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By quantization is meant some process that
reads in a system of classical mechanics, or rather of prequantum data in the form of an Lagrangian/action functional and or a phase space equipped with symplectic structure
and returns a corresponding system of quantum mechanics.
We indicate here a systematic motivation of quantization by looking at classical mechanics formalized as symplectic geometry from the point of view of Lie theory (Fiorenza-Rogers-Schreiber 13, Nuiten 13).
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Quantization of course was and is motivated by experiment, hence by observation of the observable universe: it just so happens that quantum mechanics and quantum field theory correctly account for experimental observations where classical mechanics and classical field theory gives no answer or incorrect answers. A historically important example is the phenomenon called the “ultraviolet catastrophe”, a paradox predicted by classical statistical mechanics which is not observed in nature, and which is corrected by quantum mechanics.
But one may also ask, independently of experimental input, if there are good formal mathematical reasons and motivations to pass from classical mechanics to quantum mechanics. Could one have been led to quantum mechanics by just pondering the mathematical formalism of classical mechanics? (Hence more precisely: is there a natural Synthetic Quantum Field Theory?)
The following spells out an argument to this effect. It will work for readers with a background in modern mathematics, notably in Lie theory, and with an understanding of the formalization of classical/prequantum mechanics in terms of symplectic geometry.
So to briefly recall, a system of classical mechanics/prequantum mechanics is a phase space, formalized as a symplectic manifold $(X, \omega)$. A symplectic manifold is in particular a Poisson manifold, which means that the algebra of functions on phase space $X$, hence the algebra of classical observables, is canonically equipped with a compatible Lie bracket: the Poisson bracket. This Lie bracket is what controls dynamics in classical mechanics. For instance if $H \in C^\infty(X)$ is the function on phase space which is interpreted as assigning to each configuration of the system its energy – the Hamiltonian function – then the Poisson bracket with $H$ yields the infinitesimal time evolution of the system: the differential equation famous as Hamilton's equations.
Something to take notice of here is the infinitesimal nature of the Poisson bracket. Generally, whenever one has a Lie algebra $\mathfrak{g}$, then it is to be regarded as the infinitesimal approximation to a globally defined object, the corresponding Lie group (or generally smooth group) $G$. One also says that $G$ is a Lie integration of $\mathfrak{g}$ and that $\mathfrak{g}$ is the Lie differentiation of $G$.
Therefore a natural question to ask is: Since the observables in classical mechanics form a Lie algebra under Poisson bracket, what then is the corresponding Lie group?
The answer to this is of course “well known” in the literature, in the sense that there are relevant monographs which state the answer. But, maybe surprisingly, the answer to this question is not (at time of this writing) a widely advertized fact that has found its way into the basic educational textbooks. The answer is that this Lie group which integrates the Poisson bracket is the “quantomorphism group”, an object that seamlessly leads to the quantum mechanics of the system.
Before we spell this out in more detail, we need a brief technical aside: of course Lie integration is not quite unique. There may be different global Lie group objects with the same Lie algebra.
The simplest example of this is already one of central importance for the issue of quantization, namely, the Lie integration of the abelian line Lie algebra $\mathbb{R}$. This has essentially two different Lie groups associated with it: the simply connected translation group, which is just $\mathbb{R}$ itself again, equipped with its canonical additive abelian group structure, and the discrete quotient of this by the group of integers, which is the circle group
Notice that it is the discrete and hence “quantized” nature of the integers that makes the real line become a circle here. This is not entirely a coincidence of terminology, but can be traced back to the heart of what is “quantized” about quantum mechanics.
