I recently read the following passage on page 137 in volume I of 'Quantum Fields and Strings: A course for Mathematicians' by Pierre Deligne and others (note that I am no mathematician and have not gotten too far into reading the book, so bear with me):

A physical system is usually described in terms of states and observable. In the Hamiltonian framework of classical mechanics, the states form a symplectic manifold $(M,\omega)$ and the observables are functions on $M$. The dynamics of a (time invariant) system is a one parameter group of symplectic diffeomorphisms; the generating function is the energy or Hamiltonian. The system is said to be free if $(M,\omega)$ is an affine symplectic space and the motion is by a one-parameter group of symplectic transformations. This general descriptions applies to any system that includes classical particles, fields, strings and other types of objects.

The last sentence, in particular, has really intrigued me. It implies a most general procedure for quantizing all systems encountered in physics. I haven't understood the part on symplectic diffeomorphisms or free systems. Here are my questions:

  1. Given a constraint-free phase-space, equipped with the symplectic 2-form, we can construct a Hilbert space of states and a set of observables and start calculating expectation values and probability amplitudes. Since the passage says that this applies to point particles, fields and strings, I assume this is all there is to quantization of any system. Is this true?

  2. What is the general procedure for such a construction, given $M$ and $\omega$?

  3. For classical fields and strings what does this symplectic 2-form look like? (isn't it of infinite dimension?)

  4. Also I assume for constrained systems like in loop quantum gravity, one needs to solve for the constraints and cast the system as a constraint-free before constructing the phase, am I correct?

  5. I don't know what 'the one-parameter group of symplectic diffeomorphisms' are. How are the different from ordinary diffeomorphisms on a manifold? Since diffeomorphisms may be looked at as a tiny co-ordinate changes, are these diffeomorphisms canonical transformations? (is time or its equivalent the parameter mentioned above?)

  6. What is meant by a 'free' system as given above?

  7. By 'affine' I assume they mean that the connection on $M$ is flat and torsion free, what would this physically mean in the case of a one dimensional-oscillator or in the case of systems with strings and fields?

  8. In systems that do not permit a Lagrangian description, how exactly do we define the cotangent bundle necessary for the conjugate momenta? If we can't, then how do we construct the symplectic 2-form? If we can't construct the symplectic 2-form, then how do we quantize the system?

I have asked a lot of long questions, so please answer as many as you can and link relevant articles.


2 Answers 2


The overall idea is the following. As the symplectic manifold is affine (in the sense of affine spaces not in the sense of the existence of an affine connection), when you fix a point $O$, the manifold becomes a real vector space equipped with a non-degenerate symplectic form. A quantization procedure is nothing but the assignment of a (Hilbert-) Kahler structure completing the symplectic structure. In this way the real vector space becomes a complex vector space equipped with a Hermitian scalar product and its completion is a Hilbert space where one defines the quantum theory. As I shall prove shortly in the subsequent example, symplectic symmetries becomes unitary symmetries provided the Hilbert-Kahler structure is invariant under the symmetry. In this way time evolution in Hamiltonian description gives rise to a unitary time evolution.

An interesting example is the following. Consider a smooth globally hyperbolic spacetime $M$ and the real vector space $S$ of smooth real solutions $\psi$ of real Klein-Gordon equation such that they have compactly supported Cauchy data (on one and thus every Cauchy surface of the spacetime).

A non-degenerate (well defined) symplectic form is given by: $$\sigma(\psi,\phi) := \int_\Sigma (\psi \nabla_a \phi - \phi \nabla_a \psi)\: n^a d\Sigma $$ where $\Sigma$ is a smooth spacelike Cauchy surface, $n$ its normalized normal vector future pointing and $d\Sigma$ the standard volume form induced by the metric of the spacetime. In view of the KG equation the choice of $\Sigma$ does not matter as one can easily prove using the divergence theorem.

There are infinitely many Kahler structures one can build up here. A procedure (one of the possible ones) is to define a real scalar product: $$\mu : S \times S \to R$$ such that $\sigma$ is continuous with respect to it (the factor $4$ arises for pure later convenience): $$|\sigma(\psi, \phi)|^2 \leq 4\mu(\psi,\psi) \mu(\psi,\psi)\:.$$ Under this hypotheses a Hilbert-Kahler structure can be defined as I go to summarize.

It is possible to prove that there exist a complex Hilbert space $H$ and an injective $R$-linear map $K: S \to H$ such that $K(S)+ i K(S)$ is dense in $H$ and, if $\langle | \rangle$ denotes the Hilbert space product: $$\langle K\psi | K\phi \rangle = \mu(\psi,\phi) -\frac{i}{2}\sigma(\psi,\phi) \quad \forall \psi, \phi \in S\:.$$ Finally the pair $(H,K)$ is determined up to unitary isomorphisms form the triple $(S, \sigma, \mu)$.

