When considering the Casimir effects, vacuum fluctuations, ..., the temperature can be confused by depending on spacetime + observers (Hamiltonian, accelerated observer, ...) and classical definition by entropy and energy as T-1 =∂S/∂E. In quantum mechanics, when we define a system in the state of ρ=e−βH^, the density matrices denotes the expectation value of such system is considered as thermal expectation value; this considering involves time!
When an observer is inside a defined system or accelerated, the Hamiltonian changes!
Quantum fluctuations don't have anything to do with thermal fluctuations-the baths are physically distinct. So it's necessary to specify in what order the averages are defined, if both are present.
The density matrix, exp(-βH), is an operator, so it doesn't define any particular state, anymore than the Hamiltonian, H, does.
The question of how to define temperature in vacuum fluctuations is not a trivial one, and there may be different ways of approaching it depending on the context and the interpretation of quantum mechanics. However, one possible way of thinking about it is to use the concept of thermal equilibrium and the notion of effective temperature.
Thermal equilibrium is a state in which the macroscopic properties of a system do not change with time, and there is no net flow of energy or entropy between different parts of the system. In classical thermodynamics, this state is characterized by a constant temperature that is the same for all parts of the system. However, in quantum mechanics, this concept can be generalized to include situations where the system is not in a pure state, but in a mixed state described by a density matrix. In this case, the system can be said to be in thermal equilibrium if its density matrix has the form
$$
\rho = \frac{e^{-\beta H}}{Z}
$$
where $H$ is the Hamiltonian of the system, $\beta$ is a parameter related to the inverse temperature, and $Z$ is a normalization factor called the partition function. This form of density matrix is called a thermal state or a Gibbs state, and it maximizes the entropy for a given average energy.
Now, if we consider vacuum fluctuations as quantum fluctuations of some quantum field in a vacuum state, we can ask whether this state can be described by a thermal state or not. If it can, then we can assign an effective temperature to it based on the parameter $\beta$. For example, if we consider an electromagnetic field in a vacuum state, we can show that its density matrix has the form
$$
\rho = \prod_k \frac{1}{2} \left( |0_k\rangle\langle 0_k| + |1_k\rangle\langle 1_k| \right)
$$
where $|0_k\rangle$ and $|1_k\rangle$ are the vacuum and one-photon states for each mode $k$ of the field. This density matrix cannot be written as a thermal state, because it does not depend on the energy of each mode. Therefore, we cannot assign a temperature to vacuum fluctuations of an electromagnetic field in this case.
However, there are situations where vacuum fluctuations can be described by a thermal state, and thus have an effective temperature. One such situation is when there is an external influence on the quantum field that breaks its symmetry or modifies its dynamics. For example, if we consider an electromagnetic field in a cavity with perfectly reflecting walls, we can show that its density matrix has the form
$$
\rho = \prod_k \frac{e^{-\beta \omega_k a^\dagger_k a_k}}{1 + e^{-\beta \omega_k}}
$$
where $\omega_k$ are the frequencies of each mode $k$, and $a^\dagger_k$ and $a_k$ are the creation and annihilation operators for each mode. This density matrix can be written as a thermal state with $\beta = 2\pi/\omega_c$, where $\omega_c$ is the cutoff frequency determined by the size of the cavity. Therefore, we can assign an effective temperature to vacuum fluctuations of an electromagnetic field in a cavity as
$$
T = \frac{\omega_c}{2\pi k_B}
$$
where $k_B$ is the Boltzmann constant. This temperature is called the Casimir temperature, and it depends only on the geometry of the cavity.
Another situation where vacuum fluctuations can be described by a thermal state is when there is an accelerated observer who measures the quantum field. In this case, we can show that the density matrix measured by the observer has the form
$$
\rho = \prod_k \frac{e^{-\beta \omega_k b^\dagger_k b_k}}{1 + e^{-\beta \omega_k}}
$$
where $\omega_k$ are now the frequencies measured by the observer, and $b^\dagger_k$ and $b_k$ are the creation and annihilation operators for each mode measured by the observer. This density matrix can also be written as a thermal state with $\beta = 2\pi/a$, where $a$ is the proper acceleration of the observer. Therefore, we can assign an effective temperature to vacuum fluctuations measured by an accelerated observer as
$$
T = \frac{a}{2\pi k_B}
$$
This temperature is called the Unruh temperature, and it depends only on the acceleration of the observer.
These examples show that vacuum fluctuations can have different effective temperatures depending on the context and the perspective of the observer. However, these temperatures are not intrinsic properties of the quantum field, but rather emergent phenomena that result from the interaction between the field and the environment or the observer. Therefore, they are not absolute or universal, but relative and subjective.
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