Sunday, 3 February 2013

States

In physics, one considers various quantities and functions, usually defined on spacetime, or on some assembly of particles. The quantities and functions are (usually) mixed mathematical entities, with abstract values. Laws of physics are then propositions which have semantic content: they say that these functions are related in some way -- usually via a differential equation. An important kind of value is a physical state. We have some sort of system $S$ (say a point particle, or a rigid body, or an assembly of particles, or a region of gas, etc.), and we think that it can "be" in some range of possible physical states.

I mention this only because it seems to me that there is a persistent mistake in some recent literature about applied mathematics and indispensability, in assuming that physical states are "concrete" entities, presumably of a spectacularly peculiar sort. But they are not "concrete". "Physical" does not imply "concrete". For example, a wavefunction $\Psi$ is physical, but is not a concrete entity: it's a function to $\mathbb{C}$. The wavefunction is the physical state.

For example, for an $N$-particle system in classical mechanics, the physical states are $N$-tuples of ordered pairs,
$((\mathbf{r}_i, \mathbf{v}_i) \mid i \in \{1,\dots, N\})$,
where $\mathbf{r}_i$ is the location of particle $i$, and $\mathbf{v}_i$ is the instantaneous velocity of particle $i$. Although certain kinds of equivalences may have to be taken into account, the points in the state space are the physical states. Taken together, they form a structure, which in a sense is "like" the manifold $\mathbb{R}^{6N}$, because the locations and velocities can be co-ordinatized as triples of reals (i.e., given a co-ordinate chart $\phi$ on space, and a point $\mathbf{r}$ in space, we have $\phi(\mathbf{r}) = (x, y, z) \in \mathbb{R}^3$). In classical mechanics, when one moves to the phase space, the structure is called a symplectic manifold.

Consider a 3-particle system, and consider the state
$\alpha := ((\mathbf{r}_1, \mathbf{v}_1),(\mathbf{r}_2, \mathbf{v}_2),(\mathbf{r}_3, \mathbf{v}_3))$.
This state $\alpha$ is not a concrete thing. What, for example, is the speed of this state $\alpha$? Its location? Such questions are absurd. A sequence $\alpha$ of positions and velocities is not a concrete entity (like, perhaps, the point particle itself). And despite the claims of some fictionalists, a physical state is also not like Santa Claus or Jane Eyre, because physical states are what are our scientific theories are about, quite unlike Santa and Jane Eyre.

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