Planck's constant



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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 hh is the choice of unit h ×h \in \mathbb{R}^\times in the short exact sequence h()U(1)\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 hh (see at formal deformation quantization), then Planck’s constant is the very formal expansion parameter of the deformation. This we discuss below in

As a physical constant

Planck’s constant hh 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 EE that is fixed by its frequency (cycles per second) ν\nu according to the relation E=hνE = h\nu. Thus, the energy emitted by a laser beam of fixed frequency ν\nu is an integer multiple nhνn h \nu of a packet of energy hνh\nu, where nn is the number of photons emitted.

As a fundamental physical constant, hh has dimension (mass)(length) 2(time) 1(mass)(length)^2(time)^{-1}. In meter-kilogram-second (MKS) units, its value is

h6.6260695710 34m 2kg/sh \approx 6.62606957 \cdot 10^{-34} m^2 kg / s

with an uncertainty of up to 29 in the last two digits.

The reduced Planck constant =h/2π\hbar = h/2\pi is the proportionality constant that relates energy (of a photon) to angular frequency ω\omega (radians per second as opposed to cycles per second), so that E=ωE = \hbar \omega.

In geometric quantization

Basic definition

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

\mathbb{Z} \hookrightarrow \mathbb{R}

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

h({0}) h \in (\mathbb{R}- \{0\}) \hookrightarrow \mathbb{R}
()h: (-)\cdot h \;\colon\; \mathbb{Z} \longrightarrow \mathbb{R}

and this “physical unithh is what is called Planck’s constant.

In particular the induced circle group is identified as the quotient of \mathbb{R} by hh \mathbb{Z}, in this sense

U(1)/h U(1) \simeq \mathbb{R}/h \mathbb{Z}

and under this identification its quotient map is expressed in terms of the exponential function exp:z k=0 z kk!\exp \colon z \mapsto \sum_{k = 0}^\infty \frac{z^k}{k!} \in \mathbb{C} as

exp(2πih())=exp(i()):U(1), \exp(2 \pi \tfrac{i}{h}(-)) = \exp(\tfrac{i}{\hbar} (-)) \;\colon\; \mathbb{R} \longrightarrow U(1) \,,


h/2π. \hbar \coloneqq h/2\pi \,.

The resulting short exact sequence is the real exponential exact sequence

0exp(i())U(1)0. 0 \to \mathbb{Z} \longrightarrow \mathbb{R} \stackrel{\exp(\tfrac{i}{\hbar}(-))}{\longrightarrow} U(1) \to 0 \,.

This is the source of the ubiquity of the expression exp(i())\exp(\tfrac{i}{\hbar} (-)) in quantum physics, say in the path integral, where the exponentiated action functional appears as exp(iS)\exp(\tfrac{i}{\hbar} S).

In relation to the symplectic form

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 * 2T^* \mathbb{R} \simeq \mathbb{R}^2 with standard coordinates {p,q}\{p,q\}, then the normalization of the symplectic form dpdq\sim d p \wedge dq actually needed in physics is

ω=1dpdq. \omega = \frac{1}{\hbar} d p \wedge d q \,.

This is because after geometric quantization of this form the observables will obey

[q^,p^]=i(ω p,q) 1 [\hat q, \hat p] = i (\omega_{p,q})^{-1}

and this is supposed to be

=i. \cdots = i \hbar \,.

Accordingly, it follows that if (E,)(E, \nabla) is a prequantum line bundle for ω\omega, then its kk-fold tensor product with itself, for kk \in \mathbb{N}, is a line bundle (E k, k)(E^{\otimes k}, \nabla_k) with curvature kωk \omega. By the above this corresponds to rescaling

/k. \hbar \to \hbar / k \,.

This implies in particular

  1. 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;

  2. for (E,)(E, \nabla) a given prequantum line bundle the limit of the tensor powers (E k, k)(E^{\otimes k}, \nabla_k) as kk tends to infinity roughly corresponds to taking a classical limit. See also (Donaldson 00).


Chern-Simons theory

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 =2\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 SpinSpin-principal connections is

12p^ 1:BSpin connB 3U(1) conn. \tfrac{1}{2}\hat \mathbf{p}_1 : \mathbf{B}Spin_{conn} \to \mathbf{B}^3 U(1)_{conn} \,.

Similarly for 7-dimensional String 2-group infinity-Chern-Simons theory the fundamental value is =6\hbar = 6, with the extended Lagrangian being

16p^ 2:BString connB 7U(1) conn. \tfrac{1}{6}\hat \mathbf{p}_2 : \mathbf{B}String_{conn} \to \mathbf{B}^7 U(1)_{conn} \,.

See at higher geometric quantization for more on this.

In formal deformation quantization

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

Revised on November 14, 2014 07:36:30 by Urs Schreiber (