Self-Adjoint Phase Operator

• Self-Adjoint Phase Operator is a Hermitian quantum observable that represents the classical phase variable through well-defined spectral decompositions and extended Hilbert-space constructions.
• It overcomes the one-sided number spectrum obstruction by employing quadrature formulations, Hilbert-space doubling, and alternative operator factorizations.
• The various constructions provide practical insights into phase dynamics in harmonic oscillators and photon fields, recovering classical trigonometric behavior in the large-amplitude limit.

A self-adjoint phase operator is a Hermitian quantum observable intended to represent the classical phase variable while retaining a bona fide spectral decomposition, well-defined eigenstates, and consistent time dependence. The topic arises most sharply for the harmonic oscillator and the single-mode electromagnetic field, where the lower-bounded number spectrum obstructs a naive canonical conjugate of the number operator. Modern constructions replace the direct polar decomposition of the annihilation operator by alternative operator factorizations, Hilbert-space extensions, generalized quantization on the cylinder, or regularized phase exponentials; across these approaches, the central aim is to obtain a phase observable that is self-adjoint, possesses a continuous or discrete spectral resolution as appropriate, and reproduces classical phase behavior in the relevant limit (Ma et al., 2015).

1. Foundational obstruction and the operator-theoretic problem

The central difficulty is the incompatibility between a one-sided number spectrum and an exact canonical phase conjugate defined on the same Hilbert space. In Dirac’s original attempt, the vacuum state is a boundary eigenstate of the number operator, so the would-be ladder operator $e^{i\hat\phi}$ annihilates vacuum and fails to be unitary. This is the source of the long-standing “lower-bound” and “vacuum-annihilation” obstacles in defining quantum optical phase (Prajapati et al., 2011).

Several later formulations make the obstruction explicit in different mathematical languages. In the cylindrical formulation on $L^2(S^1)$, the angle operator $\hat\Theta$ is self-adjoint and bounded, but after projection to the photon-number Hilbert space one does not obtain the exact canonical commutation relation. Instead,

$[\,N̂,\,Φ̂\,]\neq i\,\mathbb 1,$

and the commutator contains boundary or projector corrections reflecting the compactness of $S^1$ and the lower bound of the number spectrum (Przanowski et al., 2013). This is entirely consistent with the statement that exact canonical conjugacy is obstructed by the lower bound of the number spectrum.

The nonunitarity of the Susskind–Glogower operator provides the canonical example of the obstruction. In the ordinary Fock space,

$$E=\sum_{n=0}^{\infty}\lvert n\rangle\langle n+1\rvert,\qquad E\,E^\dagger=\hat1,\quad E^\dagger E=\hat1-\lvert0\rangle\langle0\rvert,$$

so $E$ is not unitary. Any phase operator obtained as a logarithm of such an operator is therefore problematic as a self-adjoint observable in the strict spectral sense (Varro, 2014).

A recurring theme in the literature is that self-adjointness can be restored either by enlarging the Hilbert space so that the number-like spectrum becomes two-sided, by constructing a different operator whose logarithm is well defined, or by representing phase first as an angle variable on $S^1$ and only then compressing to oscillator or photon-number sectors. This suggests that the self-adjoint phase problem is less a single construction than a family of operator-theoretic resolutions adapted to different physical models.

2. Harmonic-oscillator phase from quadratures

A direct construction for the harmonic oscillator was given by Ma and Rhodes, who define phase through the quadratures rather than by a direct polar decomposition of $\hat a$. Classically,

$$\cos^2\phi=\frac{q^2}{q^2+p^2},\qquad \sin^2\phi=\frac{p^2}{q^2+p^2},$$

and quantum mechanically they set

$L^2(S^1)$0

$L^2(S^1)$1

with $L^2(S^1)$2, $L^2(S^1)$3, and $L^2(S^1)$4 in units $L^2(S^1)$5 (Ma et al., 2015).

Because $L^2(S^1)$6 and $L^2(S^1)$7 are self-adjoint and $L^2(S^1)$8 is a positive self-adjoint operator with domain including all Fock states, $L^2(S^1)$9 and $\hat\Theta$0 are Hermitian and positive, and they satisfy

$\hat\Theta$1

The associated doubled-angle operator is

$\hat\Theta$2

The construction is explicitly designed so that the operator identities mirror the classical trigonometric relations (Ma et al., 2015).

In the Fock basis, the action of the dressed ladder operators introduces the ratio

$\hat\Theta$3

This yields tridiagonal matrix elements for $\hat\Theta$4,

$\hat\Theta$5

with all other off-diagonal elements zero and diagonal elements vanishing. These matrix elements fully characterize the operator in the number basis (Ma et al., 2015).

