nLab uncertainty of fluxes

Idea

In (higher) gauge theory, the total (generalized) electric/magnetic fluxes through given surfaces (or higher submanifolds) should be observables and hence have Poisson brackets in the classical field theory To the extent that these induce non-trivial operator commutators in the corresponding quantum field theory, this means that the corresponding kinds of fluxes would be subject to the Heisenberg uncertainty principle.

A subtlety here is that the usual phase space-methods for Lagrangian field theories directly provide these brackets only for the differential forms which are the flux densities. These however, provide only a rational image of the fluxes, which in general may furthermore have torsion-cohomology-components.

In plain electromagnetism, the canonical momentum of the gauge potential $A$ is proportional (by constants to be ignored in the following) to the electric flux density $\star F$ (where $F$ is the Faraday tensor flux density and $\star$ is the Hodge star operator on the given spacetime pseudo-Riemannian manifold), which however means that the the Poisson bracket of the magnetic flux density $F \equiv \mathrm{d}_{dR} A$ with the electric flux density is a total derivative (cf. eg. Blaschke & Gieres 2021 (5.5)), so that the bracket of the integrated fluxes $\Phi_E$ and $\Phi_B$ through given orientable closed surfaces $S_E$ and $S_B$, respectively, actually vanishes (cf. FMS07b (3.6) and Prop. below):

(1)$$\big\{ \Phi_E ,\, \Phi_B \big\} \;\equiv\; \Big\{ \textstyle{\int}_{S_B} \star F ,\, \textstyle{\int}_{S_E} F \Big\} \;=\; 0 \,.$$

The thrust of FMS 07a, 07b is the claim that this bracket becomes non-vanishing if torsion-components of the fluxes (through their enhancement to ordinary differential cohomology) are retained, but it seems that this is ultimately by definition (FMS07a, Def. 1.29; FMS07b (3.28)) more than by derivation from first principles.

On the other hand, for non-abelian gauge group, a careful analysis of the (somewhat subtle) Poisson brackets reveals (Cattaneo & Perez 2017) that the electric fluxes – understood as flux densities integrated against Lie algebra valued functions $\alpha$ – have a non-trivial bracket among themselves (Prop. ):

$$\big\{ \Phi^\alpha_E ,\, \Phi^{\alpha'}_E \big\} \;\propto\; \Phi_E^{[\alpha, \alpha']}$$

(where $[-,-]$ is the pointwise Lie bracket). This result has found a lot of attention (only) in the context of first-order formulations of gravity.

Details

Consider a metric Lie algebra $\mathfrak{g}$ – such as $\mathfrak{u}(1)$ or $\mathfrak{su}(n)$, $\mathfrak{so}(n)$ via their Killing forms, or any direct sum of these.

For 3+1 dimensional Yang-Mills theory with gauge Lie algebra $\mathfrak{g}$, we discuss (below) the Poisson bracket of integrals (against Lie algebra-valued functions) of flux densities (curvature 2-forms) over oriented closed surfaces, following Cattaneo & Perez 2017 (where this is discussed for the special case $\mathfrak{g} =$ $\mathfrak{su}(2)$, cf. Ex. ).

The Poisson bracket of electric fluxes among themselves is equation (7) in CP17, and to warm-up we spell out the relevant computation in detail (Prop. below, which also provides a rigorous proof of the abelian case (1)). Then we similarly compute the bracket between electric and magnetic fluxes (Prop. below, which seems not to discussed elsewhere in the literature).

The computations are standard and straightforward, but since, as usual, the results depend crucially on delicate signs, we go into full detail.

Lie theoretic preliminaries

Our ground field is the real numbers.

