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In many courses, specially when studying the structure of matter, we write the Hamiltonian of a given system thinking of it classically and then admit an implicit change of classical dynamic functions to quantum observables in order to obtain the Hamiltonian of the quantum system, and this seems to work just fine in most cases (except for when there is a product of two observables which are non-commuting in QM). Therefore, I would like to know whether this "analogy" can be taken as far as to claim:

"For every quantum system admitting a classical analogue, if a quantity is conserved in the classical system then it is also conserved in the quantum system".

This can be written mathematically as:

$$\frac{d}{dt}\Omega_{classic}=0\Longrightarrow \frac{d}{dt}\langle \Omega_{QM}\rangle=0$$

Of course, the implication arrow could not go the other way, since there are quantum observables which lack a classical analogue, such as spin.

Qmechanic
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Lagrangiano
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2 Answers2

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"For every quantum system admitting a classical analogue, if a quantity is conserved in the classical system then it is also conserved in the quantum system".

This is not true.

A classical conservation law, by Noether's theorem, is given by a classical symmetry. Now a classical symmetry of the Lagrangian need not be preserved by quantisation - in this case we say the symmetry is anomalous. Two important such anomalies are the chiral anomaly in QFT and the conformal anomaly in conformal field theory and hence also in string theory.

The chiral anomaly originally referred to the 'anomalous' decay rate of the neutral pion. Early calculations suggested that the decay rate of the neutral pion was suppressed. However, this was contradicted by experiment. Eventually, Adler-Bell-Jackiw traced this anomaly to the breaking of axial symmetry in QED by quantum corrections.

The conformal anomaly in string theory cancels when the dimension of spacetime is 26d in the bosonic theory and 10d in the supersymmetric theory.

Mozibur Ullah
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In classical physics the equation of a physical quantity such as momentum $\mathbf{p}$ is written in terms of a function $\mathbf{p}(t)$ where the value of the momentum at time $t$ is $\mathbf{p}(t)$. Since momentum is conserved if you add up the momenta of all systems involved in an interaction that sum doesn't change over time.

In quantum physics the evolution of a physical quantity is described by a linear operator called an observable. The possible results of measurements of an observable are its eigenvalues. Quantum physics predicts the probability of each of the possible values and so the expectation value of the observable. In general the results of a measurement of an observable depend on what has happened to all of its possible values during an experiment: quantum interference. For an example, see Section 2 of

https://arxiv.org/abs/math/9911150

When information is copied out of a quantum system, interference is suppressed, this is called decoherence:

https://arxiv.org/abs/1911.06282

The systems you see in everyday life have information copied out of them on scales of space and time much smaller than those over which they change significantly and as a result you don't see interference in everyday life: you don't diffract when you walk through a doorway. Decoherence makes such systems evolve approximately according to classical equations of motion.

A quantity is conserved in quantum theory when it commutes with the Hamiltonian and in such cases decoherence selects histories in which those quantities are conserved:

https://arxiv.org/abs/gr-qc/9410006

https://arxiv.org/abs/0903.1802

alanf
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