The same work emphasizes that the resulting $\hat\Theta$6 is Hermitian on the full infinite-dimensional Hilbert space. Its operator algebra is therefore not introduced through a truncated space or an auxiliary POV framework alone, but by an explicit self-adjoint operator whose trigonometric functions satisfy the expected identities.

3. Spectrum, eigenfunctions, and phase states

The spectral analysis of the Ma–Rhodes operator proceeds through the eigenvalue problem

$\hat\Theta$7

Expanding $\hat\Theta$8 produces a three-term recurrence relation. Because parity is conserved, the Hilbert space separates into even and odd sectors, and the coefficients are expressed through Gegenbauer ultraspherical polynomials: $\hat\Theta$9 The orthogonality relation

$[\,N̂,\,Φ̂\,]\neq i\,\mathbb 1,$0

determines the normalization of the continuous eigenfamilies (Ma et al., 2015).

Two orthonormal continuous families $[\,N̂,\,Φ̂\,]\neq i\,\mathbb 1,$1 and $[\,N̂,\,Φ̂\,]\neq i\,\mathbb 1,$2 are then combined into phase eigenstates $[\,N̂,\,Φ̂\,]\neq i\,\mathbb 1,$3 by setting $[\,N̂,\,Φ̂\,]\neq i\,\mathbb 1,$4, with $[\,N̂,\,Φ̂\,]\neq i\,\mathbb 1,$5. In this representation,

$[\,N̂,\,Φ̂\,]\neq i\,\mathbb 1,$6

Accordingly, the spectrum of $[\,N̂,\,Φ̂\,]\neq i\,\mathbb 1,$7 is continuous on $[\,N̂,\,Φ̂\,]\neq i\,\mathbb 1,$8 (Ma et al., 2015).

A structurally different spectral construction appears in the regular phase operator of Varró. There one introduces, for $[\,N̂,\,Φ̂\,]\neq i\,\mathbb 1,$9,

$S^1$0

equivalently

$S^1$1

The phase operator is defined as the strongly convergent series

$S^1$2

on the dense domain

$S^1$3

This series is also expressible through logarithms of $S^1$4 and $S^1$5, again on $S^1$6 (Varro, 2014).

The same construction admits a generalized spectral decomposition

$S^1$7

with $S^1$8 and

$S^1$9

The eigenstates of $$E=\sum_{n=0}^{\infty}\lvert n\rangle\langle n+1\rvert,\qquad E\,E^\dagger=\hat1,\quad E^\dagger E=\hat1-\lvert0\rangle\langle0\rvert,$$0 are normalized SU(1,1) coherent states in the Holstein–Primakoff realization,

$$E=\sum_{n=0}^{\infty}\lvert n\rangle\langle n+1\rvert,\qquad E\,E^\dagger=\hat1,\quad E^\dagger E=\hat1-\lvert0\rangle\langle0\rvert,$$1

which diagonalize $$E=\sum_{n=0}^{\infty}\lvert n\rangle\langle n+1\rvert,\qquad E\,E^\dagger=\hat1,\quad E^\dagger E=\hat1-\lvert0\rangle\langle0\rvert,$$2 and support a diagonal representation of the phase operator (Varro, 2014).

These two lines of work illustrate distinct spectral strategies. One produces a continuous spectrum through explicit oscillator eigenfunctions in the number basis; the other regularizes the exponential phase operator so that the phase operator exists as a strongly convergent series with a covariant positive-operator-valued measure. A plausible implication is that self-adjointness alone does not uniquely determine the preferred phase observable; the operator model and the chosen functional calculus remain decisive.

4. Photon phase via Hilbert-space extension and two-mode structure

A particularly direct route to self-adjointness is to enlarge the photon Hilbert space so that the number spectrum becomes two-sided. Prajapati and Ranganathan take

$$E=\sum_{n=0}^{\infty}\lvert n\rangle\langle n+1\rvert,\qquad E\,E^\dagger=\hat1,\quad E^\dagger E=\hat1-\lvert0\rangle\langle0\rvert,$$3

with $$E=\sum_{n=0}^{\infty}\lvert n\rangle\langle n+1\rvert,\qquad E\,E^\dagger=\hat1,\quad E^\dagger E=\hat1-\lvert0\rangle\langle0\rvert,$$4, identify right-circular states with $$E=\sum_{n=0}^{\infty}\lvert n\rangle\langle n+1\rvert,\qquad E\,E^\dagger=\hat1,\quad E^\dagger E=\hat1-\lvert0\rangle\langle0\rvert,$$5 and left-circular states with $$E=\sum_{n=0}^{\infty}\lvert n\rangle\langle n+1\rvert,\qquad E\,E^\dagger=\hat1,\quad E^\dagger E=\hat1-\lvert0\rangle\langle0\rvert,$$6, and thereby construct a single integer-labeled basis $$E=\sum_{n=0}^{\infty}\lvert n\rangle\langle n+1\rvert,\qquad E\,E^\dagger=\hat1,\quad E^\dagger E=\hat1-\lvert0\rangle\langle0\rvert,$$7. The total photon-number operator is then

$$E=\sum_{n=0}^{\infty}\lvert n\rangle\langle n+1\rvert,\qquad E\,E^\dagger=\hat1,\quad E^\dagger E=\hat1-\lvert0\rangle\langle0\rvert,$$8

Because the eigenvalues run over all integers, every basis state has both an upward and a downward neighbor (Prajapati et al., 2011).