Consider a Lie algebra $\mathfrak{g}$ with Lie bracket

(2)$$[-,-] \,\colon\, \mathfrak{g} \otimes \mathfrak{g} \longrightarrow \mathbb{R} \,.$$

We aim to mostly write intrinsic expressions, but it is still useful to have the component expression at hand. For that purpose we choose once and for all any linear basis of the underlying vector space of the Lie algebra

$$\mathfrak{g} \;\simeq_{{}_\mathbb{R}}\; \mathbb{R} \big\langle t_1, \cdots, t_{{}_{dim(\mathfrak{g})}} \big\rangle$$

in terms of which the Lie bracket (2) is given by the structure constants $\{f^i{}_{j k}\}$ defined by the following equation (using Einstein summation convention, throughout):

$$[t_j, t_k] \;=\; f^i{}_{j k} \, t_i \,.$$

The following discussion is all about $\mathfrak{g}$-valued differential forms, being elements of the tensor product $\Omega^\bullet_{dR} \otimes \mathfrak{g}$ of the de Rham algebra on a given smooth manifold with the given Lie algebra.

More generally, we should be dealing with differential forms with values in sections of a $\mathfrak{g}$-adjoint bundle $\mathfrak{g}_P$ associated to a principal bundle $P$ (underlying a principal connection modelling a background Yang-Mills gauge potential). Since the adjoint bundle $\mathfrak{g}_P$ is a Lie algebra-fiber bundle, we may think of $\Omega^\bullet_{dR}(-; \mathfrak{g})$ as $\Omega^\bullet(-;\mathfrak{g}_P)$ in all of the following, without otherwise changing much of the discussion (the definition of the flux density $F_A$ picks up one correction term). But since, in the end, we consider only $\mathrm{ad}$-invariant combinations of such $\mathfrak{g}_P$-valued forms (namely the invariant flux densities $\langle \alpha, E\rangle$ and $\langle \beta, F_A\rangle$ from Prop. below) the result of the following discussion is actually independent of the background field $P$, so that we may as well take it to be trivial — as usual in most of the physics literature on the subject.

The primary example are $\mathfrak{g}$-valued 1-forms

(3)$$A \;\equiv\; A^i \otimes t_i \;\in\; \Omega^1_{dR} \otimes \mathfrak{g} \,,$$

which are going to represent the (local) gauge potential of $\mathfrak{g}$-Yang-Mills theory.

In writing (Lie- or Poisson-)brackets of such $\mathfrak{g}$-valued differential forms, we mean to (1.) take the wedge product of the bare differential forms in the given order and (2.) apply the bracket to the coefficients,

(4)$$\omega, \gamma \in \Omega^\bullet_{dR} \otimes \mathfrak{g} \;\;\;\;\;\;\; \vdash \;\;\;\;\;\;\; [ \omega , \gamma ] \,\coloneqq\, \omega^i \wedge \gamma^j \otimes [t_i, t_j] \,.$$

This bracket makes $\mathfrak{g}$-valued forms into a super Lie algebra:

(5)$$\begin{array}{l} \omega \,\in\, \Omega_{dR}^p \otimes \mathfrak{g} \\ \omega' \,\in\, \Omega_{dR}^{p'} \otimes \mathfrak{g} \end{array} \;\;\;\;\;\;\;\;\;\; \vdash \;\;\;\;\;\;\;\;\;\; \big[ \omega , \omega' \big] \;=\; -(-1)^{p p'} \big[ \omega' , \omega \big] \,.$$

For example, the Lie bracket of the gauge potential 1-form (3) with itself is:

(6)$$[ A , A ] \;\equiv\; A^j \wedge A^k \otimes [t_i, t_j] \;=\; f^i{}_{j k} \, A^j \wedge A^k \otimes t_i \,.$$

The Jacobi identity of $\mathfrak{g}$ says that $[t_i,-]$ is a derivation of the Lie bracket:

(7)$$\big[t_i, [t_j, t_k] \big] \;=\; \big[ [t_i, t_j], t_k] \big] \,+\, \big[ t_j, [t_i, t_k] \big] \;\;\;\;\; \Leftrightarrow \;\;\;\;\; f^\bullet{}_{i l} f^{l}{}_{j k} \;=\; f^\bullet{}_{l k} f^l{}_{i j} \,+\, f^\bullet{}_{ j l } f^l{}_{ i k } \,.$$

For any $X \in \Omega^n_{dR} \otimes \mathfrak{g}$ this means for instance, with (6), that

