this algebra. To get back the geometry we can use the representation theorem mentioned above to first construct a space which can be interpreted as the phase space and measures on this space each observable corresponding to a state.( I did not write about the time evolution. Even the time evolution can be completely described in terms of the algebra. Time evolution is defined by a one parameter group of automorphisms of the algebra).