Namely, one finds that the Poisson bracket Lie algebra $\mathfrak{poiss}(X,\omega)$ of the classical observables on phase space is (for $X$ a connected manifold) a Lie algebra extension of the Lie algebra $\mathfrak{ham}(X)$ of Hamiltonian vector fields on $X$ by the line Lie algebra:
This means that under Lie integration the Poisson bracket turns into an central extension of the group of Hamiltonian symplectomorphisms of $(X,\omega)$. And either it is the fairly trivial non-compact extension by $\mathbb{R}$, or it is the interesting central extension by the circle group $U(1)$. For this non-trivial Lie integration to exist, $(X,\omega)$ needs to satisfy a quantization condition which says that it admits a prequantum line bundle. If so, then this $U(1)$-central extension of the group $Ham(X,\omega)$ of Hamiltonian symplectomorphisms exists and is called… the quantomorphism group $QuantMorph(X,\omega)$:
While important, for some reason this group is not very well known, which is striking because it contains a small subgroup which is famous in quantum mechanics: the Heisenberg group.
More precisely, whenever $(X,\omega)$ itself has a compatible group structure, notably if $(X,\omega)$ is just a symplectic vector space (regarded as a group under addition of vectors), then we may ask for the subgroup of the quantomorphism group which covers the (left) action of phase space $(X,\omega)$ on itself. This is the corresponding Heisenberg group $Heis(X,\omega)$, which in turn is a $U(1)$-central extension of the group $X$ itself:
At this point it is worth pausing for a second to note how the hallmark of quantum mechanics has appeared as if out of nowhere simply by applying Lie integration to the Lie algebraic structures in classical mechanics:
if we think of Lie integrating $\mathbb{R}$ to the interesting circle group $U(1)$ instead of to the uninteresting translation group $\mathbb{R}$, then the name of its canonical basis element $1 \in \mathbb{R}$ is canonically “$i$”, the imaginary unit. Therefore one often writes the above central extension instead as follows:
in order to amplify this. But now consider the simple special case where $(X,\omega) = (\mathbb{R}^2, d p \wedge d q)$ is the 2-dimensional symplectic vector space which is for instance the phase space of the particle propagating on the line. Then a canonical set of generators for the corresponding Poisson bracket Lie algebra consists of the linear functions $p$ and $q$ of classical mechanics textbook fame, together with the constant function. Under the above Lie theoretic identification, this constant function is the canonical basis element of $i \mathbb{R}$, hence purely Lie theoretically it is to be called “$i$”.
With this notation then the Poisson bracket, written in the form that makes its Lie integration manifest, indeed reads
Since the choice of basis element of $i \mathbb{R}$ is arbitrary, we may rescale here the $i$ by any non-vanishing real number without changing this statement. If we write “$\hbar$” for this element, then the Poisson bracket instead reads
This is of course the hallmark equation for quantum physics, if we interpret $\hbar$ here indeed as Planck's constant. We see it arises here merely by considering the non-trivial (the interesting, the non-simply connected) Lie integration of the Poisson bracket.
This is only the beginning of the story of quantization, naturally understood and indeed “derived” from applying Lie theory to classical mechanics. From here the story continues. It is called the story of geometric quantization. We close this motivation section here by some brief outlook.
The quantomorphism group which is the non-trivial Lie integration of the Poisson bracket is naturally constructed as follows: given the symplectic form $\omega$, it is natural to ask if it is the curvature 2-form of a $U(1)$-principal connection $\nabla$ on complex line bundle $L$ over $X$ (this is directly analogous to Dirac charge quantization when instead of a symplectic form on phase space we consider the the field strength 2-form of electromagnetism on spacetime). If so, such a connection $(L, \nabla)$ is called a prequantum line bundle of the phase space $(X,\omega)$. The quantomorphism group is simply the automorphism group of the prequantum line bundle, covering diffeomorphisms of the phase space (the Hamiltonian symplectomorphisms mentioned above).