(8)$$\big[A, [A, X]\big] \;=\; \big[ [A,A], X \big] - \big[A, [A, X]\big] \;\;\;\; \text{and hence} \;\;\;\; \big[A, [A, X]\big] \;=\; \tfrac{1}{2} \big[ [A,A], X \big] \,.$$

Given a gauge potential $A$ as above (3), the covariant derivative is the sum of the de Rham differential with the operation $[A,-]$ (4):

(9)$$\mathrm{d}_{A} X \;\equiv\; \mathrm{d} X + [A, X] \;=\; \mathrm{d} X^\bullet + f^\bullet{}_{j k} A^j \wedge X^k \,.$$

Since the de Rham differential $\mathrm{d}$ is a degree=1 graded derivation on the de Rham algebra $\Omega^\bullet_{dR}$, the Jacobi identity (7) implies that the covariant derivative (9) is a degree=1 graded derivation on the Lie bracket (4) of Lie algebra-valued differential forms:

(10)$$\begin{array}{l} \omega \,\in\, \Omega^p_{dR} \otimes \mathfrak{g} \\ \gamma \,\in\, \Omega^\bullet_{dR} \otimes \mathfrak{g} \end{array} \;\;\;\;\;\;\; \vdash \;\;\;\;\;\;\; \mathrm{d}_A [ \omega, \gamma ] \;=\; \big[ \mathrm{d}_A \omega ,\, \gamma \big] + (-1)^{p} \big[ \omega ,\, \mathrm{d}_A \gamma \big] \,.$$

Another key point is that applying the covariant derivative (9) twice is equal to applying the Lie bracket $[F_A, -]$ with the curvature 2-form

$$F_A \;\equiv\; \mathrm{d}A + \tfrac{1}{2}[A, A] \,,$$

representing the magnetic flux density carried by the gauge field $A$, because:

(11)$$\begin{array}{l} \mathrm{d}_A \mathrm{d}_A X \\ \;=\; \mathrm{d}_A \big( \mathrm{d}X + [A,X] \big) \\ \;=\; [A, \mathrm{d}X] + \mathrm{d}[A,X] + \big[ A, [A, X] \big] \\ \;=\; [A, \mathrm{d}X] + [\mathrm{d} A,X] - [A, \mathrm{d} X] + \tfrac{1}{2}\big[ [A, A], X \big] \\ \;=\; \big[ \underset{ \equiv \, F_A }{ \underbrace{ \mathrm{d} A + \tfrac{1}{2}[A, A] } } ,\, X \big] \,, \end{array}$$

where in the third step we used (8).

Similarly, from applying the covariant derivative three times follows the Bianchi identity, $\mathrm{d}_A F_A = 0$:

(12)$$\begin{array}{l} \mathrm{d}_A \mathrm{d}_A \mathrm{d}_A X \\ \;=\; \mathrm{d}_A \big[ F_A ,\, X \big] \\ \;=\; \big[ \mathrm{d}_A F_A ,\, X \big] + \big[ F_A ,\, \mathrm{d}_A X \big] \\ \;=\; \big[ \mathrm{d}_A F_A ,\, X \big] + \mathrm{d}_A \mathrm{d}_A \mathrm{d}_A X \end{array} \;\;\;\;\;\;\;\;\; \Rightarrow \;\;\;\;\;\;\;\;\; \mathrm{d}_A F_A \,=\, 0 \,.$$

Our assumption that $\mathfrak{g}$ is a metric Lie algebra means that it is equipped with an binary invariant polynomial, namely a bilinear map

$$\langle -,-\rangle \;\colon\; \mathfrak{g} \otimes \mathfrak{g} \to \mathbb{R} \,, \;\;\;\;\;\; k_{i j} \;\equiv\; \langle t_i, t_j\rangle$$

which is symmetric

(13)$$\langle t_i, t_j\rangle \;=\; \langle t_j, t_i\rangle$$

and ad-invariant, in that

(14)$$\big\langle [t, -] ,\, - \big\rangle + \big\langle - ,\, [t,-] \big\rangle \;=\; 0 \,.$$

For example:

