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What is called Planck’s constant in physics and specifically in quantum physics (after Max Planck) is a physical unit of “action” which sets the scale at which effects of quantum physics are genuinely important and physics is no longer well approximated by classical mechanics/classical field theory. This we discuss below at
In the mathematical formulation of the theory, Planck’s constant $h$ is the choice of unit $h \in \mathbb{R}^\times$ in the short exact sequence $\mathbb{Z}\stackrel{h\cdot(-)}{\longrightarrow} \mathbb{R} \to U(1)$ which governs the prequantization lift from real (differential) cohomology to (differential) integral cohomology. The integrality of $\mathbb{Z}$ here is the very “quantum”-ness of quantum theory, and this is what Planck’s constant parameterizes. This we discuss below in
Finally, when infinitesimally approximating this quantization step in perturbation theory in $h$ (see at formal deformation quantization), then Planck’s constant is the very formal expansion parameter of the deformation. This we discuss below in
Planck’s constant $h$ is a quantum of action. It may be illustrated in the case of the electromagnetic field by the fact that each of its quanta – a photon – carries an energy $E$ that is fixed by its frequency (cycles per second) $\nu$ according to the relation $E = h\nu$. Thus, the energy emitted by a laser beam of fixed frequency $\nu$ is an integer multiple $n h \nu$ of a packet of energy $h\nu$, where $n$ is the number of photons emitted.
As a fundamental physical constant, $h$ has dimension $(mass)(length)^2(time)^{-1}$. In meter-kilogram-second (MKS) units, its value is
with an uncertainty of up to 29 in the last two digits.
The reduced Planck constant $\hbar = h/2\pi$ is the proportionality constant that relates energy (of a photon) to angular frequency $\omega$ (radians per second as opposed to cycles per second), so that $E = \hbar \omega$.
The step of prequantization is about refining data in (differential) real cohomology to (differential) integral cohomology. Often this is understood in terms of the canonical inclusion
of the integers as an addiditve subgroup of the real numbers. But since strictly speaking what appears in physics is the real line on which a unit is chosen as part of the identification of mathematical formalism with physical reality, one should really consider all possible additive group homomorphisms $\mathbb{Z}\to \mathbb{R}$. These are parameterized by
and this “physical unit” $h$ is what is called Planck’s constant.
In particular the induced circle group is identified as the quotient of $\mathbb{R}$ by $h \mathbb{Z}$, in this sense
and under this identification its quotient map is expressed in terms of the exponential function $\exp \colon z \mapsto \sum_{k = 0}^\infty \frac{z^k}{k!} \in \mathbb{C}$ as
where
The resulting short exact sequence is the real exponential exact sequence
This is the source of the ubiquity of the expression $\exp(\tfrac{i}{\hbar} (-))$ in quantum physics, say in the path integral, where the exponentiated action functional appears as $\exp(\tfrac{i}{\hbar} S)$.
In the context of geometric quantization Planck’s constant appears as the inverse scale of the symplectic form.
For instance in the simple case that phase space is $T^* \mathbb{R} \simeq \mathbb{R}^2$ with standard coordinates $\{p,q\}$, then the normalization of the symplectic form $\sim d p \wedge dq$ actually needed in physics is
This is because after geometric quantization of this form the observables will obey
and this is supposed to be
Accordingly, it follows that if $(E, \nabla)$ is a prequantum line bundle for $\omega$, then its $k$-fold tensor product with itself, for $k \in \mathbb{N}$, is a line bundle $(E^{\otimes k}, \nabla_k)$ with curvature $k \omega$. By the above this corresponds to rescaling
This implies in particular
a global rescaling of the periods of the symplectic form may be absorbed in a rescaling of Planck’s constant, see at geometric quantization of non-integral forms;
for $(E, \nabla)$ a given prequantum line bundle the limit of the tensor powers $(E^{\otimes k}, \nabla_k)$ as $k$ tends to infinity roughly corresponds to taking a classical limit. See also (Donaldson 00).
In Chern-Simons theory Planck’s constant corresponds to the inverse level of the theory, hence the inverse of the characteristic class that defines the theory, regarded as an element in $\mathbb{Z}$.
Similarly for infinity-Chern-Simons theory. For instance ordinary spin group Chern-Simons theory may be taken to have as the fundamental value $\hbar = 2$, because the first Pontryagin class that defines the theory is divisible by 2, the prequantum 3-bundle that defines the theory of the moduli stack of $Spin$-principal connections is
Similarly for 7-dimensional String 2-group infinity-Chern-Simons theory the fundamental value is $\hbar = 6$, with the extended Lagrangian being
See at higher geometric quantization for more on this.
In formal deformation quantization, Planck’s constant is that very formal deformation parameter. See there for more.
Simon Donaldson, Planck’s constant in complex and almost-complex geometry, XIIIth International Congress on Mathematical Physics (London, 2000), 63–72, Int. Press, Boston, MA, 2001
Wikipedia, Planck’s constant