As such, the quantomorphism group naturally acts on the space of sections of $L$. Such a section is like a wavefunction, except that it depends on all of phase space, instead of just on the “canonical coordinates”. For purely abstract mathematical reasons (which we won’t discuss here, but see at motivic quantization for more) it is indeed natural to choose a “polarization” of phase space into canonical coordinates and canonical momenta and consider only those sections of the prequantum line bundle which depend only on the former. These are the actual wavefunctions of quantum mechanics, hence the quantum states. And the subgroup of the quantomorphism group which preserves these polarized sections is the group of exponentiated quantum observables. For instance in the simple case mentioned before where $(X,\omega)$ is the 2-dimensional symplectic vector space, this is the Heisenberg group with its famous action by multiplication and differentiation operators on the space of complex-valued functions on the real line.
An interesting subclass of quantum field theories, especially of perturbative quantum field theories, is thought to arise from prequantum field theory via a process of quantization. This process reads in certain – usually differential geometric – data (Lagrangian field theory), interprets this data as specifying the dynamics of some physical system, and spits out the perturbative quantum field theory that encodes the time evolution of this system.
For more on this see at Wick algebra and causal perturbation theory.
Historically, it was an approximation to the true time evolution that was originally found and studied, by Newton, Maxwell, Einstein and others. This is now known as “classical physics”. The true dynamics in turn is “quantum physics”.
In view of this, quantization is often understood as a right inverse to the procedure that sends the full quantum dynamics to its classical limit. As such it is not well defined, i.e. unique, when it exists. Additional structures sometimes make it unique.
algebraic deformation quantization
dimension | classical field theory | Lagrangian BV quantum field theory | factorization algebra of observables |
---|---|---|---|
general $n$ | P-n algebra | BD-n algebra? | E-n algebra |
$n = 0$ | Poisson 0-algebra | BD-0 algebra? = BD algebra | E-0 algebra? = pointed space |
$n = 1$ | P-1 algebra = Poisson algebra | BD-1 algebra? | E-1 algebra? = A-∞ algebra |
There have been found some formalizations of at least some aspects of quantization, notably there is algebraic deformation quantization and geometric quantization.
duality between $\;$algebra and geometry
$\phantom{A}$geometry$\phantom{A}$ | $\phantom{A}$category$\phantom{A}$ | $\phantom{A}$dual category$\phantom{A}$ | $\phantom{A}$algebra$\phantom{A}$ |
---|---|---|---|
$\phantom{A}$topology$\phantom{A}$ | $\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$ | $\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/Gelfand-Kolmogorov+theorem">Gelfand-Kolmogorov</a>}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}$$\phantom{A}$ | $\phantom{A}$commutative algebra$\phantom{A}$ |
$\phantom{A}$topology$\phantom{A}$ | $\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$ | $\phantom{A}$$\overset{\text{<a class="existingWikiWord" href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand duality</a>}}{\simeq} TopAlg^{op}_{C^\ast, comm}$$\phantom{A}$ | $\phantom{A}$comm. C-star-algebra$\phantom{A}$ |
$\phantom{A}$noncomm. topology$\phantom{A}$ | $\phantom{A}$$NCTopSpaces_{H,cpt}$$\phantom{A}$ | $\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}$$\phantom{A}$ | $\phantom{A}$general C-star-algebra$\phantom{A}$ |