On this extended space, the Susskind–Glogower exponential operator is defined to act as a perfect bilateral shift,

$$E=\sum_{n=0}^{\infty}\lvert n\rangle\langle n+1\rvert,\qquad E\,E^\dagger=\hat1,\quad E^\dagger E=\hat1-\lvert0\rangle\langle0\rvert,$$9

and one checks

$E$0

The phase operator is then introduced through

$E$1

so $E$2 is Hermitian because $E$3 is unitary. The commutation relations become

$E$4

The eigenstates $E$5 of $E$6, with $E$7, are complete and delta-normalized, and $E$8 (Prajapati et al., 2011).

A related but physically different two-mode construction was given for the electromagnetic vector potential. Starting from the $E$9 modes, one introduces cosine and sine combinations

$S^1$0

so that the free Hamiltonian becomes a two-dimensional harmonic oscillator. In the associated oscillator plane one defines

$S^1$1

and then the phase-factor operator

$S^1$2

Its unitarity follows from the positivity of the intermediate operator $S^1$3, and the phase operator $S^1$4 is self-adjoint with spectrum $S^1$5 (Damgaard, 2017).

When this two-mode operator is compressed to the forward-only subspace, one obtains

$S^1$6

with $S^1$7 and $S^1$8 as $S^1$9. This operator is not unitary, so one cannot define $\hat a$0 as a self-adjoint operator. The failure is concentrated at low occupation number and reproduces the one-sided Fock-space obstruction in explicit operator form (Damgaard, 2017).

These photon constructions agree on a common point: self-adjointness becomes available when the phase factor is realized as a genuine unitary shift on a two-sided or effectively doubled Hilbert space. They differ, however, in what plays the role of the conjugate quantity: the integer-labeled action operator $\hat a$1 in the polarization-doubled model, and the mode-difference or angular-momentum operator in the $\hat a$2 oscillator-plane construction.

5. Weyl quantization on the cylinder and group-theoretical formulations

A mathematically distinct route begins from the cylindrical phase space $\hat a$3. In generalized Weyl quantization, the Hilbert space is

$\hat a$4

with

$\hat a$5

Using a generalized Stratonovich–Weyl quantizer $\hat a$6, the classical phase symbol $\hat a$7 is mapped to

$\hat a$8

For any admissible real kernel $\hat a$9 satisfying $$\cos^2\phi=\frac{q^2}{q^2+p^2},\qquad \sin^2\phi=\frac{p^2}{q^2+p^2},$$0, the resulting bounded operator is kernel-independent and has matrix elements

$$\cos^2\phi=\frac{q^2}{q^2+p^2},\qquad \sin^2\phi=\frac{p^2}{q^2+p^2},$$1

in the $$\cos^2\phi=\frac{q^2}{q^2+p^2},\qquad \sin^2\phi=\frac{p^2}{q^2+p^2},$$2-eigenbasis (Przanowski et al., 2013).

In the angle representation, $$\cos^2\phi=\frac{q^2}{q^2+p^2},\qquad \sin^2\phi=\frac{p^2}{q^2+p^2},$$3 acts simply by multiplication,

$$\cos^2\phi=\frac{q^2}{q^2+p^2},\qquad \sin^2\phi=\frac{p^2}{q^2+p^2},$$4

with spectral resolution

$$\cos^2\phi=\frac{q^2}{q^2+p^2},\qquad \sin^2\phi=\frac{p^2}{q^2+p^2},$$5

Because it is bounded and symmetric, $$\cos^2\phi=\frac{q^2}{q^2+p^2},\qquad \sin^2\phi=\frac{p^2}{q^2+p^2},$$6 is self-adjoint on all of $$\cos^2\phi=\frac{q^2}{q^2+p^2},\qquad \sin^2\phi=\frac{p^2}{q^2+p^2},$$7. Its compression

$$\cos^2\phi=\frac{q^2}{q^2+p^2},\qquad \sin^2\phi=\frac{p^2}{q^2+p^2},$$8

to the nonnegative-mode subspace $$\cos^2\phi=\frac{q^2}{q^2+p^2},\qquad \sin^2\phi=\frac{p^2}{q^2+p^2},$$9 remains bounded and self-adjoint, but its commutator with number is modified rather than canonical (Przanowski et al., 2013).