(15)$$X, Y \,\in\, \Omega^0 \otimes \mathfrak{g} \;\;\;\;\;\;\;\; \vdash \;\;\;\;\;\;\;\; \langle [A, X] ,\, Y \rangle + \langle X ,\, [A , Y ] \rangle \;=\; 0 \,.$$

The induced trilinear pairing

(16)$$\langle -,-,- \rangle \;\coloneqq\; \big\langle -,[-,-] \big\rangle \;\colon\; \mathfrak{g} \otimes \mathfrak{g} \otimes \mathfrak{g} \to \mathbb{R} \,, \;\;\;\;\;\;\; f_{i j k} \;\equiv\; \langle t_i, t_j, t_k\rangle \;=\; k_{i i'} f^{i'}{}_{j k}$$

is also ad-invariant:

(17)$$\begin{array}{l} \big\langle [t,-], -,- \big\rangle + \big\langle -, [t,-], - \big\rangle + \big\langle -, -, [t,-] \big\rangle \\ \;\equiv\; \big\langle [t,-], [-,-] \big\rangle + \Big\langle -, \big[ {[t,-]}, -\big] \Big\rangle + \Big\langle -, \big[-, [t,-]\big] \Big\rangle \\ \;=\; \big\langle [t,-], [-,-] \big\rangle + \Big\langle -, \big[ t, [-, -] \big] \Big\rangle \\ \;=\; 0 \,, \end{array}$$

(where we used first the Jacobi identity (7) and then the invariance (14) of the bilinear pairing)

and is invariant under cyclic permutation of its arguments:

(18)$$\begin{array}{l} \langle t_i, t_j , t_k \rangle \\ \;\equiv\; \phantom{-\;} \big\langle t_i, [t_j , t_k] \big\rangle \\ \;=\; - \big\langle [t_j , t_i ], t_k \big\rangle \\ \;=\; \phantom{-\;} \big\langle [t_i, t_j ], t_k \big\rangle \\ \;=\; \phantom{-\;} \big\langle t_k, [t_i, t_j ] \big\rangle \\ \;\equiv\; \phantom{-\;} \langle t_k, t_i, t_j \rangle \,, \end{array}$$

(where we used ad-invariance (14), and symmetry (13) of the pairing and skew-symmetry of the Lie bracket).

Example

In the case $\mathfrak{g} =$ $\mathfrak{su}(2)$ equipped with its Killing form and with linear basis taken to be the Pauli matrices, we have:

• $f_{i j k} = \epsilon_{i j k}$ the Levi-Civita symbol,

• $k_{i j} = \delta_{i j}$ the Kronecker delta.

This is the case considered in Cattaneo & Perez 2017.

The phase space of Yang-Mills theory

We consider now such Lie algebra-valued differential forms on a given smooth 3-manifold $X$, thought of as a chosen Cauchy surface in a 3+1-dimensional Minkowski spacetime.

On the space of $\mathfrak{g}$-valued 1-forms $A$ (3) and 2-forms $E$, which on any coordinate chart $\mathbb{R}^3 \hookrightarrow X$ we may expand as

$$A \,=\, A_a^i \, \mathrm{d}x^a \otimes t_u \,, \;\;\;\; E \,=\, E_{a b}^i \, \mathrm{d} x^a \wedge \mathrm{d} x^b \otimes t_u \,,$$

we declare the distributional Poisson bracket:

(19)$$\begin{array}{l} \big\{ E^i_{a b}(x), E^j_{c d}(x') \big\} \;=\; 0 \\ \big\{ A^j_a(x), A^j_b(x') \big\} \;=\; 0 \\ \big\{ E^i_{a b}(x), A^j_c(x') \big\} \;\equiv\; k^{ i j } \epsilon_{a b c} \delta(x,x') \end{array}$$

together with the first class constraint that the covariant derivative (9) of $E$ vanishes:

(20)$$\mathrm{d}_A E \;\approx\; 0 \,.$$

This is the phase space of Yang-Mills theory [eg. Friedman & Papastamatiou 1983, §3; Bassetto, Lazzizzera & Soldati 1984, §2] with:

• the canonical coordinate being the gauge potential $A$ in temporal gauge,

• its canonical momentum being the electric field flux density $E$,

• the constraint (20) expressing the Gauss law.

