Temperature operator in quantum mechanics
There are two kinds of quantities in quantum mechanics. Firstly, there are quantities like mass and charge that can be known to any degree of precision without disturbing the wave-function of the particle. Such quantities are not quantum operators, but appear as parameters in quantum mechanics. Secondly, quantities such as position and energy are defined as Hermitian operators in quantum mechanics, which correspond to physical observables. Temperature can be easily measured with a thermometer. Should temperature be a quantum operator?
Temperature is conventionally defined in terms of the number of micro-states a system can possibly be in with its energy, volume, and particle number fixed. Temperature is conventionally treated as a parameter in both, classical and quantum mechanics.
Temperature is experimentally measured by observing the height of a mercury column or the electric current through a thermocouple. Position and current are quantum operators, and a system can be in a superposition of different positions. Can a system be in a superposition of different temperatures? More interestingly, the two different temperatures may correspond to different macroscopic phases of the material, such as paramagnetism and ferromagnetism. Can any material be in a superposition of being a "paramagnet" and a "ferromagnet"?
The conventional view of temperature has a difficulty that I will illustrate with a thought experiment.
Consider a box containing several particles that are all entangled through interactions. Now split this box into two parts and take them far away from each other. Now measure the energy of one particle in one of the boxes. If the particle is found to be in a higher (or lower) than average energy, it will thermalize with its neighbors and that box will be slightly warmer/cooler.
Due to the entanglement, the other box will cool down (or warm up) to conserve energy. Assume for now that temperature is a parameter, and like other parameters can be known precisely at all times without disturbing the system. If the temperatures of the two boxes are being monitored continuously by two observers, one of them will instantaneously know that a measurement was made far away by the other observer. But there cannot be any communication faster than the speed of light! This is a thermal version of the E.P.R. paradox.
There can be two ways to resolve the paradox: either change the axioms of quantum mechanics so that there are fundamental limitations to measuring any parameter, including mass and charge. This is too radical a proposition as it would change the basic equations of quantum mechanics including the Schrodinger equation, for which we have ample experimental evidence. The second way, and in my opinion the correct way, to resolve the paradox is to conceive temperature as a quantum operator. It must be impossible to continuously monitor the temperature of a system.
Like other observables, the measurement of temperature always accompanies a wavefunction collapse. This resolves the problem of superluminal communication posed above. The fact that temperature is a quantum observable immediately leads us to the problem of defining a quantum operator for temperature. Like any other operator, the temperature operator depends on the system. Just as the Hamiltonian of a system of harmonic oscillators is different from that of a chain of spins, so is the temperature operator.
One way of defining the temperature operator is to assign to every energy eigenvector, the eigenvalue of the corresponding temperature. Thus, the temperature operator will be a function of the Hamiltonian operator. Another way is to model a real position based thermometer and assign to position eigenvectors, the eigenvalues of the indicated temperature. We have discussed such thermometers in detail in our paper.
The most important requirement for any theory is that it should be verifiable. The view of temperature presented here also has testable consequences. Consider a magnetic material maintained very close to the temperature where it gets demagnetized. A source emits a sufficiently energetic photon to slightly raise the temperature of the material. A half silvered mirror is kept between the source and the material. After the photon is emitted, the material will be in a superposition of two different temperatures, and thus two different values of magnetization. Another source throws tiny magnetic dipoles close to the material. Upon scattering, they will be in a superposition of two different scattered states. If we see an interference pattern on a screen placed behind slits, this will support the theory that temperature is a quantum observable. If just spots are observed behind the slits, temperature operator may not be the correct resolution to the paradox described above.
If you, the reader are an experimental physicist, I hope you will find time (and funding!) to do this experiment. Do tell us how it went.