Hilbert Positivity and Unitary Evolution

A positive physical inner product and self-adjoint dynamics answer two separate questions. After any required null quotient, positivity makes norm squares and Born probabilities nonnegative and excludes nonzero null vectors from the physical Hilbert space. Self-adjointness makes the Hamiltonian generate a unitary one-parameter group, so inner products and total probability are preserved in a closed system. Gauge-fixed covariant descriptions may violate positivity on an auxiliary space; the probability interpretation is recovered only after the physical condition, null quotient or cohomology, and observable restriction are specified. This page develops that distinction for a regulated free scalar and the covariantly quantized free photon, without deriving scattering cuts, Euclidean reconstruction, or interacting operator-domain theorems.

Required background. Vacua, States, and Representations supplies the distinction among an algebra, a state, and a Hilbert-space representation. Canonical Quantization: Algebra, Representation, and State supplies the canonical commutators and free-scalar Fock representation used below.

Helpful background. Bilinear and Hermitian Forms, Adjoints, and Isometries supplies signatures, adjoints, and null subspaces. Self-Adjointness, Extensions, and Unitary Evolution supplies the domain-sensitive form of Stone’s theorem.

Physical positivity supports a probability interpretation

On a physical Hilbert space $\mathcal H_{\mathrm{phys}}$, positivity means

$$\langle\psi|\psi\rangle\ge0, \qquad \langle\psi|\psi\rangle=0 \Longleftrightarrow |\psi\rangle=0.$$

Positive definiteness makes $\|\cdot\|$ a genuine norm, which induces the distance $d(\chi,\psi)=\|\chi-\psi\|$; the inner product satisfies the Cauchy–Schwarz inequality. Completion is the additional step that makes the space Hilbert. For normalized vectors, a rank-one transition probability obeys

$$0\le |\langle\chi|\psi\rangle|^2\le1.$$

For a density operator, the same statement is encoded by

$$\rho\ge0, \qquad \operatorname{Tr}\rho=1, \qquad \Pr(\Pi)=\operatorname{Tr}(\rho\Pi)\ge0$$

for every orthogonal projector $\Pi$. These are statements about the physical space and its physical observables, not about every component field introduced in a convenient gauge or coordinate description.

There is a representation-independent version. A state on a $*$-algebra satisfies

$$\omega(A^*A)\ge0.$$

It first defines a positive semidefinite form on algebra elements. Quotienting the null left ideal

$$\mathcal N_\omega=\{A:\omega(A^*A)=0\}$$

and completing produces the Gelfand–Naimark–Segal Hilbert space. Thus “positive state,” “positive semidefinite pre-Hilbert form,” and “positive-definite physical Hilbert space” are related stages, not interchangeable words. Fewster and Rejzner 2019, arXiv v2, §§ 2.1–2.3, printed pp. 4–9 (PDF) develop this state-to-representation passage.

The inequality applies to positive elements such as $A^*A$, not to an arbitrary operator $A$. It is also unrelated to whether the energy spectrum is positive.

Self-adjointness generates unitary evolution

For a closed system, a self-adjoint Hamiltonian $H$ generates the strongly continuous group

$$U(t)=e^{-iHt}, \qquad U(t)^\dagger=U(-t), \qquad U(t)^\dagger U(t)=\mathbf1.$$

Here $U(t)$ denotes Schrödinger evolution. With the chapter’s translation convention $U_{\mathrm{trans}}(a)=e^{iP\cdot a}$ and $H=P^0$,

$$U(t)=e^{-iHt}=U_{\mathrm{trans}}(-t,\mathbf0).$$

Hence

$$\langle U(t)\chi|U(t)\psi\rangle =\langle\chi|\psi\rangle, \qquad \|U(t)\psi\|=\|\psi\|.$$

On an invariant domain where differentiation is legitimate, the local check is

$$\frac{\mathrm d}{\mathrm dt} \langle\psi(t)|\psi(t)\rangle =i\langle H\psi(t)|\psi(t)\rangle -i\langle\psi(t)|H\psi(t)\rangle =0.$$

The global conclusion needs self-adjointness, not merely a formally Hermitian differential expression. Boundary conditions and operator domains can change which self-adjoint realization exists. A symmetric operator with unresolved deficiency spaces is not yet a complete dynamics.

