Historically, this was not quite Werner Heisenberg's route to obtaining quantum mechanics, but Paul Dirac introduced it in his 1926 doctoral thesis, the "method of classical analogy" for quantization,[1] and detailed it in his classic text Principles of Quantum Mechanics.
By contrast, in quantum mechanics, all significant features of a particle are contained in a state
As in classical mechanics, all dynamical operators can be represented by functions of the position and momentum ones,
This relation encodes (and formally leads to) the uncertainty principle, in the form Δx Δp ≥ ħ/2.
A fundamental difference between classical and quantum mechanics concerns the concept of indistinguishability of identical particles.
The usual wave function is obtained using the Slater determinant and the identical particles theory.
Dirac's book[2] details his popular rule of supplanting Poisson brackets by commutators:
In its simplest form, Groenewold's theorem says that there is no map satisfying the above ground rules and also the bracket condition
[6][7] Suppose we first try to find a quantization map on polynomials of degree less than or equal to three satisfying the bracket condition whenever
Accepting properties 1 & 2, along with a weaker condition that 3 be true only asymptotically in the limit ħ→0 (see Moyal bracket), leads to deformation quantization, and some extraneous information must be provided, as in the standard theories utilized in most of physics.
Accepting properties 1 & 2 & 3 but restricting the space of quantizable observables to exclude terms such as the cubic ones in the above example amounts to geometric quantization.
Quantum mechanics was successful at describing non-relativistic systems with fixed numbers of particles, but a new framework was needed to describe systems in which particles can be created or destroyed, for example, the electromagnetic field, considered as a collection of photons.
Thus, the normal modes comprising the amplitude of the field are simple oscillators, each of which is quantized in standard first quantization, above, without ambiguity.
Historically, quantizing the classical theory of a single particle gave rise to a wavefunction.
Alternatively, the Feynman integral approach is available for quantizing relativistic fields, and is manifestly invariant.
For non-relativistic field theories, such as those used in condensed matter physics, Lorentz invariance is not an issue.
In a non-interacting (free) field theory, the vacuum is normally identified as a state containing zero particles.
Operators constructed from φ and π can then formally be defined at other times via the time-evolution generated by the Hamiltonian,
Many of these issues can be sidestepped using the Feynman integral as described for a particular V(φ) in the article on scalar field theory.
It is the φks that have become operators obeying the standard commutation relations, [φk, πk†] = [φk†, πk] = iħ, with all others vanishing.
is the Hilbert space constructed by applying any combination of the infinite collection of creation operators ak† to
If there is a gauge symmetry, then the number of independent components of the field must be carefully analyzed to avoid over-counting equivalent configurations, and gauge-fixing may be applied if needed.
It turns out that commutation relations are useful only for quantizing bosons, for which the occupancy number of any state is unlimited.
When quantizing fermions, the fields are expanded in creation and annihilation operators, θk†, θk, which satisfy
annihilated by the θk, and the Fock space is built by applying all products of creation operators θk† to |0⟩.
In cases involving spontaneous symmetry breaking, it is possible to have a non-zero VEV, because the potential is minimized for a value φ = v .
This construction is utilized in the Higgs mechanism in the standard model of particle physics.
The classical theory is described using a spacelike foliation of spacetime with the state at each slice being described by an element of a symplectic manifold with the time evolution given by the symplectomorphism generated by a Hamiltonian function over the symplectic manifold.
The quantum algebra of "operators" is an ħ-deformation of the algebra of smooth functions over the symplectic space such that the leading term in the Taylor expansion over ħ of the commutator [A, B] expressed in the phase space formulation is iħ{A, B} .
In general, for the quantities (observables) involved, and providing the arguments of such brackets, ħ-deformations are highly nonunique—quantization is an "art", and is specified by the physical context.