You see that, as a matter of fact, $H$ is a Hilbertian complexfication of $S$ whose antisymmetric part of the scalar product is the symplectic form. (It is also possible to write down the almost complex structure of the theory that is related with the polar decomposition of the operator representing $\sigma$ in the closure of the real vector space $S$ equipped with the real scalar product $\mu$.)

What is the physical meaning of $H$?

It is that the physicists call one-particle Hilbert space. Indeed consider the bosonic Fock Space, ${\cal F}_+(H)$, generated by $H$.

$${\cal F}_+(H)= C \oplus H \oplus (H\otimes H)_S \oplus (H\otimes H\otimes H)_S \oplus \cdots\:,$$ and we denote by $|vac_\mu\rangle$ the number $1$ in $C$ viewed as a vector in ${\cal F}_+(H)$

One may define of ${\cal F}_+(H)$ a faithful representation of bosonic CCR by defining the field operator:

$$\Phi(\psi) := a_{K\psi} + a^*_{K\psi}$$

where $a_f$ is the standard annihilation operator referred to the vector $f\in H$ and $a_f^*$ the standard creation operator referred to the vector $f\in H$. It turns out that, with that definition the vacuum expectation values: $$\langle vac_\mu| \Phi(\psi_1)\cdots \Phi(\psi_n) |vac_\mu\rangle $$ satisfy the standard Wick's prescription and thus all them can be computed in terms of the two-point function only: $$\langle vac_\mu| \Phi(\psi) \Phi(\phi) |vac_\mu\rangle $$ Moreover they are in agreement with the formula valid for Gaussian states (like free Minkowski vacuum in Minkowski spacetime) $$ \langle vac_\mu | e^{i \Phi(\psi)} |vac_\mu \rangle = e^{-\mu(\psi,\psi)/2}$$

Actually, in view of the GNS theorem the constructed representation of the CCR is uniquely determined by $\mu$, up to unitary equivalences.

The field operator $\Phi$ is smeared with KG solutions instead of smooth supportly compacted functions $f$ as usual. However the "translation" is simply obtained. If $E : C_0^{\infty}(M) \to S$ denotes the causal propagator (the difference of the advanced and retarded fundamental solution of KG equation) the usual field operator smeared with $f\in C_0^{\infty}(M)$ is: $$\hat{\phi}(f) := \Phi(Ef)\:.$$

The CCR can be stated in both languages. Smearing fields with KG solutions one has:

$$[\Phi(\psi), \Phi(\phi)] = i \sigma(\psi,\phi)I\:,$$

smearing field operators with functions, one instead has:

$$[\hat{\phi}(f), \hat{\phi}(g)] = i E(f,g) I$$

Every one-parameter group of symplectic diffeomorphisms $\alpha_t :S \to S$ (for instance continuous Killing isometries of $M$) give rise to an action on the algebra of the quantum fields $$\alpha^*_t(\Phi(\psi)) := \Phi(\psi \circ \alpha_t)\:.$$ If the state $|vac_\mu\rangle$ is invariant under $\alpha_t$, namely $$\mu\left(\psi \circ \alpha_t,\psi \circ \alpha_t\right) = \mu\left(\psi ,\psi \right)\quad \forall t \in R,$$ then, essentially using Stone's theorem, one sees that the said continuous symmetry admits a (strongly continuous) unitary representation: $$U_t \Phi(\psi) U^*_t =\alpha^*_t(\Phi(\psi))\:.$$ The self-adjoint generator of $U_t= e^{-itH}$ is an Hamiltonian operator for that symmetry. Actually this interpretation is suitable if $\alpha_t$ arises by a timelike continuous Killing symmetry. Minkowki vacuum is constructed in this way requiring that the corresponding $\mu$ is invariant under the whole orthochronous Poincaré group.

All the picture I have sketched is intermediate between the "practical" QFT and the so-called algebraic formulation. I only would like to stress that choosing different $\mu$ one generally obtain unitarily inequivalent representations of bosonic CCR.

