# Which temperature transformation does QFT allow?

How light is a feather? Light = color, or light = weight? Those are different questions, so of course they have different answers. When I skim through the literature about how Lorentz trnasformations affect temperature, I see people arguing about how light a feather is.

We can debate how the words should be used, but the physics is straightforward.

Consider a Lorentz-symmetric QFT. To reduce clutter, I'll use units in which the speed of light $c$ and Boltzmann's constant $k$ are both equal to $1$. Let $P_0,P_1,P_2,P_3$ be the operators representing the system's total energy $(P_0)$ and total momentum ($P_1,P_2,P_3$). Start with the density matrix $$ \rho\propto \exp(-\beta P_0) \tag{1} $$ for some positive constant $\beta$. If we apply a boost in the $1$-direction with rapidity $\theta$, then the resulting density matrix is $$ \rho'\propto \exp\big( -\beta P_0 \cosh\theta -\beta P_1 \sinh\theta \big) \tag{2} $$ with the same constant $\beta$ as before. This is straightforward and uncontroversial. But does it answer the question?

If the question is "How does the state (1) transform under a boost?" then the answer is (2). That's the QFT content of this post.

If the question is "What state should I use?" then the answer is another question: what situation are you trying to model, and which details do you want to assume are known?

If the question is "When physicists say 'temperature', what do they mean?" then I'll list a couple of possibilities below...

If the state is (1), then the quantity $1/\beta$ is typically called "temperature." More generally, if the state is $\exp(-\beta_0 P_0-\beta_1 P_1)$, then the quantity $1/\beta_0$ is a reasonable guess about what the author/speaker might mean by "temperature." Applied to (2), this would give $1/(\beta\cosh\theta)$ for the temperature. Or the author/speaker might use "temperature" to refer instead to the appearance of the radiation emitted by the thermal body, as perceived by an observer in some specific location. Then the temperature depends on the observer's location: the radiation in the boosted frame is not isotropic, because it depends on the boost direction. We could also consider other definitions, which may give other answers.

Words are not always used consistently, and nothing I say here is going to fix that eternal problem.

As the excellent answer by Chiral Anomaly already states, QFT takes no particular stance on the issue unless you specify more precisely what you mean by "temperature". I'll just elaborate on why this would be a hard thing to do.

A very good definition of temperature is that it's whatever a thermometer measures, just like how time is whatever a clock measures in relativity. Now, forgetting about boosts for the moment, why should we expect such a definition to work at all? Why is it that in everyday life, all properly calibrated thermometers and clocks precisely agree with each other, even if they're made of completely different materials?

For time, it's because the time elapsed for any clock can be calculated geometrically, as the length of its path through spacetime. And for temperature, it's because thermal equilibrium has the deeper meaning of maximizing entropy when energy exchange is allowed. When a thermometer is placed in a hot system, *whatever* that system is, the system will transfer energy to the thermometer, until the rate it loses entropy equals the rate the thermometer gains entropy. At the end, you get a temperature reading
$$\frac{1}{T} = \frac{\partial S}{\partial E}.$$
The result only depends on the relationship between entropy and energy of the system you're measuring. If you put a thermometer in your oven, it'll give the same reading in thermal equilibrium whether it's shiny, dirty, painted white, or painted red, even though it'll interact completely differently with the radiation field in each case.

More generally, there is an associated temperature-like quantity for every conserved quantity that can be exchanged to reach equilibrium. If two systems can exchange particle number, then in equilibrium they have equal $\partial S / \partial N$, and we call this quantity $\mu/T$. If two systems can exchange volume, then they match their $\partial S / \partial V$, and we call this quantity $P/T$. The reason we don't have to worry about $\mu$ and $P$ for thermometers is because most thermometers we use, to a good approximation, don't exchange particles with their environment, or change in volume. (And if we had a thermometer that did, we probably wouldn't even call it a thermometer.)

So what about boosts? The problem is that once you boost systems, they pick up another conserved quantity: the overall momentum, which leads to three more quantities $\mathbf{R}/T = \partial S / \partial \mathbf{P}$. And while it's easy to build a thermometer that exchanges energy but not particles, it's basically impossible to imagine one that exchanges energy but not momentum. (It would have to somehow never experience any net force, yet also not just be transparent.)

The result is that different thermometer designs can give completely different results, depending on how they couple to momentum, making it meaningless to speak of a unique temperature at all. Quantum field theory doesn't change this, because the problem has nothing to do with describing the thermodynamic system, and everything to do with how it's measured.