# Canonical quantization of time-dependent lagrangians

I have a lagrangian

$$L(x^{a}, \dot{x}^{a}, t),$$

which is non-degenerate, quadratic in the fields, and contains an explicit dependence on the evolution parameter $t$.

If $L$ was time-independent, I would follow the following algorithm to deduce the quantum theory:

1. Define the canonical momenta as $p_a = \partial L / \partial \dot{x}^{a}$ and the phase space as a decompactified limit of the symplectic manifolds with the canonical symplectic form given by $\left\{ x^{a}, p_{b} \right\} = \delta^{a}_{b}$.
2. Find a linear combination $a^{a} = \alpha x^{a} + \beta p^{a}$ such that $\left\{a^{a}, a^{*\,b}\right\} = (i \hbar)^{-1} \delta^a_b.$
3. Promote $a$ and $a^{*}$ to operators $\hat{a}$ and $\hat{a}^{\dagger}$ and define $\left|0 \right>$ as an element of the Hilbert space annihilated by all $\hat{a}$.
4. Build the direct representation of $\hat{a}$, $\hat{a}^{\dagger}$ on $\mathcal{H}$ using the commutation relation $[\hat{a}, \hat{a}^{\dagger}] = 1$ and re-interpret $\hat{a}$, $\hat{a}^{\dagger}$ as annihilation and creation operators.
5. For any classical observable use the Weyl quantization map to assign an operator to it. Because $a$ and $a^{*}$ are linear in $x$ and $p$, the Weyl quantization map ensures that the same relation holds for operators, thus proving consistency of the quantization scheme. Re-express all resulting operators in terms of $\hat{a}$ and $\hat{a}^{\dagger}$ to get their explicit representation on $\mathcal{H}$.

That works well for non-degenerate quadratic in $x$, $\dot{x}$ lagrangians with no explicit $t$ dependence.

I want to know how the recipe above changes (or if it is possible to use it at all) when $L$ contains an explicit time dependence. For simplicity, let's assume that it is still non-degenerate and quadratic in $x$, $\dot{x}$.

• If you want a general recipe for canonical quantisation (one where the variables may have any Grassmann parity; be defined on an arbitrary manifold; where the Lagrangian is completely arbitrary; the system may have any sort of constraint and/or gauge symmetries; etc.), you need to introduce the Peierls-DeWitt bracket, which replaces the Poisson bracket and is much more powerful. The procedure is essentially equivalent, but promoting the P-DW bracket to a super-commutator instead of the Poisson one. You will find a great discussion on DeWitt's The global approach to Quantum Field Theory. – AccidentalFourierTransform Aug 10 '18 at 22:42
• Out of curiosity, why do you promote $a,a^*$ to operators at step 3? I noticed that this is what everyone does, but to me it makes much more sense to promote $x,p$ to operators right at step 1. Why wait? Formulate the theory as a quantum theory from the onset! – AccidentalFourierTransform Aug 10 '18 at 22:48
• @AccidentalFourierTransform thanks for the reference, but the book is behind a paywall. I would appreciate if you could sketch how to use it in the answer to this question. I would be particularly happy if you could first explain how to introduce the bracket in the most general case (arbitrary spacetime manifold, any kind of lagrangian, any parity, etc.), and then briefly explain the aspects relevant to my question here. – Prof. Legolasov Aug 10 '18 at 22:49
• @AccidentalFourierTransform if you are satisfied with the algebraic approach to quantum theory, then yes, you can promote $x$ and $p$ to a $C^{*}$ algebra straight away and then deduce its representation. However, in some cases (not related to the question at hand) that last part is very hard: there could be multiple representations, and it is always hard to find the one which is physically relevant. Using $a$ and $a^{*}$ ensures that the Hilbert space is built along the quantization correctly, and all operators are represented on it explicitly. I find it nice. – Prof. Legolasov Aug 10 '18 at 22:51
• Ah, I wish I could. The general definition of the P-DW bracket is much more elaborate than that of the P bracket, and while I am familiar with the general picture, I don't remember the details. Also, the concept is so rich and interesting that it deserves a discussion more thorough that what one may write in an answer here. Just so that you know, if you are willing to download the pdf, it is on Library Genesis (it is not like DeWitt is going to get any money if you buy it anyway...) – AccidentalFourierTransform Aug 10 '18 at 22:53

Quantization is a huge topic, cf. e.g. Ref. 1. Since OP seems interested in the classical Hamiltonian formulation in its own right, it seems natural to split the task in 2 parts:

1. Legendre transformation from Lagrangian to classical Hamiltonian formulation.

2. Quantize the Hamiltonian formulation, using e.g. real variables $(q^i,p_i)$ or complex variables $(a_i,a^{\ast}_i)$.

Explicit time dependence does not fundamentally alter this recipe.

References:

1. M. Henneaux & C. Teitelboim, Quantization of Gauge Systems, 1994.
• 1. Could you be more specific, please? 2. Are you suggesting that the phase space (space of solutions of equations of motions) is still described by $(x,p)$ for initial time with the canonical Poisson bracket? 3. And the same definition of $p$? – Prof. Legolasov Aug 11 '18 at 20:05
• 2. That's one way to look at it. 3. Since I assume a Hamiltonian formulation, the $p_i$ are independent variables. – Qmechanic Aug 12 '18 at 10:20