Mixed states evolve as

$$\rho(t)=U(t)\rho(0)U(t)^\dagger.$$

Conjugation by $U(t)$ preserves positivity and the trace, so total probability remains one. It also preserves the spectrum of $\rho$, including purity and von Neumann entropy, for exact closed-system evolution.

Individual measurement probabilities can still vary with time. Unitarity preserves their normalized total and the inner products of consistently evolved states; it does not freeze every expectation value.

Self-adjointness does not imply that $H$ is bounded below. Poincaré Covariance and the Spectrum Condition treats the lower spectral bound and forward-spectrum condition as separate stability inputs. Nor does unitary time evolution by itself imply Lorentz covariance, locality, the existence of scattering states, or asymptotic completeness. Schwartz 2014, § 8.1.1, pp. 111–113 distinguishes the positive-norm unitary transformations acting on states from the finite-dimensional Lorentz transformations carried by covariant field components; the domain-sensitive self-adjoint-generator theorem remains in the linked Mathematical Methods treatment.

The regulated free scalar passes both checks

Put a free real scalar in a finite spatial box and retain finitely many modes. This makes the first application an ordinary oscillator calculation before any continuum or infinite-volume limit. For mode energies $E_{\mathbf k}>0$,

$$[a_{\mathbf k},a_{\mathbf q}^\dagger] =\delta_{\mathbf k\mathbf q}, \qquad H=\sum_{\mathbf k}E_{\mathbf k} \left(a_{\mathbf k}^\dagger a_{\mathbf k}+\frac12\right).$$

The Fock inner product is positive. In particular,

$$\|a_{\mathbf k}^\dagger|0\rangle\|^2 =\langle0|a_{\mathbf k}a_{\mathbf k}^\dagger|0\rangle =1,$$

and the normalized occupation basis has nonnegative norms. The Hamiltonian defined on the finite-particle subspace is essentially self-adjoint; its closure is self-adjoint on its standard dense diagonal domain in oscillator Fock space and is bounded below. A one-particle energy eigenstate therefore evolves only by a phase:

$$e^{-iH(t-t_i)}a_{\mathbf k}^\dagger|0\rangle =e^{-i(E_0+E_{\mathbf k})(t-t_i)} a_{\mathbf k}^\dagger|0\rangle.$$

After subtracting the regulated vacuum energy $E_0$, the phase contains just $E_{\mathbf k}$. Its norm and every transition probability are unchanged. In the Heisenberg description the same check reads

$$a_{\mathbf k}(t) =e^{iH(t-t_i)}a_{\mathbf k}e^{-iH(t-t_i)} =e^{-iE_{\mathbf k}(t-t_i)}a_{\mathbf k}.$$

More generally, if

$$|\psi(0)\rangle=\sum_{\mathbf n}c_{\mathbf n}|\mathbf n\rangle,$$

then every coefficient acquires a phase of modulus one and

$$\|\psi(t)\|^2 =\sum_{\mathbf n}|c_{\mathbf n}|^2 =\|\psi(0)\|^2.$$

Removing the regulator requires control of domains and limits, but it does not change which two structural claims were tested: the Fock representation is positive, and the chosen self-adjoint Hamiltonian generates unitary evolution.

The covariant photon begins in an auxiliary space

The free photon shows why those claims must name their space. In covariant Feynman-gauge quantization with the site metric, the four oscillator components satisfy

$$[a^\mu(\mathbf k),a^{\nu\dagger}(\mathbf q)] =-(2\pi)^3\,2\omega_{\mathbf k}\, \eta^{\mu\nu}\delta^{(3)}(\mathbf k-\mathbf q).$$

Here $\dagger$ denotes the adjoint for the auxiliary indefinite-metric construction, not an adjoint defined by a positive Hilbert norm.

