It can be shown that the total angular momentum of a free electromagnetic field is given by (for example, in the book A Modern Introduction to quantum field theory by Maggiore, page 98, Eq. (4.82)) $$J^{jk}=L^{jk}+S^{jk}\tag{1}$$ where $$L^{jk}=\int d^3x[\partial_0A^i(x^j\partial^k-x^k\partial^j)A^i]\tag{2}$$ represents the components of the orbital angular momentum and $$S^{jk}=\int d^3x[A^j\partial_0A^k-A^k\partial_0A^j]\tag{3}$$ represents the components of the spin angular momentum.

The expression (2), with normal ordering, has been expressed in terms of creation and annihilation operators (in Eq. 4.84, of the same reference) as $$S^{ij}=i\int \frac{d^3q}{(2\pi)^3}\sum\limits_{\lambda,\lambda^\prime}\Big[\epsilon^i(\textbf{q},\lambda^\prime)\epsilon^{j*}(\textbf{q},\lambda)-\epsilon^{i*}(\textbf{q},\lambda)\epsilon^{j}(\textbf{q},\lambda^\prime)\Big]a^\dagger_{\textbf{q},\lambda}a_{\textbf{q},\lambda^\prime}\tag{4}$$ which is then used to act on one-particle photon states to determine the helicity or spin projection along the direction of propagation.

However, nothing is said about $L^{ij}$ i.e., whether it can be expressed in terms of creation and annihilation operators. This is important since a quantized expression for $L^{ij}$ can be similarly operated on one-particle photon states, to determine whether it has any non-vanishing orbital angular momentum unambiguously.

Question : Can someone give a reference which expresses $L^{jk}$ in terms of the creation and annihilation operators? The presence of $x^j$ and $\partial^k$ in it makes the process of quantizing $L^{jk}$ non-trivial compared to that of $S^{jk}$. Therefore, any hint on how to obtain $L^{jk}$, especially what to substitute for $x^j$ and $\partial^k$, will also be helpful. The ultimate motive is to act it on one-photon states and draw some conclusion.


1 Answer 1


Let me first give you 3 references: Van Enk and NieNhuis , Leader , and, Santamato.

The problem with the decomposition given in the question of the photon angular momentum into spin and orbital angular momenta is that this decomposition is not gauge invariant, thus do not correspond to physical observables.

A remedy for the situation was proposed by Van Enk and NieNhuis (first reference) by restricting the fields in the decomposition to be transversal. The expressions (of both spin and orbital angular momenta) after this restriction become gauge invariant. Their components are actually being measured in quantum optics experiments. The expressions of the spin and orbital angular momenta (not their densities !) in terms of creation and anihilation operators are given by Van Enk and NieNhuis equations (8), for arbitrarily polarized waves. Van Enk and NieNhuis checked that these operators do not satisfy the canonical commutaion relations (only their sum does). This is the price that we pay by the restriction to the physical transverse components. Thus these observables are not really angular momenta although they are legitimate observables.

Satamato (the third reference) gives an expression of angular momenta after the restriction in term of the fields only. This is possible since these expressions are gauge invariant.

Leader (the second reference) summrises this subject.

The fact that the free electromagnetic field has many observables which are not equivalent to linear or angular momenta should not come as a surprise because it is an infinite dimensional integrable system.


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