Similarity in form of time-evolution and Gibbs weight?

In summary, the time-evolution operator in quantum mechanics ##\exp{iHt}## and the Gibbs-weight operator in statistical physics ##\exp{-H/T}## have the same functional form due to the equations of motion of QT and the maximum-entropy principle. This similarity is useful in perturbation theory for finite-temperature QFT. However, the difference in boundary conditions leads to a different approach in thermal field theory. The connection between these operators may be related to generating functions.
  • #1
muscaria
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Why do the time-evolution operator in quantum mechanics ##\exp{iHt}## and the Gibbs-weight operator in statistical physics ##\exp{-H/T}## have the same functional form? – i.e. both exponentials of the Hamiltonian operator.
The Matsubara trick/method just takes this as a fact in thermal QFT; but one might wonder whether there’s a deeper reason why these two operators (with apparently different physical meanings) look so similar. My feeling is that it is related to generating functions in some way, but not sure how.
Thanks
 
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  • #2
The time evolution operator for states in the Schrödinger picture is
$$\hat{U}=\exp(-\mathrm{i} \hat{H} t).$$
This is due to the equations of motion of QT.

The statistical operator in the canonical ensemble is
$$\hat{\rho}(T)=\frac{1}{Z} \exp(-\beta \hat{H}),$$
where ##\beta=1/T## is the inverse temperature due to the maximum-entropy principle for the case that the average energy of the system is given. ##\beta## is the Lagrange parameter for this constraint.

It's just luck that both operators are so similar, and you can do perturbation theory along the Matsubara temperature to calculate finite-temperature QFT almost as in vacuum QFT. However in thermal field theory you have (anti-)periodic boundary conditions for the field operators, so that instead of an energy integral in the vacuum QFT you have the sum over the Matsubara frequency in thermal QFT. For more details, see

http://th.physik.uni-frankfurt.de/~hees/publ/off-eq-qft.pdf
 

1. What is similarity in form of time-evolution and Gibbs weight?

Similarity in form of time-evolution and Gibbs weight refers to the relationship between the dynamics of a physical system and the statistical properties of that system. It describes how the behavior of a system over time is related to the distribution of its microscopic states, as described by the Gibbs weight.

2. How does similarity in form of time-evolution and Gibbs weight impact the study of physical systems?

This concept is important in understanding the behavior of physical systems, as it allows scientists to make predictions about the macroscopic properties of a system based on its microscopic states. It also helps to bridge the gap between the microscopic and macroscopic levels of description.

3. Can you provide an example of similarity in form of time-evolution and Gibbs weight?

One example of this concept is the Ising model, which is used to study the behavior of magnetic materials. The time-evolution of the system is described by the dynamics of the spins of individual particles, while the Gibbs weight describes the probability of each spin configuration. These two aspects are closely related and can be used to predict the macroscopic behavior of the material.

4. What are the implications of similarity in form of time-evolution and Gibbs weight for statistical mechanics?

Similarity in form of time-evolution and Gibbs weight is a key concept in statistical mechanics, as it allows for the calculation of macroscopic properties of a system from its microscopic states. It also allows for the use of statistical methods to study complex systems, as the relationship between time-evolution and the Gibbs weight can reveal important information about the behavior of the system.

5. How does similarity in form of time-evolution and Gibbs weight relate to entropy?

Entropy, which is a measure of the disorder or randomness in a system, is closely related to the Gibbs weight. As the system evolves over time, its entropy will tend to increase, and this increase is related to the change in the Gibbs weight. This relationship between entropy and the Gibbs weight is a fundamental aspect of statistical mechanics and plays a crucial role in understanding the behavior of physical systems.

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