### Beyond the CPT theorem

Virtually all known laws of physics are invariant under the CPT transformation — that is, the combined operations of Charge conjugation (C), Parity or "mirror flipping" (P), and Time reversal (T). What that means is the following. Start with a trajectory $$\psi(t)$$ through state space, which represents some possible way for a system to change over time according to the known laws of physics. Now transform that trajectory, by reversing the order of all the states, and then applying C, P and T to each of them:

$\psi(t) \mapsto CPT\psi(-t).$

$$CPT$$-invariance means that the resulting trajectory $$CPT\psi(-t)$$ will also be a possible according to the laws. One can check that this is equivalent to the statement that $$CPT$$ commutes with the Hamiltonian, $$[CPT,H]=0.$$

Why is $$CPT$$ so often symmetry? There is a theorem that explains it: if we characterize quantum field theory in a very plausible and general way (such as by the Wightman axioms or Haag axioms), and in particular assume that it admits a unitary representation of the Poincaré group, then $$CPT$$-invariance is guaranteed. This result is called the CPT theorem. See Borchers & Yngvason for a very readable proof in the Haag framework.

Ok, that's the background for today. Now:

Question: as we move forward, and begin to adopt theories that go beyond the standard model of particle physics, will we continue to have a CPT theorem or something like it?

It is widely believe that the CPT may fail in generic extensions of quantum theory. In particular, the requirement of a unitary representation of the Poincaré group is pretty strong, and may not hold in the kind of general context of interest in quantum gravity. Just search for CPT-violation on the arxiv to see what I mean.

But there is a sense in which something "like" a CPT theorem probably will hold in physics beyond the standard model. That sense is this: every unitary dynamics admits infinitely many "time reversing transformations" (i.e., time reversal plus some other linear symmetries) under which the dynamics is invariant. Here's a more careful statement of this fact.

Fact. Let $$\mathcal{H}$$ be a separable Hilbert space with a unitary group $$U_t = e^{-itH}$$ describing the quantum dynamics, and let $$T$$ be the (antiunitary) time reversal operator. Then there exists a unitary operator $$\Theta$$ such that the dynamics is $$\Theta T$$-invariant, in that $$[\Theta T,H]=0$$.

Think of $$\Theta$$ as some kind of generalized symmetry transformation, similar to $$CP$$, but something else entirely. It is in this sense that this fact expresses something like the $$CPT$$-theorem, although unlike the $$CPT$$-theorem the mathematics is completely trivial.

There are two steps to seeing why this "Fact" is true. The first is to observe that, for every self-adjoint operator $$H$$, there is something called a conjugation operator $$K_H$$ such that $$[K_H,H]=0$$. Here's how it's defined. The self-adjoint operator $$H$$ comes with its own basis set for the Hilbert space, $${v_1,v_2,v_3,\dots}$$. That's because of the spectral theorem. So for every vector $$\psi$$ in the Hilbert space there are complex constants $$c_i$$ that allow you to write that vector,

$\psi = c_1v_1 + c_2v_2 + c_3v_3 + \cdots.$

The conjugation operator $$K_H$$ is just the operator defined by conjugating all the complex constants of a vector written in the $$H$$-basis,

$K_H\psi = c_1^*v_1 + c_2^*v_2 + c_3^*v_3 + \cdots.$

So that's pretty easy. And it's easy to check that $$K_H$$ satisfies some special properties: it is antilinear $$K_H(a\psi+b\phi)=a^*K_H\psi+b^*K_H\phi$$, antiunitary $$\langle K_H\psi,K_H\phi\rangle = \langle \psi,\phi \rangle^*$$, and it commutes with the $$H$$ that we used to define it $$[K_H,H]=0$$. We will use all of these properties in the next step.

The second step to seeing why our "Fact" is true is to recognize that if $$K_1$$ and $$K_2$$ are any two antiunitary operators, then they are related by a unitary operator, $$K_2 = UK_1$$. It's a nice exercise to check for yourself that this is true, but if you get stuck, try here.