$\phantom{A}$algebraic geometry$\phantom{A}$ | $\phantom{A}$$\phantom{NC}Schemes_{Aff}$$\phantom{A}$ | $\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/affine+scheme#AffineSchemesFullSubcategoryOfOppositeOfRings">almost by def.</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{fin}$$\phantom{A}$ | $\phantom{A}$fin. gen.$\phantom{A}$ $\phantom{A}$commutative algebra$\phantom{A}$ |
$\phantom{A}$noncomm. algebraic$\phantom{A}$ $\phantom{A}$geometry$\phantom{A}$ | $\phantom{A}$$NCSchemes_{Aff}$$\phantom{A}$ | $\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}$$\phantom{A}$ | $\phantom{A}$fin. gen. $\phantom{A}$associative algebra$\phantom{A}$$\phantom{A}$ |
$\phantom{A}$differential geometry$\phantom{A}$ | $\phantom{A}$$SmoothManifolds$$\phantom{A}$ | $\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">Milnor's exercise</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}$$\phantom{A}$ | $\phantom{A}$commutative algebra$\phantom{A}$ |
$\phantom{A}$supergeometry$\phantom{A}$ | $\phantom{A}$$\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}$$\phantom{A}$ | $\phantom{A}$$\array{ \overset{\phantom{\text{Milnor's exercise}}}{\hookrightarrow} & Alg^{op}_{\mathbb{Z}_2 \phantom{AAAA}} \\ \mapsto & C^\infty(\mathbb{R}^n) \otimes \wedge^\bullet \mathbb{R}^q }$$\phantom{A}$ | $\phantom{A}$supercommutative$\phantom{A}$ $\phantom{A}$superalgebra$\phantom{A}$ |
$\phantom{A}$formal higher$\phantom{A}$ $\phantom{A}$supergeometry$\phantom{A}$ $\phantom{A}$(super Lie theory)$\phantom{A}$ | $\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}$ | $\phantom{A}\array{ \overset{ \phantom{A}\text{<a href="https://ncatlab.org/nlab/show/L-infinity-algebra#ReformulationInTermsOfSemifreeDGAlgebra">Lada-Markl</a>}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}$ | $\phantom{A}$differential graded-commutative$\phantom{A}$ $\phantom{A}$superalgebra $\phantom{A}$ (“FDAs”) |
in physics:
$\phantom{A}$algebra$\phantom{A}$ | $\phantom{A}$geometry$\phantom{A}$ |
---|---|
$\phantom{A}$Poisson algebra$\phantom{A}$ | $\phantom{A}$Poisson manifold$\phantom{A}$ |
$\phantom{A}$deformation quantization$\phantom{A}$ | $\phantom{A}$geometric quantization$\phantom{A}$ |
$\phantom{A}$algebra of observables | $\phantom{A}$space of states$\phantom{A}$ |
$\phantom{A}$Heisenberg picture | $\phantom{A}$Schrödinger picture$\phantom{A}$ |
$\phantom{A}$AQFT$\phantom{A}$ | $\phantom{A}$FQFT$\phantom{A}$ |
$\phantom{A}$higher algebra$\phantom{A}$ | $\phantom{A}$higher geometry$\phantom{A}$ |
$\phantom{A}$Poisson n-algebra$\phantom{A}$ | $\phantom{A}$n-plectic manifold$\phantom{A}$ |
$\phantom{A}$En-algebras$\phantom{A}$ | $\phantom{A}$higher symplectic geometry$\phantom{A}$ |
$\phantom{A}$BD-BV quantization$\phantom{A}$ | $\phantom{A}$higher geometric quantization$\phantom{A}$ |
$\phantom{A}$factorization algebra of observables$\phantom{A}$ | $\phantom{A}$extended quantum field theory$\phantom{A}$ |
$\phantom{A}$factorization homology$\phantom{A}$ | $\phantom{A}$cobordism representation$\phantom{A}$ |
These traditional formulations are geared towards mechanics (as opposed to field theory). Even in that restricted application, it seems that a more comprehensive understanding is still missing
The goal is to get closer to a systematic theory of quantization.
In the context of field theory the conceptual issues become even more severe. For example the class of field theories called Yang-Mills theory is a core ingredient in the standard model of particle physics but the quantization of Yang-Mills theory (see there for more) poses famous open problems – of which of course that there isn’t yet a comprehensive theory of what this even means is not the least.
Here we survey aspects of quantization which give rise to an index map or something similar, that is, a push-forward in generalized cohomology. More on this is in (Nuiten 13) and at motivic quantization.