On this phase space, we are concerned with observables expressing electromagnetic flux through closed oriented surface submanifolds (not necessarily connected)

(21)$$S \xhookrightarrow{\phantom{---}} X \,.$$

Taken at face value, the linear such observables are the surface integrals of the flux densities over $S$ against $\mathfrak{g}$-valued “smearing”-functions:

(22)$$\begin{array}{c} \Phi_E^\alpha \;\overset{?}{\equiv}\; \int_S \langle \alpha, E \rangle \,, \\ \Phi_B^\beta \;\overset{?}{\equiv}\; \int_S \langle \alpha, F_A \rangle \,, \end{array} \;\;\;\;\;\;\;\; \text{for} \;\; \alpha, \beta \;\in\; \Omega^0(S) \otimes \mathfrak{g}$$

and more general flux observables ought to be taken to be the polynomials in these linear observables.

However, these expressions (22) need to be corrected (“regularized”) in order to become actual observables, since as given they do not have associated smooth Hamiltonian vector fields. This is the point explained in Cattaneo & Perez 2017: Instead, one needs to consider 3-dimensional “smearing” of the canonical observables in Prop. below.

Using the orientation of $S$ we consider any one-sided tubular neighbourhood $\hat S$ of $S$ inside $X$, extending to the “exterior” of $S$ (a non-compact submanifold with boundary $S$),

(CP17 speak of extending to the “interior” and seem to imagine that $S$ is actually the boundary of a compact sub-manifold – which however is generally not what one wants to assume in discussion of fluxes: Typical choices of $S$ coincide with actual (asymptotic) boundaries of the ambient spacetime, such as around the singular locus of a magnetic monopole that has been removed from spacetime. Our choice of extending to a neighbourhood of $S$ towards the exterior, instead of the interior, works just as well and allows $S$ to actually sit on a boundary of spacetime.)

and extending the coefficient functions to this neighbourhood with compact support (i.e. such that they vanish some finite distance from $S$):

Proposition

(CP17 (6)) Well-defined linear flux-observables equivalent to the naïve observables (22) are of this form:

(23)$$\begin{array}{l} \Phi_E^\alpha \;\equiv\; \int_{\widehat S} \big\langle \mathrm{d}_A \alpha ,\, E \big\rangle \,, \\ \Phi_B^\beta \;\equiv\; \int_{\widehat S} \big\langle \mathrm{d}_A \alpha ,\, F_A \big\rangle \,, \end{array} \;\;\;\; \text{for} \;\; \alpha, \beta \;\in\; \Omega^0\big(\widehat S \big)_{cpt} \otimes \mathfrak{g} \,.$$

Proof

To check that these are indeed equivalent to the naïve observables, in that the difference is proportional to the left hand side of the Gauss law (20), so that they coincide on the locus where the Gauss law holds:

$$\begin{array}{l} \int_{\widehat{S}} \big\langle \mathrm{d}_A \alpha ,\, E \big\rangle \\ \;\approx\; \int_{\widehat{S}} \Big( \big\langle \mathrm{d}_A \alpha ,\, E \big\rangle + \underset{ \approx 0 }{ \underbrace{ \big\langle \alpha ,\, \mathrm{d}_A E \big\rangle } } \Big) \\ \;=\; \int_{\widehat{S}} \Big( \big\langle \mathrm{d} \alpha ,\, E \big\rangle \,+\, \big\langle \alpha ,\, \mathrm{d} E \big\rangle \Big) \\ \;=\; \int_{\widehat{S}} \mathrm{d} \big\langle \alpha ,\, E \big\rangle \\ \;=\; \int_{S} \big\langle \alpha ,\, E \big\rangle \mathrlap{\,,} \end{array}$$

where we used:

1. the Gauss law

2. (15)

3. derivation property of $\mathrm{d}$

4. Stokes

In the computation below, of the Poisson brackets of these flux observables, we repeatedly need the following identities, which also serve as good examples for how to compute with the Poisson brackets (19):