For a wave-packet state

$$|f\rangle =\int_{\mathbf k}f_\mu(\mathbf k) a^{\mu\dagger}(\mathbf k)|0\rangle, \qquad \int_{\mathbf k}:= \int\frac{\mathrm d^3\mathbf k} {(2\pi)^3\,2\omega_{\mathbf k}},$$

the preserved auxiliary form is

$$(f,f)_{\mathrm{aux}} =-\int_{\mathbf k} f_\mu^*(\mathbf k)\eta^{\mu\nu}f_\nu(\mathbf k).$$

A purely timelike polarization has negative auxiliary norm, while a transverse spatial polarization has positive auxiliary norm. This indefinite space keeps Lorentz covariance manifest, but it is not yet a Hilbert space of photon probabilities. Evolution may preserve this indefinite form, but that is preservation of an auxiliary sesquilinear form—not physical unitarity on a positive Hilbert space.

In the Gupta–Bleuler construction, the subsidiary condition selects a subspace $\mathcal V_{\mathrm{GB}}$ of the auxiliary Fock space. The form restricted there is positive semidefinite. Its radical and the completed physical space are

$$\mathcal N =\mathcal V_{\mathrm{GB}}\cap\mathcal V_{\mathrm{GB}}^\perp, \qquad \mathcal H_{\mathrm{phys}} =\overline{\mathcal V_{\mathrm{GB}}/\mathcal N}.$$

In its one-photon sector, at momentum $k=(\omega,0,0,\omega)$, the condition $k^\mu f_\mu=0$ gives $f_3=-f_0$, so

$$(f,f)_{\mathrm{aux}} =|f_1|^2+|f_2|^2.$$

The time and longitudinal components cancel; their remaining common null direction is removed by the quotient. Two positive transverse photon polarizations remain in this sector. Because $\partial\cdot A$ obeys the free wave equation, free evolution preserves $\mathcal V_{\mathrm{GB}}$; preservation of the auxiliary form then preserves its radical $\mathcal N$. The evolution therefore descends to a unitary map on $\mathcal H_{\mathrm{phys}}$.

On the corresponding full-Fock quotient, let $\mathcal D$ be a common invariant dense domain for the auxiliary fields. An unbounded operator $O$ induces an operator on quotient representatives only if

$$O(\mathcal D\cap\mathcal V_{\mathrm{GB}}) \subset\mathcal V_{\mathrm{GB}}, \qquad O(\mathcal D\cap\mathcal N)\subset\mathcal N.$$

Appropriately smeared free field strengths $F_{\mu\nu}$ satisfy this criterion, whereas the potential $A_\mu$ generally does not. Thus gauge-invariant observables, rather than every auxiliary field component, act on the physical quotient.

This calculation is limited to the free Abelian theory; it is not a general gauge-theory construction. Steinmann 1989, p. 300 states the indefinite-space, subsidiary-condition, and null-quotient pattern. Covariant Free-Photon Quantization and Propagator derives the complete free Gupta–Bleuler calculation. Ghosts, non-Abelian constraints, and Becchi–Rouet–Stora–Tyutin cohomology require their own formulations.

Four positivity and unitarity claims remain distinct

Four distinct positivity or unitarity claims

Claim	Space or object	Direct consequence	What does not follow
Physical Hilbert positivity	Physical states or an observable-state representation	Nonnegative probabilities and Cauchy–Schwarz bounds	A lower energy bound or a dynamics
Unitary time evolution	A physical Hilbert space with a self-adjoint Hamiltonian	Preservation of inner products in a closed system	Scattering states, locality, or amplitude cuts
Scattering-matrix unitarity	The in/out scattering space, when constructed	Probability balance among asymptotic channels	Existence or completeness of those channels
Euclidean reflection positivity	A reflected Euclidean quadratic form	A positive-semidefinite form and null quotient used in Osterwalder–Schrader reconstruction	A Lorentzian QFT or unitary dynamics without the remaining reconstruction hypotheses

The rows distinguish claims made on different mathematical objects. A covariant auxiliary photon form belongs to none of the physical-probability rows until the subsidiary condition and quotient have been imposed. Likewise, an optical-theorem relation belongs to scattering theory, not to the mere statement that $e^{-iHt}$ preserves norms.