The general pattern here is this:
The kinetic action part of the action functional of a prequantum field theory is really part of the measure/orientation in generalized cohomology on the moduli space of fields
The (gauge coupling-)interaction in the action functional of a prequantum field theory induces a cocycle in some cohomology which serves as a twist for the generalized cohomology theory in which the above integration/push-forward takes place.
the push-forward itself is the path integral quantization, the construction of partition functions.
For instance the archetypical example of the quantization of the charged particle propagating on a Riemannian manifold $X$ proceeds (in Wick rotated form as it appears in the worldline formalism for corresponding QFT) like so: write $[\exp(-S_{kin}(\gamma)) D\gamma]$ for the Wiener measure on paths in $X$, then the integral kernel of the time evolution propagator of the quantum particle is given by the path integral (see there for more)
which is the integration of the parallel transport/holonomy functional (of the given connection that models the background gauge field) against the Wiener measure.
More generally, for given any phase space equipped with a prequantum line bundle $(L_\omega, \nabla)$, the corresponding geometric quantization is, if $X$ admits a compatible spin^c structure, the index of the spin^c Dirac operator coupled to this bundle, hence the space of quantum states is
For more on this see at geometric quantization – As Index of the Spin^c Dirac operator.
Here the Dirac operator represents a class in K-homology
and this is now the “measure” against which the K-theory class
of the prequantum line bundle is being integrated, by the index map hence the composition in KK-theory:
By the discussion at index this is equivalently the partition function of an auxiliary supersymmetric quantum mechanics system with background gauge field $\nabla$:
The appearance of an auxiliary and already quantized field theory here may seem circular, but is in fact part of a deeper pattern of quantization by the holographic principle, where rich quantum theories arise as boundary field theories of higher dimensional topological field theories. In the above case the higher dimensional theory is secretly the non-perturbative version of the Poisson sigma-model associated with the original phase space Poisson manifold. More details on this are indicated at extended geometric quantization of 2d Chern-Simons theory. In one dimension higher the Witten genus arises this way as the quantization of the heterotic string 2d QFT regarded as the boundary field theory of the M2-brane in Horava-Witten theory. (See (Nuiten13) for more.)
The choice of spin^c structure in the above is really the choice of a Poincaré duality (see at Poincaré duality algebra for more) which exhibits the orientation in generalized cohomology by a combined Atiyah duality/Thom isomorphism.
The analogue of such a choice of Poincaré duality between cohomology and homology in BV-formalism is the choice of a volume form that identified the de Rham complex with the BV-complex of multivector fields. See at BV-BRST formalism – Homological integration.
Again, here the kinetic action functional is part of the measure, now under the duality part of the BV-operator. The quantum master equation is now the analog of the orientability condition. Homological BV-quantization is then obtain by assing to the homology groups of this BV-operator, hence again by “pushing it to the point”.
This is the setup in which one can derive Feynman diagram rules form cohomologicalquantization, see at Feynman diagram – Refereces – In homological BV-quantization.
quantization
A general survey is in
A general geometrically inclined introduction can be found in
Sean Bates, Alan Weinstein, Lectures on the geometry of quantization, pdf
Klaas Landsman, Mathematical topics between classical and quantum mechanics, Springer Monographs in Math. 1998. xx+529 pp.
A proposal for a full formalization of the notion of quantization for “finite” theories such as Dijkgraaf-Witten theory is in
A historical discussion by one of the $n$labizants is here: mathlight:quantization. See also Urs’s manifesto at Mathematical Foundations of Quantum Field and Perturbative String Theory.
Marc Henneaux, Claudio Teitelboim, Quantization of Gauge Systems, Princeton University Press 1992
Nikolai Reshetikhin, Lectures on quantization of gauge systems, arxiv/1008.1411
The quantization via the A-model-method is described in
The perspective of geometric pre-quantization as a canonical construction in higher Lie theory is discussed in
A discussion of quantization as cohomological/motivic quantization of higher prequantum field theory is in
Some discussion of quantization in terms of Bohr toposes is in