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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{Planck's constant} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{physics}{}\paragraph*{{Physics}}\label{physics} [[!include physicscontents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{Idea}{Idea}\dotfill \pageref*{Idea} \linebreak \noindent\hyperlink{AsAPhysicalConstant}{As a physical constant}\dotfill \pageref*{AsAPhysicalConstant} \linebreak \noindent\hyperlink{InGeometricQuantization}{In geometric quantization}\dotfill \pageref*{InGeometricQuantization} \linebreak \noindent\hyperlink{BasicDefinition}{Basic definition}\dotfill \pageref*{BasicDefinition} \linebreak \noindent\hyperlink{in_relation_to_the_symplectic_form}{In relation to the symplectic form}\dotfill \pageref*{in_relation_to_the_symplectic_form} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{InFormalDeformationQuantization}{In perturbative quantization}\dotfill \pageref*{InFormalDeformationQuantization} \linebreak \noindent\hyperlink{in_perturbative_quantum_field_theory}{In perturbative quantum field theory}\dotfill \pageref*{in_perturbative_quantum_field_theory} \linebreak \noindent\hyperlink{InFeynmanPerturbationSeries}{In the Feynman perturbation series}\dotfill \pageref*{InFeynmanPerturbationSeries} \linebreak \noindent\hyperlink{History}{History}\dotfill \pageref*{History} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{Idea}{}\subsection*{{Idea}}\label{Idea} What is called \emph{Planck's constant} in [[physics]] and specifically in [[quantum physics]] (after [[Max Planck]]) is a [[physical unit]] of ``[[action functional|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 \begin{itemize}% \item \emph{\hyperlink{AsAPhysicalConstant}{As a physical constant}} \end{itemize} In the [[mathematics|mathematical]] formulation of the [[theory (physics)|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|differential]]) [[cohomology]] to ([[differential cohomology|differential]]) [[integral cohomology]]. The [[integer|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 \begin{itemize}% \item \emph{\hyperlink{InGeometricQuantization}{In geometric quantization}}. \end{itemize} 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 geometry|formal expansion parameter]] of the [[deformation theory|deformation]]. This we discuss below in \begin{itemize}% \item \emph{\hyperlink{InFormalDeformationQuantization}{In perturbative quantization}}. \end{itemize} 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 \begin{itemize}% \item \emph{\hyperlink{InFeynmanPerturbationSeries}{In the Feynman perturbation series}} \end{itemize} \hypertarget{AsAPhysicalConstant}{}\subsection*{{As a physical constant}}\label{AsAPhysicalConstant} Planck's constant $h$ is a quantum of [[action functional|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) [[physical units|units]], its value is \begin{displaymath} h \approx 6.62606957 \cdot 10^{-34} m^2 kg / s \end{displaymath} 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 $E = \hbar \omega$. \hypertarget{InGeometricQuantization}{}\subsection*{{In geometric quantization}}\label{InGeometricQuantization} \hypertarget{BasicDefinition}{}\subsubsection*{{Basic definition}}\label{BasicDefinition} The step of [[prequantization]] is about refining data in ([[differential cohomology|differential]]) real [[cohomology]] to ([[differential cohomology|differential]]) [[integral cohomology]]. Often this is understood in terms of the canonical inclusion \begin{displaymath} \mathbb{Z} \hookrightarrow \mathbb{R} \end{displaymath} 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 \emph{all} possible additive group homomorphisms $\mathbb{Z}\to \mathbb{R}$. These are parameterized by \begin{displaymath} h \in (\mathbb{R}- \{0\}) \hookrightarrow \mathbb{R} \end{displaymath} \begin{displaymath} (-)\cdot h \;\colon\; \mathbb{Z} \longrightarrow \mathbb{R} \end{displaymath} and this ``physical [[unit]]'' $h$ is what is called \emph{Planck's constant}. In particular the induced [[circle group]] is identified as the [[quotient]] of $\mathbb{R}$ by $h \mathbb{Z}$, in this sense \begin{displaymath} U(1) \simeq \mathbb{R}/h \mathbb{Z} \end{displaymath} 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 \begin{displaymath} \exp(2 \pi \tfrac{i}{h}(-)) = \exp(\tfrac{i}{\hbar} (-)) \;\colon\; \mathbb{R} \longrightarrow