The group-theoretical approach organizes phase operators through representations of $L^2(S^1)$00, the generalized oscillator algebra with

$L^2(S^1)$01

For $L^2(S^1)$02, one obtains a finite-dimensional SU(2) representation; for $L^2(S^1)$03, an infinite-dimensional SU(1,1) representation. In the compact SU(2) case, one can write

$L^2(S^1)$04

where the operator $L^2(S^1)$05 is unitary. Since $L^2(S^1)$06, its spectrum is discrete,

$L^2(S^1)$07

The Hermitian phase operator is then

$L^2(S^1)$08

with an orthonormal phase basis and exact closure relation (Atakishiyev et al., 2010).

In the SU(2) setting one finds from $L^2(S^1)$09 that

$L^2(S^1)$10

so $L^2(S^1)$11 plays the role of a canonical phase conjugate to $L^2(S^1)$12, up to the “bracket-of-truncation subtleties” that disappear as $L^2(S^1)$13. By contrast, in the noncompact SU(1,1) case the analogous operator $L^2(S^1)$14 is only a partial isometry, not unitary, even though its continuous-angle eigenstates admit a closure relation. Thus the compact case yields a self-adjoint phase operator directly, while the noncompact case reproduces the familiar infinite-dimensional obstruction without additional regularization (Atakishiyev et al., 2010).

6. Dynamics, classical limit, and comparative assessment

The dynamical behavior of a self-adjoint phase operator is a key criterion for whether it mirrors the classical phase variable. For the Ma–Rhodes oscillator operator under

$L^2(S^1)$15

the phase states evolve as

$L^2(S^1)$16

so the instantaneous phase distribution shifts rigidly by $L^2(S^1)$17. At $L^2(S^1)$18,

$L^2(S^1)$19

which mirrors the classical half-period inversion (Ma et al., 2015).

The same construction reproduces the classical limit in coherent states. For $L^2(S^1)$20 with $L^2(S^1)$21,

$L^2(S^1)$22

and

$L^2(S^1)$23

Thus the Poisson-bracket relations are recovered in the large-amplitude limit (Ma et al., 2015).

The regular operator of Varró addresses a different criterion: convergence and domain control. The original Susskind–Glogower operator yields only weak convergence in naive phase series, and the Garrison–Wong phase operator likewise converges only weakly; by dressing $L^2(S^1)$24 with $L^2(S^1)$25, the resulting $L^2(S^1)$26-based phase series converges strongly on the natural domain $L^2(S^1)$27. No truncation is required, unlike the Pegg–Barnett approach, and the construction comes equipped with a covariant positive-operator-valued measure $L^2(S^1)$28 (Varro, 2014).

The cylindrical and doubled-space constructions clarify a common misconception. A self-adjoint phase operator does not necessarily imply an exact canonical commutator with the standard one-sided number operator. In $L^2(S^1)$29, $L^2(S^1)$30 is self-adjoint while the compressed oscillator phase has a modified commutator (Przanowski et al., 2013). In the doubled photon space, exact canonical commutation is recovered only because the number spectrum is extended to all integers (Prajapati et al., 2011). In the $L^2(S^1)$31 vector-potential construction, the most natural canonical conjugate is the mode-difference operator or angular momentum $L^2(S^1)$32, not necessarily the total photon number (Damgaard, 2017).

The following summary captures the main operator strategies discussed in the literature.

Construction	Hilbert-space setting	Defining feature
Ma–Rhodes (Ma et al., 2015)	Infinite-dimensional oscillator Fock space	Phase from dressed quadratures $L^2(S^1)$33
Prajapati–Ranganathan (Prajapati et al., 2011)	Polarization-doubled photon space with $L^2(S^1)$34	Unitary bilateral shift $L^2(S^1)$35
Generalized Weyl (Przanowski et al., 2013)	$L^2(S^1)$36, then compression to $L^2(S^1)$37	Angle operator on the cylinder
Vector-potential correction (Damgaard, 2017)	Two-mode $L^2(S^1)$38 oscillator plane	Unitary $L^2(S^1)$39 from $L^2(S^1)$40
Varró regular operator (Varro, 2014)	Harmonic-oscillator Fock space	Strongly convergent $L^2(S^1)$41-series and POV measure
SU(2) approach (Atakishiyev et al., 2010)	Finite-dimensional compact representation	Unitary phase operator from polar decomposition

Taken together, these results show that the phrase “self-adjoint phase operator” does not designate a single universal object. It denotes a class of mathematically rigorous phase observables whose precise form depends on whether one prioritizes exact unitarity of the phase exponential, direct operator realization in Fock space, quantization on $L^2(S^1)$42, strong convergence of phase series, or group-theoretical compactification. The common achievement is the replacement of the naive nonunitary phase factor by constructions whose spectral properties, algebraic identities, and in several cases dynamical behavior are controlled at the operator level.