Since time reversal $$T$$ and the conjugation operator $$K_H$$ for the Hamiltonian are both anitunitary, this means that $$K_H$$ is related to $$T$$ by some unitary operator $$\Theta$$:

$K_H = \Theta T.$

So, there is always a unitary operator $$\Theta$$ such that $$\Theta T$$ commutes with the Hamiltonian $$H$$.

Above, I said there were actually infinitely many such operators. Puzzle: Can you work out why?

If you're impatient, here's the reason. Let $$f:\mathbb{R}\rightarrow\mathbb{C}$$ be a function (a Borel function if we're being pedantic), and let $$f(H)$$ be the corresponding Hilbert space operator as a function of the Hamiltonian $$H$$. (For example, if $$f(x)=x^2$$, then $$f(H)=H^2=H\circ H$$.) Every such function $$f(H)$$ commutes with $$H$$. And we already know that $$K_H$$ does as well. So their composition commutes with $$H$$ as well:

$0 = [f(H)K_H,H] = [f(H)\Theta T,H] = [\Theta^\prime T, H],$

where $$\Theta^\prime = f(H)\Theta$$. Since there are infinitely many such functions, this means that there are infinitely many such operators $$\Theta^\prime$$.

### Four 1975 Lectures by Paul Dirac

Slobodan Perovic (via Chris Joas) pointed this gem out to me. Four wonderful lectures given by Paul Dirac in 1975, on various topics in the foundations of quantum theory and cosmology.

"Quantum Mechanics"

"Quantum Electrodynamics"

"Magnetic Monopoles"

"Large Number Hypothesis"

Dear academic publishers: your business model runs completely counter to the aims of the academic community, for this reason: academic publishing is not like commercial publishing. Stop conflating the two.

I know of nary an academic that is publishing for the bling. So, stop thinking of us as obscure niche counterparts to J. K. Rowling. Scholarly authors would be crazy to write books for the tiny (or often non-existent) monetary compensation. They do it to disseminate information as widely as possible. So, stop treating academic work as if it were commercial. You're running completely off the rails.

Here's an example. You put a \$229 USD price-tag on an important textbook, Souriau's (1970) Structure of Dynamical Systems. I'm sure you've done the calculation: how many people can be expected buy the textbook at that price? Not many. Not to mention that we could pick up two copies of J. K. Rowling's "complete works" for this royal sum. This is not dissemination of information. This is you failing the academic community.

Because of your silliness, Souriau's scholarship is not being widely shared in the way that the academic community needs. In this case, the author himself is taking steps to overcome your failing, by posting the French edition on his website. (Souriau's stated motto, translated from French: "I wanted this site to distribute my work as widely as possible.") Unfortunately, an English version of the book is not freely available. At least, not anymore.

It was freely available, on underground websites like Library.nu (rest in peace). Such websites came into existence because you did not meet the aims of the academic community. While we worked to share information, providing publishers with free content to put in their books and journals, you turned around and sold that content at commercially high prices. You actively prevented the dissemination of scholarly knowledge. On the other hand, by making half a million scholarly books publicly available, Library.nu actively enabled it. The end of this service amounted to a huge loss for the scholarly community.

Academics and publishers alike are beginning to recognize that we have a problem. Fortunately, there are many other publishing models on the table, many of which might go far to meet our aims. Take a long hard look in the mirror, academic publishers. It's time for a major change.

### Sellars and the Philosophy of Physics

In a letter to Chisholm, Wilfred Sellars wrote:
Thus, while I agree with you that
'. . .' means - - -

is not constructable in Rylean terms ('Behaviorese,' I have called it), I also insist that it is not to be analyzed in terms of
'. . .' expresses t, and t is about - - -.

My solution is that "'. . .' means - - -" is the core of a unique mode of discourse which is as distinct from the description and explanation of empirical fact, as is the language of prescription and justification.
Although Sellars was concerned with the philosophy of mind, there is something important here for philosophers of physics to learn as well. A major activity of physics is the collection of empirical facts. Another is the prediction and justification of these facts. But the activity of investigating meaning is a distinct activity altogether. This last activity includes much of what concerns the philosophy of physics, when it is done well.

Whether it be causation, equivalence, gauge, prediction, or simultaneity -- among many examples -- I think much of what distinguishes philosophy of physics from physics is a central concern with the (particularly philosophical) activity of explicating meaning.