U(1) \,, \end{displaymath} where \begin{displaymath} \hbar \coloneqq h/2\pi \,. \end{displaymath} The resulting [[short exact sequence]] is the real [[exponential exact sequence]] \begin{displaymath} 0 \to \mathbb{Z} \longrightarrow \mathbb{R} \stackrel{\exp\left(\tfrac{i}{\hbar}(-)\right)}{\longrightarrow} U(1) \to 0 \,. \end{displaymath} 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 $\exp(\tfrac{i}{\hbar} S)$. \hypertarget{in_relation_to_the_symplectic_form}{}\subsubsection*{{In relation to the symplectic form}}\label{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 $T^* \mathbb{R} \simeq \mathbb{R}^2$ with standard coordinates $\{p,q\}$, then the normalization of the symplectic form $\sim d p \wedge dq$ actually needed in physics is \begin{displaymath} \omega = \frac{1}{\hbar} d p \wedge d q \,. \end{displaymath} This is because after [[geometric quantization]] of this form the [[observables]] will obey \begin{displaymath} [\hat q, \hat p] = i (\omega_{p,q})^{-1} \end{displaymath} and this is supposed to be \begin{displaymath} \cdots = i \hbar \,. \end{displaymath} 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 \begin{displaymath} \hbar \to \hbar / k \,. \end{displaymath} This implies in particular \begin{enumerate}% \item a global rescaling of the [[periods]] of the symplectic form may be absorbed in a rescaling of Planck's constant, see at \emph{[[geometric quantization of non-integral forms]]}; \item 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 (\hyperlink{Donaldson00}{Donaldson 00}). \end{enumerate} \hypertarget{examples}{}\subsubsection*{{Examples}}\label{examples} In [[Chern-Simons theory]] Planck's constant corresponds to the inverse \emph{level} of the theory, hence the inverse of the [[characteristic class]] that defines the theory, regarded as an element in $\mathbb{Z}$. Similarly for [[schreiber: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 circle n-bundle|prequantum 3-bundle]] that defines the theory of the [[moduli stack]] of $Spin$-[[principal connections]] is \begin{displaymath} \tfrac{1}{2}\hat \mathbf{p}_1 : \mathbf{B}Spin_{conn} \to \mathbf{B}^3 U(1)_{conn} \,. \end{displaymath} Similarly for 7-dimensional [[String 2-group]] [[schreiber:infinity-Chern-Simons theory]] the fundamental value is $\hbar = 6$, with the extended Lagrangian being \begin{displaymath} \tfrac{1}{6}\hat \mathbf{p}_2 : \mathbf{B}String_{conn} \to \mathbf{B}^7 U(1)_{conn} \,. \end{displaymath} See at \emph{[[higher geometric quantization]]} for more on this. \hypertarget{InFormalDeformationQuantization}{}\subsection*{{In perturbative quantization}}\label{InFormalDeformationQuantization} 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 series|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. \hypertarget{in_perturbative_quantum_field_theory}{}\subsection*{{In perturbative quantum field theory}}\label{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 \emph{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 \hyperlink{RelationToPowersInPlancksConstant}{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 $\hbar \to 0$. Indeed tree level Feynman diagrams yield [[perturbation theory|perturbative]] solutions of the [[classical field theory|classical]] [[equations of motion]] (see \hyperlink{Helling}{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 \hyperlink{Cacciari05}{Cacciari 05, slide 5}, e.g. in [[Higgs field]] computation, see \hyperlink{ADDHM15}{ADDHM 15}). In rare cases higher loop orders are used (for instance in the computation of the [[anomalous magnetic moments]] \hyperlink{AHKN12}{AHKN 12}, but this is not a scattering experiment). This usefulness of low loop order is forturnate because \begin{enumerate}% \item the [[S-matrix]] [[formal power series]] for all [[theory (physics)|theories]] of interest has \emph{vanishing} [[radius of convergence]] (\href{perturbation+theory#Dyson52}{Dyson 52}), hence is at best an [[asymptotic series]] for which the [[sum]] of more than some low order terms is meaningless; \item the computational effort increases immensely with loop order. \end{enumerate} \hypertarget{InFeynmanPerturbationSeries}{}\subsection*{{In the Feynman perturbation series}}\label{InFeynmanPerturbationSeries} In the computation of [[scattering amplitudes]] for [[field (physics)|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 $\Gamma$ is a finite labeled [[graph]], and the order in $\hbar$ to which this graph contributes is \begin{displaymath} \hbar^{ E(\Gamma) - V(\Gamma) } \end{displaymath} where \begin{enumerate}% \item $V(\Gamma) \in \mathbb{N}$ is the number of [[vertices]] of the graph \item $E(\Gamma) \in \mathbb{N}$ is the number of [[edges]] in the graph. \end{enumerate} This comes about (see at \emph{\href{S-matrix#ExistenceAndRenormalization}{S-matrix -- Feynman diagrams and Renormalization}} for details) because \begin{enumerate}% \item the explicit $\hbar$-dependence of the [[S-matrix]] is \begin{displaymath} 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}}} ) \end{displaymath} \item the further $\hbar$-dependence of the [[time-ordered product]] $T(\cdots)$ is \begin{displaymath} 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} ) \,, \end{displaymath} \end{enumerate} where $\omega_F$ denotes the [[Feynman propagator]] and $\phi(x)$ the field observable at point $x$ (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 \begin{enumerate}% \item the number of factors of the [[interaction]] $L_{int}$, these are the [[vertices]] of the corresponding Feynman diagram and hence each contibute with $\hbar^{-1}$ \item the number of integrals over the Feynman propagator $\omega_F$, which correspond to the edges of the Feynman diagram, and each contribute with $\hbar^1$. \end{enumerate} 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 \emph{faces} here thought of as the number of ``loops'', is \begin{displaymath} L(\Gamma) = 1 + E(\Gamma) - V(\Gamma) \,. \end{displaymath} Hence a planar Feynman diagram $\Gamma$ contributes with the ``[[loop order]]'' $L(\Gamma)$ as \begin{displaymath} \hbar^{L(\Gamma)-1} \,. \end{displaymath} So far this is the discussion for internal edges. An actual scattering matrix element is of the form \begin{displaymath} \langle \psi_{out} \vert S\left(\tfrac{g}{\hbar} L_{int} \right) \vert \psi_{in} \rangle \,, \end{displaymath} where \begin{displaymath} \vert \psi_{in}\rangle \propto \tfrac{1}{\sqrt{\hbar^{n_{in}}}} \phi^\dagger(k_1) \cdots \phi^\dagger(k_{n_{in}}) \vert vac \rangle \end{displaymath} is a state of $n_{in}$ free field quanta and similarly \begin{displaymath} \vert \psi_{out}\rangle \propto \tfrac{1}{\sqrt{\hbar^{n_{out}}}} \phi^\dagger(k_1) \cdots \phi^\dagger(k_{n_{out}}) \vert vac \rangle \end{displaymath} is a state of $n_{out}$ field quanta. The normalization of these states, in view of the commutation relation $[\phi(k), \phi^\dagger(q)] \propto \hbar$, yields the given powers of $\hbar$. This means that an actual [[scattering amplitude]] given by a [[Feynman diagram]] $\Gamma$ with $E_{ext}(\Gamma)$ external vertices scales as \begin{displaymath} \hbar^{L(\Gamma) - 1 + E_{ext}(\Gamma)/2 } \,. \end{displaymath} (For the analogous discussion of the dependence on the actual [[quantum observables]] on $\hbar$ given by [[Bogoliubov's formula]], see \href{Bogoliubov's+formula#PowersInPlancksConstant}{there}.) \hypertarget{History}{}\subsection*{{History}}\label{History} [[Max Planck]] introduced the constant named after him in the discussion of [[black body radiation]]. In [[classical field theory]] black body radiation comes out completely wrong (``[[ultraviolet catastrophe]]''). Planck fixed this in an ad-hoc but succesful manner by postulating that [[energy]]/[[frequency]] of the [[harmonic oscillators]] in the black body ([[atoms]], [[molecules]]) is quantized in units measured by some quantum of action. Eventually this ad hoc postulate led to a change of the foundations of physics from [[classical physics]] to [[quantum physics]], which now predicts the quantization of energy/frequency from more conceptual, fundamental principles. \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[Compton wavelength]] \item [[theory (physics)]] \item [[quantization]] \item [[coupling constant]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} \begin{itemize}% \item [[Simon Donaldson]], \emph{Planck's constant in complex and almost-complex geometry}, XIIIth International Congress on Mathematical Physics (London, 2000), 63--72, Int. Press, Boston, MA, 2001 \item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Planck_constant}{Planck's constant}} \end{itemize} [[!redirects Planck constant]] \end{document}