Example

A key consequence of the corrected flux observables (23), is a non-trivial Poisson bracket between electric field observables and plain smearing functions:

(24)$$\begin{array}{l} \Big\{ \textstyle{\int} \big(\mathrm{d}_A \alpha_i\big) E^i ,\, \mathrm{d}_A \beta \Big\} \\ \;\equiv\; \Big\{ \textstyle{\int} \big(\mathrm{d}_A \alpha_i\big) E^i ,\, \mathrm{d}\beta + [A, \beta] \Big\} \\ \;=\; \textstyle{\int} \big(\mathrm{d}_A \alpha_i\big) \Big[ \big\{ E^i ,\, A \big\} ,\, \beta \Big] \\ \;=\; \big(\mathrm{d}_A \alpha_i\big) \big[ t^i ,\, \beta \big] \\ \;=\; \Big[ \big(\mathrm{d}_A \alpha\big) ,\, \beta \Big] \,. \end{array}$$

This is the relation needed for the computation of the bracket among electric fluxes in Prop. below.

Example

The Poisson bracket of an electric flux observable with a magnetic flux density results in the covariant derivative of the electric smearing function:

(25)$$\begin{array}{l} \Big\{ \textstyle{\int} \big(\mathrm{d}_A \alpha_i\big) \wedge E^i ,\, F_A(x') \Big\} \;=\; d_A \big(\mathrm{d}_A \alpha\big)(x') \end{array}$$

This is the main relation needed in the computation of the bracket between electric and magnetic fluxes, in Prop. below.

Proof

For the first summand in $F_A \,\equiv\, \mathrm{d}A + \tfrac{1}{2}[A, A]$ we have simply:

$$\begin{array}{l} \Big\{ \textstyle{\int} \big(\mathrm{d}_A \alpha_i \big) \wedge E^i ,\, \mathrm{d} A(x') \Big\} \\ \;=\; \mathrm{d} \Big\{ \textstyle{\int} \big(\mathrm{d}_A \alpha_i \big) \wedge E^i ,\, A(x') \Big\} \\ \;=\; \mathrm{d} \big(\mathrm{d}_A \alpha(x') \big) \,. \end{array}$$

For the second summand, considering any $\omega \in \Omega^1_{dR} \otimes \mathfrak{g}$, we have:

$$\begin{array}{l} \Big\{ \textstyle{\int} \omega_i E^i ,\, \tfrac{1}{2} [A, A] \Big\} \\ \;=\; \tfrac{1}{2} \Big[ \big\{ \textstyle{\int} \omega_i E^i ,\, A \big\} ,\, A \Big] + \tfrac{1}{2} \Big[ A ,\, \big\{ \textstyle{\int} \omega_i E^i ,\, A \big\} \Big] \\ \;=\; \big[ A ,\, \omega \big] \,. \end{array}$$

Together this yields the claim.

It may be instructive to state the last computation with more components made explicit:

(combinatorial prefactors currently not shown properly, but they don’t affect the sign, of course)

$$\begin{array}{l} \Big\{ \textstyle{\int} \omega^i E_i ,\, \tfrac{1}{2}[A, A] \Big\} \\ \;=\; \textstyle{\int} \omega_{i a} \epsilon^{a b c} \tfrac{1}{2} \big\{ E^i_{b c} ,\, A^j_{b'} A^k_{c'} \big\} [t_j, t_k] \mathrm{d} x^{b'} \wedge \mathrm{d} x^{c'} \\ \;=\; \textstyle{\int} \tfrac{1}{2} \omega^i_a \epsilon^{a b c} \big( k^{i j} \epsilon_{b c b'} A^k_{c'} + A^j_{b'} k^{i k} \epsilon_{b c c'} \big) [t_j, t_k] \mathrm{d} x^{b'} \wedge \mathrm{d} x^{c'} \\ \;=\; \tfrac{1}{2} \big( A^k_{c'} \omega^j_{b'} + A^j_{b'} \omega^k_{c'} \big) [t_j, t_k] \mathrm{d} x^{b'} \wedge \mathrm{d} x^{c'} \\ \;=\; A^j_{b'} \omega^k_{c'} [t_j, t_k] \mathrm{d} x^{b'} \wedge \mathrm{d} x^{c'} \\ \;=\; [A, \omega] \,. \end{array}$$