Failure modes and limits

• A positive vector space without a self-adjoint generator has no specified unitary dynamics. Positivity and time evolution are independent inputs.
• A self-adjoint Hamiltonian need not be stable. Its spectrum may be unbounded below unless a separate lower-bound or spectrum condition is imposed.
• A zero-norm vector is not automatically gauge. In a general indefinite space, all zero-norm vectors need not form a linear subspace. The quotient uses the specific null subspace inside the subsidiary physical subspace.
• Negative auxiliary norm is not a negative probability. It signals that the auxiliary representation is not the physical Hilbert space. Physical probability claims must wait for the restriction or quotient.
• An effective non-Hermitian Hamiltonian is not exact closed-system evolution. It may describe a subsystem, resonance approximation, absorbing boundary, or postselection, with the omitted degrees of freedom carrying the lost norm.
• A unitary approximation is not automatic. Truncations, gauge-breaking regulators, and inconsistent physical-state projections can violate identities that the exact theory obeys.
• Time-evolution unitarity is not S-matrix unitarity. The latter requires in/out states and a well-defined scattering operator; asymptotic completeness is stronger still.

Check your understanding

Check 1: separate a lower bound from self-adjointness

Let $H$ be multiplication by $x$ on $L^2(\mathbb R,\mathrm dx)$ with its standard self-adjoint domain. Does it generate unitary evolution, and is it bounded below?

Answer. It is self-adjoint, so $(U(t)\psi)(x)=e^{-ixt}\psi(x)$ is unitary. Its spectrum is all of $\mathbb R$, so it has no lower bound. Norm preservation and stability are therefore distinct.

Check 2: locate positivity in the photon example

Why is the equation $(f,f)_{\mathrm{aux}}=|f_1|^2+|f_2|^2$ not yet a proof that every auxiliary photon state has positive norm?

Answer. The formula follows only after the Gupta–Bleuler condition has selected the subsidiary subspace, and a null direction still remains. Positivity becomes definite only on the quotient $\mathcal V_{\mathrm{GB}}/\mathcal N$; the original four-component auxiliary space remains indefinite.

Check 3: identify the unitarity claim

An optical-theorem calculation checks a sum over asymptotic channels. Which row of the comparison table does it test?

Answer. It tests scattering-matrix unitarity, assuming the relevant in/out state space and normalization already exist. It is not merely the norm-preservation statement for finite-time evolution.

Where to continue

• Inspect the detailed free gauge example: Covariant Free-Photon Quantization and Propagator constructs the subsidiary subspace, null quotient, two transverse polarizations, and covariant propagator.
• Resolve operator domains: Self-Adjointness, Extensions, and Unitary Evolution develops the functional-analytic conditions behind the exponential $e^{-iHt}$.
• Keep Euclidean positivity separate: Reflection Positivity within Osterwalder–Schrader Reconstruction explains the reflected Euclidean form and the additional reconstruction hypotheses.
• Continue the structural sequence: Microcausality and Relativistic Compatibility asks a different question—compatibility of spacelike-localized observables—without treating locality as a consequence of unitarity.
• Enter scattering only after asymptotic states: S-Matrix Unitarity develops channel probability conservation, the optical theorem, and cuts on the scattering space.
• Return to the chapter map: Structural Principles and Axiom Maps compares physical positivity and unitary evolution with the other independent structural inputs.

References

• Fewster, Christopher J., and Kasia Rejzner. “Algebraic Quantum Field Theory—an Introduction.” arXiv:1904.04051v2, 2019, 47 pp.; published in Progress and Visions in Quantum Theory in View of Gravity, edited by Felix Finster, Domenico Giulini, Johannes Kleiner, and Jürgen Tolksdorf, 1–61. Birkhäuser, 2020. DOI. Open manuscript PDF, v2.

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge University Press, 2014. DOI.

• Steinmann, Othmar. “On the Characterization of Physical States in Gauge Theories.” Annales de l’Institut Henri Poincaré. Physique Théorique 51, no. 3 (1989): 299–321. NUMDAM.