# How does conservation of energy manifest itself quantum mechanically?

We know that classically, if we have some theory $\mathcal{L}$ such that the action $\int d^4 x \mathcal{L}$ is invariant under time translation, then we can use Noether's theorem to find that (the spatial integral over)

$$\sum\limits_i \partial_t \phi_i \frac{\partial \mathcal{L}}{\partial (\partial_t \phi_i)} - \mathcal{L}$$

is classically a conserved quantity, and we happen to call it "energy".

This is a classical field theory result. But how do we go about showing a similar conservation law in QFT?

• Noether's theorem. A Lagrangian is a Lagrangian, whether it's classical or quantum mechanical. There are so-called anomalies, which arise in path integrals because the $d[\phi]$ in the path integral have a determinant which can be conserved as well.
– webb
Commented Oct 30, 2014 at 18:25
• @webb part of the derivation of Noether's theorem requires the assumption that the Euler-Lagrange equation is satisfied. Maybe somebody can correct me if I'm wrong, but I believe this implies that any conservation law derived from Noether's theorem is strictly classical. Commented Oct 30, 2014 at 18:33
• Dirac equation, Schrodinger equation, etc. are all derived from minimizing an action that has conserved currents associated with symmetries in the action. For example, the invariance of the Schrodinger equation Lagrangian under unitary transformations of $\psi$ implies conservation of the norm.
– webb
Commented Oct 30, 2014 at 18:39
• The precise quantum analogue to Noether's theorem are the Ward-Takahashi identities. Commented Oct 30, 2014 at 20:11
• Bridgeburners, well, not exactly. Quantum operator equations are known to look like classical ones, so in most cases the operator 'conservation' is present. Commented Oct 30, 2014 at 20:12

$$\Psi(t) = \sum_i e^{-i E_i t / \hbar} \cdot \Psi_i.$$
Note that $$E_i$$ and $$\Psi_i$$ do not depend on $$t$$, so energy is in fact conserved (this applies to all QM with time-independent Hamiltonian and not just to QFT).