The sub-phase space of integrated fluxes

Theorem

The reduced sub-phase space of integral fluxes through a given surface $S \hookrightarrow X$ (21) in $\mathfrak{g}$-Yang-Mills theory is isomorphic to the Fréchet Lie-Poisson manifold

(26)$$C^\infty\bigg( S ,\, \Big( \underset{el}{ \underbrace{ \big( \mathfrak{g} ,\, [-,-] \big) } } \ltimes_{ad} \underset{mag}{ \underbrace{ \mathfrak{g} } } \Big)^\ast \bigg)$$

given by the Lie algebra of smooth maps (with pointwise Lie bracket) from $S$ into the linear dual of the semidirect product Lie algebra of $\mathfrak{g}$ with its underlying abelian Lie algebra via the adjoint action.

Proof

That the underlying Fréchet manifold is as claimed is just a restatement of the form of the linear flux observables in Prop. . We need to check that the Poisson brackets (19) of these linear observables is equivalently the Lie bracket of their smearing functions $\alpha_{el}$, $\alpha_{mag}$, regarded as elements of (26). This is the content of Prop. and Prop. below.

Proposition

(CP17 (7))
The Poisson bracket (19) of electric flux observables (23) among each other is the Lie bracket on their smearing functions:

$$\Big\{ \Phi_E^{\alpha} ,\, \Phi_E^{\beta} \Big\} \;=\; \Phi_E^{ [\alpha,\beta] } \,.$$

Proof

We compute as follows:

(27)$$\begin{array}{l} \Big\{ \textstyle{\int} \big( \mathrm{d}_A \alpha_i \big) E^i ,\, \textstyle{\int} \big( \mathrm{d}_A \beta_j \big) E^j \Big\} \\ \;=\; \textstyle{\int} \Big[ \big(\mathrm{d}_A \alpha\big) ,\, \beta \Big]_i E^i - \textstyle{\int} \Big[ \big(\mathrm{d}_A \beta\big) ,\, \alpha \Big]_i E^i \\ \;=\; \textstyle{\int} \Big[ \big(\mathrm{d}_A \alpha\big) ,\, \beta \Big]_i E^i + \textstyle{\int} \Big[ \alpha ,\, \big(\mathrm{d}_A \beta\big) \Big]_i E^i \\ \;=\; \textstyle{\int} \big( \mathrm{d}_A [\alpha,\beta]_i \big) E^i \mathrlap{\,,} \end{array}$$

where we used, in order of appearance:

1. (24)

2. (5)

3. (10).

Proposition

The Poisson bracket (19) of an electric flux observable with a magnetic flux observable (23) is the magnetic flux observable smeared by the Lie bracket of the given smearing functions:

$$\Big\{ \Phi_E^\alpha ,\, \Phi_B^\beta \Big\} \;=\; \Phi_B^{[\alpha, \beta]} \,.$$

Proof

We compute as follows:

(28)$$\begin{array}{l} \Big\{ \textstyle{\int} \big( \mathrm{d}_A \alpha_i \big) \wedge E^i ,\, \textstyle{\int} \big( \mathrm{d}_A \beta_j \big) \wedge F_A^j \Big\} \\ \;=\; \textstyle{\int} \Big\langle \big[ \mathrm{d}_A \alpha ,\, \beta_j \big] ,\, F_A^j \Big\rangle + \textstyle{\int} \Big\langle \big( \mathrm{d}_A \beta \big) ,\, \mathrm{d}_A \big( \mathrm{d}_A \alpha \big) \Big\rangle \\ \;=\; \textstyle{\int} \big\langle \mathrm{d}_A \alpha ,\, \beta ,\, F_A \big\rangle + \textstyle{\int} \big\langle \mathrm{d}_A \beta ,\, F_A ,\, \alpha \big\rangle \\ \;=\; \textstyle{\int} \big\langle \mathrm{d}_A \alpha ,\, \beta ,\, F_A \big\rangle + \textstyle{\int} \big\langle \alpha ,\, \mathrm{d}_A \beta ,\, F_A \big\rangle \\ \;=\; \textstyle{\int} \big\langle \mathrm{d}_A [\alpha, \beta] ,\, F_A \big\rangle \,. \end{array}$$

Here we used, in order of appearance:

1. (24) and (25);

2. (18) and (11)

3. (18)

4. (10).