  • $\begingroup$ +1, Informative answer (though OP likely doesn't know Kähler). / Does the same work if you view classical mechanics as a field theory over the $\mathbb R$ time axis? / You pop out one concise definition after the other and it seems to work. What are general obstacles to have injective and dense $K$? For computational purposes, this map seems essentially bijective. Does one need to know much about solving the classical equations of motion (Klein Gordon here, although it's never used in your answer as far as I can see) to find the right Hilber space of could one start ad hoc on the other side? $\endgroup$
    – Nikolaj-K
    Jan 6, 2014 at 14:59
  • $\begingroup$ It works on every manifold of course, I think it works with classical mechanics, but I never tried to do it. $K$ is always injective because $\sigma$ is non-degenerate (the proof is trivial). It is possible to prove that $K$ is dense (not $K+iK$) if and only if the state on the CCR $^*$-algebra is pure (i.e. extremal). Actually the classical equation do not play a fundamental role, the crucial object is the symplectic structure that, here, arises from the classical equation of motion. $\endgroup$ Jan 6, 2014 at 15:14
  • $\begingroup$ There is no a "right" Hilbert space: roughly speaking, you have as many Hilbert spaces as many the scalar products $\mu$ are. At most a "right" Hilbert space can be fixed by requiring that it supports certain unitary representations of symmetry groups. And it is done by requiring that $\mu$ is invariant under the corresponding symplectic symmetries. $\endgroup$ Jan 6, 2014 at 15:18
  • $\begingroup$ I do not think there is something like Darboux' theorem when you consider infinite dimensional symplectic spaces. It is better to think of the symplectic form, even in the finite dimensional case, as an antisymmetric non-degenerate map $\omega : TM\times TM \to R$ without fixing a preferred coordinate system to describe it. Essentially $M$ identifies with its tangent space at a point $T_pM$ as $M$ is supposed to be affine. $\endgroup$ Jan 6, 2014 at 19:46
  • $\begingroup$ OK. The vector space $V$ is that of the 2n-ple $(x^1,\ldots,x^n,p_1,\ldots,p_n)$. You have to define a real symmetric scalar product $\mu$ on $V$ that satisfies (I used an apparently weaker requirement in my answer, but is equivalent to this if $V$ is finite dimensional) $4\mu(z,z) = \max_{z'\neq 0} |\sigma(z,z')|^2/\mu(z',z')$. With these choices one sees that there is a complex vector space $H$ (subspace of $V+iV$) equipped with an Hermitean scalar product and a $R$-linear map $K: V \to H$ that verifies the two conditions I wrote in my answer (actually $K(V)=H$ in this case). $\endgroup$ Jan 7, 2014 at 9:02

Here are some comments on the literature, maybe serving to put Valter Moretti's more concrete response into a broader perspective.

The question asked is a surprisingly good question. It is "good", because it is indeed true that there is this very general prescription for quantization; and "surprisingly" because, while the general idea has been around for ages, this has been understood in decent generality only last year!

Namely, on the one hand it is long appreciated in the context of quantum mechanics that what physicists sweepingly call "canonical quantization" is really this: the construction of the covariant phase space as a (pre-)symplectic manifold, and then the quantization this by the prescription of either algebraic deformation quantization or geometric quantization.

In contrast, it has been understood only surprisingly more recently that established methods of perturbative quantization of field theories, especially in the guise of Epstein-Glaser's causal peruturbation theory (such as QED, QCD, and also perturbative quantum gravity, as in Scharf's textbooks) are indeed also examples of this general method.

For free fields (no interactions), this was first understood in

  • J. Dito. "Star-product approach to quantum field theory: The free scalar field". Letters in Mathematical Physics, 20(2):125–134, 1990.

and then amplified in a long series of articles on locally covariant perturbative quantum field theory

by Klaus Fredenhagen and collaborators, starting with

  • M. Dütsch and K. Fredenhagen. "Perturbative algebraic field theory, and deformation quantization". In R. Longo (ed), "Mathematical Physics in Mathematics and Physics, Quantum and Operator Algebraic Aspects", volume 30 of Fields Institute Communications, pages 151–160. American Mathematical Society, 2001.

Curiously, despite this insight, these authors continued to treat interacting quantum field theory by the comparatively ad hoc Bogoliubov formula, instead of similarly deriving it from a quantization of the (pre-)symplectic structure of the phase space of the interacting theory.

That last step, to show that the traditional construction of interacting peturbative quantum field theory via time-ordered products and Bogoliubov's formula also follows from the general prescription of deformation/geometric quantization of (pre-)symplectic phase space was made, unbelievably, only last year, in the highly recommendable thesis

  • Giovanni Collini, Fedosov Quantization and Perturbative Quantum Field Theory arXiv:1603.09626

Just read the introduction of this thesis, it is very much worthwhile.

(I learned about this article from Igor Khavkine and Alexander Schenkel for which I am grateful.)

In a similar spirit a little later appeared

  • Eli Hawkins, Kasia Rejzner, The Star Product in Interacting Quantum Field Theory arxiv:1612.09157

which disucsses the situation in a bit more generality than Collini does, but omitting the technical details of renormalization in this perspective.


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