Quantum fluctuation

Inquantum field theory, fields undergo quantum fluctuations. A reasonably clear distinction can be made between quantum fluctuations and thermal fluctuations of a quantum field (at least for a free field; for interacting fields, renormalization substantially complicates matters). An illustration of this distinction can be seen by considering quantum and classical Klein–Gordon fields:^[8] For the quantized Klein–Gordon field in the vacuum state, we can calculate the probability density that we would observe a configuration ${\displaystyle \varphi _{t}($x)}   at a time t in terms of its Fourier transform ${\displaystyle {\tilde {\varphi }}_{t}($k)}   to be

${\displaystyle \rho _{0}[\varphi _{t}]=\exp {\left[-{\frac {1}{\hbar }}\int {\frac {d^{3}k}{(2\pi )^{3}}}{\tilde {\varphi }}_{t}^{*}($k){\sqrt {|k|^{2}+m^{2}}}\,{\tilde {\varphi }}_{t}(k)\right]}.}  

In contrast, for the classical Klein–Gordon field at non-zero temperature, the Gibbs probability density that we would observe a configuration ${\displaystyle \varphi _{t}($x)}   at a time ${\displaystyle t}$ is

${\displaystyle \rho _{E}[\varphi _{t}]=\exp {\big [}-H[\varphi _{t}]/k_{\text{B}}T{\big ]}=\exp {\left[-{\frac {1}{k_{\text{B}}T}}\int {\frac {d^{3}k}{(2\pi )^{3}}}{\tilde {\varphi }}_{t}^{*}($k){\frac {1}{2}}\left(|k|^{2}+m^{2}\right)\,{\tilde {\varphi }}_{t}(k)\right]}.}  

These probability distributions illustrate that every possible configuration of the field is possible, with the amplitude of quantum fluctuations controlled by the Planck constant ${\displaystyle \hbar }$ , just as the amplitude of thermal fluctuations is controlled by ${\displaystyle k_{\text{B}}T}$ , where k_B is the Boltzmann constant. Note that the following three points are closely related:

1. the Planck constant has units of action (joule-seconds) instead of units of energy (joules),
2. the quantum kernel is ${\displaystyle {\sqrt {|k|^{2}+m^{2}}}}$  instead of ${\displaystyle {\tfrac {1}{2}}{\big (}|k|^{2}+m^{2}{\big )}}$  (the quantum kernel is nonlocal from a classical heat kernel viewpoint, but it is local in the sense that it does not allow signals to be transmitted),^{[citation needed]}
3. the quantum vacuum state is Lorentz-invariant (although not manifestly in the above), whereas the classical thermal state is not (the classical dynamics is Lorentz-invariant, but the Gibbs probability density is not a Lorentz-invariant initial condition).

Aclassical continuous random field can be constructed that has the same probability density as the quantum vacuum state, so that the principal difference from quantum field theory is the measurement theory (measurement in quantum theory is different from measurement for a classical continuous random field, in that classical measurements are always mutually compatible – in quantum-mechanical terms they always commute).