The algebra of quantum observables on fluxes

By Thm. , the quantum observables on fluxes in Yang-Mills theory should constitute a quantization of the Lie-Poisson phase space (26).

The strict (non-perturbative) deformation quantization of phase spaces which are Lie-Poisson manifolds of a Lie algebra is well-known [Rieffel 1990; Landsman 1999, Ex. 2; Landsman & Ramazan 2001, Ex. 11.1 ]: The $C^\ast$-algebra of quantum observables is a group algebra of a corresponding Lie group (i.e. with the convolution product), with due care about analytic issue.

Hence consider $G$ a Lie group (not necessarily connected) with Lie algebra $\big(\mathfrak{g}, [-,-]\big)$ which serves as the actual gauge group of our Yang-Mills theory.

Given a linear representation of $G$ on $\mathfrak{g}$ which on the connected component $G_{\mathrm{e}} \subset G$ of the neutral element restricts to the adjoint action and which preserves a lattice $\Lambda \subset \mathfrak{g}$ (possibly empty, generally not maximal-dimensional), we may take the Lie integration of the underlying abelian Lie algebra of $\mathfrak{g}$ to be the torus $\mathfrak{g}/\Lambda$ equipped with a group action by $G$.

With these choices, the semidirect product group

$$G \,\ltimes_{ad}\, (\mathfrak{g}/\Lambda)$$

exists and has as Lie algebra the semidirect product Lie algebra defining the Lie-Poisson structure (26).

Therefore a non-perturvative algebra of quantum observables is given by the $C^\ast$-group algebra, suitably defined, of the group of maps $C^\infty\big(S, G \,\ltimes_{ad}\, (\mathfrak{g}/\Lambda)\big)$.

To isolate the topological sector of fluxes, consider the connected components of this group given by the homotopy classes of maps

$$\pi_0 \Big( C^\infty\big( S ,\, G \,\ltimes_{ad}\, (\mathfrak{g}/\Lambda) \big) \Big) \;\simeq\; \pi_0 \bigg( Map\Big( \esh S ,\, \esh \big( G \,\ltimes_{ad}\, (\mathfrak{g}/\Lambda) \big) \Big) \bigg)$$

This is now a discrete group. Its group algebra serves as an algebra of observables on the topological sectors of fluxes.

(…)

Related concepts

• flux

  • flux tube

  • flux quantization

  • flux compactification

References

In electromagnetism, with focus on torsion components that are argued to not generally commute:

• Daniel Freed, Greg Moore, Graeme Segal, The Uncertainty of Fluxes, Commun. Math. Phys. 271 (2007) 247-274 [arXiv:hep-th/0605198, doi:10.1007/s00220-006-0181-3]

• Daniel Freed, Greg Moore, Graeme Segal, Heisenberg Groups and Noncommutative Fluxes, Annals Phys. 322 (2007) 236-285 [arXiv:hep-th/0605200, doi:10.1016/j.aop.2006.07.014]

In SU(2)-gauge theory (highlighted for the case of the first-order formulation of gravity but applying verbatim also to Yang-Mills theory, cf. there):

• Alberto S. Cattaneo, Alejandro Perez, A note on the Poisson bracket of 2d smeared fluxes, Class. Quant. Grav. 34 (2017) 107001 [arXiv:1611.08394, doi:10.1088/1361-6382/aa69b4]

Proposal of experiments potentially measuring Heisenberg uncertainty relations of fluxes:

• Alexei Kitaev, Gregory W. Moore, Kevin Walker, Noncommuting Flux Sectors in a Tabletop Experiment [arXiv:0706.3410]

See also:

• Mikhail A. Savrov, Commutator of Electric Charge and Magnetic Flux [arXiv:2003.02225]