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 \hbar (see at formal deformation quantization), then Planck’s constant is the very formal expansion parameter of the deformation. This we discuss below in

Applied to the key example of perturbative quantum field theory it turns out that the powers of \hbar in contributions to the Feynman perturbation series essentially correspond to the loop order of the given Feynman diagram. This we discuss 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).


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 perturbative quantization

In perturbative quantum field theory Planck’s constant (together with the coupling constant, which indicates the strength of interactions) is regarded as tiny, in fact as infinitesimal, in that all observables are expressed as (generally non-converging, asymptotic) formal power series in the coupling constant.

This is explicitly realized by formal deformation quantization, which regards quantization as as deformation of the classical algebra of observables to a non-commutative algebra on formal power series with coefficients the original observables.

In perturbative quantum field theory

In perturbative quantum field theory the scattering amplitudes in the S-matrix are expressed as formal power series in (the coupling constant and) in Planck's constant \hbar. This formal power series may be expressed as a formal sum of contributions labeled by Feynman diagrams. The loop order refers to something like the “number of loops” of edges in the Feynman diagram that contibutes to a given scattering amplitude. It turns out that the loop order corresponds to the order in \hbar that is contributed by this diagram (see below).

Therefore contributions of graphs at zero without loops (these are trees, and hence these contributions are referred to as being at “tree level”) correspond to the limit of classical field theory with 0\hbar \to 0. Indeed tree level Feynman diagrams yield perturbative solutions of the classical equations of motion (see Helling).

Most predictions of the standard model of particle physics have very good agreement with experiment already to very low loop order, first or second; inclusion of third loop order is used (at least in QCD) to constrain theoretical uncertainties of the result (see Cacciari 05, slide 5, e.g. in Higgs field computation, see ADDHM 15). In rare cases higher loop orders are used (for instance in the computation of the anomalous magnetic moments AHKN 12, but this is not a scattering experiment).

This usefulness of low loop order is forturnate because

  1. the S-matrix formal power series for all theories of interest has vanishing radius of convergence (Dyson 52), hence is at best an asymptotic series for which the sum of more than some low order terms is meaningless;

  2. the computational effort increases immensely with loop order.

In the Feynman perturbation series

In the computation of scattering amplitudes for fields/particles via perturbative quantum field theory the scattering matrix (Feynman perturbation series) is a formal power series in (the coupling constant and) Planck's constant \hbar whose contributions may be labeled by Feynman diagrams. Each Feynman diagram Γ\Gamma is a finite labeled graph, and the order in \hbar to which this graph contributes is

E(Γ)V(Γ) \hbar^{ E(\Gamma) - V(\Gamma) }


  1. V(Γ)V(\Gamma) \in \mathbb{N} is the number of vertices of the graph

  2. E(Γ)E(\Gamma) \in \mathbb{N} is the number of edges in the graph.

This comes about (see at S-matrix – Feynman diagrams and Renormalization for details) because

  1. the explicit \hbar-dependence of the S-matrix is

    S(gL int)=kg k kk!T(L intL intkfactors) S\left(\tfrac{g}{\hbar} L_{int} \right) = \underset{k \in \mathbb{N}}{\sum} \frac{g^k}{\hbar^k k!} T( \underset{k \, \text{factors}}{\underbrace{L_{int} \cdots L_{int}}} )
  2. the further \hbar-dependence of the time-ordered product T()T(\cdots) is

    T(L intL int)=prodexp(ω F(x,y)δδϕ(x)δδϕ(y))(L intL int), T(L_{int} L_{int}) = prod \circ \exp\left( \hbar \int \omega_{F}(x,y) \frac{\delta}{\delta \phi(x)} \otimes \frac{\delta}{\delta \phi(y)} \right) ( L_{int} \otimes L_{int} ) \,,

where ω F\omega_F denotes the Feynman propagator and ϕ(x)\phi(x) the field observable at point xx (where we are notationally suppressing the internal degrees of freedom of the fields for simplicity, writing them as scalar fields, because this is all that affects the counting of the \hbar powers).

The resulting terms of the S-matrix series are thus labeled by

  1. the number of factors of the interaction L intL_{int}, these are the vertices of the corresponding Feynman diagram and hence each contibute with 1\hbar^{-1}

  2. the number of integrals over the Feynman propagator ω F\omega_F, which correspond to the edges of the Feynman diagram, and each contribute with 1\hbar^1.

Now the formula for the Euler characteristic of planar graphs says that the number of regions in a plane that are encircled by edges, the faces here thought of as the number of “loops”, is

L(Γ)=1+E(Γ)V(Γ). L(\Gamma) = 1 + E(\Gamma) - V(\Gamma) \,.

Hence a planar Feynman diagram Γ\Gamma contributes with the “loop orderL(Γ)L(\Gamma) as

L(Γ)1. \hbar^{L(\Gamma)-1} \,.

So far this is the discussion for internal edges. An actual scattering matrix element is of the form

ψ out|S(gL int)|ψ in, \langle \psi_{out} \vert S\left(\tfrac{g}{\hbar} L_{int} \right) \vert \psi_{in} \rangle \,,


|ψ in1 n inϕ (k 1)ϕ (k n in)|vac \vert \psi_{in}\rangle \propto \tfrac{1}{\sqrt{\hbar^{n_{in}}}} \phi^\dagger(k_1) \cdots \phi^\dagger(k_{n_{in}}) \vert vac \rangle

is a state of n inn_{in} free field quanta and similarly

|ψ out1 n outϕ (k 1)ϕ (k n out)|vac \vert \psi_{out}\rangle \propto \tfrac{1}{\sqrt{\hbar^{n_{out}}}} \phi^\dagger(k_1) \cdots \phi^\dagger(k_{n_{out}}) \vert vac \rangle

is a state of n outn_{out} field quanta. The normalization of these states, in view of the commutation relation [ϕ(k),ϕ (q)][\phi(k), \phi^\dagger(q)] \propto \hbar, yields the given powers of \hbar.

This means that an actual scattering amplitude given by a Feynman diagram Γ\Gamma with E ext(Γ)E_{ext}(\Gamma) external vertices scales as

L(Γ)1+E ext(Γ)/2. \hbar^{L(\Gamma) - 1 + E_{ext}(\Gamma)/2 } \,.

(For the analogous discussion of the dependence on the actual quantum observables on \hbar given by Bogoliubov's formula, see there.)


  • 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 October 9, 2017 12:48:46 by Urs Schreiber (