Unitarity, Normalization, and Largest-Time Identities

Microscopic unitarity and a trace-one initial state constrain a closed-time-path theory more strongly than ordinary Hermiticity: equal branch sources give the integrated identity $Z[J,J]=1$ for every source profile. In the $r/a$ basis this eliminates all-$a$ connected correlators, yields largest-time identities, and supplies practical tests for regulators and approximations. The doubled bulk action vanishes pointwise at $\phi_a=0$; a raw initial density kernel generally does not.

Required background. Use the closed-time-path generating functional and the distinction between unitary evolution and Hilbert-space positivity.

Helpful background. Trace, positivity, and causal consistency in open QFT shows how these conditions change after degrees of freedom are traced out.

Three exact identities

For a normalized $\rho_0$ and Hermitian Hamiltonian,

$$Z[J_+,J_-]=\operatorname{Tr}(U_{J_+}\rho_0U_{J_-}^\dagger)$$

obeys:

$$Z[J,J]=1, \qquad Z[J_+,J_-]^*=Z[J_-,J_+], \qquad Z[0,0]=1.$$

The first uses unitarity, trace cyclicity, and $\operatorname{Tr}\rho_0=1$; the second uses Hermiticity; the third is the zero-source special case of the first. A regulator or effective description must preserve the appropriate version of these relations. An anomaly can modify a classical symmetry identity, but it does not license failure of trace normalization for a closed unitary system.

With $Z=e^{iW}$ and the chapter’s source rotation, equal sources mean $J_a=0$. The corresponding operator and generating-functional identities are derived in Crossley, Glorioso, and Liu 2017, § II.A, pp. 32–35. Therefore

$$W[J_r,J_a=0]=0, \qquad \left.\frac{\delta^nW}{\delta J_r(x_1)\cdots\delta J_r(x_n)}\right|_{J_a=0}=0.$$

Because $J_r$ couples to $\phi_a$, connected all-$a$ correlators vanish. These statements concern the fully integrated generating functional. They do not require the bare contour integrand to equal one on every coincident field configuration.

For the microscopic dynamical action, the stronger pointwise relation is immediate:

$$S_{\mathrm{dyn}}[\phi_r,\phi_a] =S[\phi_r+\phi_a/2]-S[\phi_r-\phi_a/2], \qquad S_{\mathrm{dyn}}[\phi_r,0]=0.$$

Likewise, at tree level a local trace-preserving dynamical or influence effective action is organized so that every bulk term contains at least one $a$ field. Beyond tree level, this pointwise condition is necessary but need not by itself enforce the full path-integral identity; the measure and any auxiliary sector must preserve it as well Crossley, Glorioso, and Liu 2017, § I.D, pp. 16–20. None of these pointwise statements applies term by term to $S_0=-i\log(\rho_0/\mathcal N)$: a Gaussian preparation normally contains a pure-$r$ boundary term.

Largest-time identity

Consider a contour correlator expanded as a sum over branch assignments. If an insertion at the strictly latest time is changed from $+$ to $-$, all later forward and backward evolution has already cancelled; the two terms are equal up to the branch sign. Their difference—the $a$ insertion—vanishes. Hence

$$\langle\phi_a(t_{\max})\,\mathcal O_{r/a}(t_i<t_{\max})\rangle=0,$$

apart from specified equal-time contact terms. At two points,

$$-i\langle\phi_r(x)\phi_a(y)\rangle=G^R(x,y), \qquad -i\langle\phi_a(x)\phi_r(y)\rangle=G^A(x,y).$$

If the $a$ time is latest, both causal functions vanish. This is the causal content of the largest-time equation.

Equal times require care. Canonical commutators and derivative couplings can generate contact terms, while a lattice time contour requires a definite placement prescription. One must not infer that a nonzero equal-time response violates unitarity before accounting for these terms.

The largest-time identity is an identity of the contour trace after the initial state has been inserted. It constrains later correlators and dynamical vertices, not the earliest-time density kernel pointwise.

The two dashed parts of the schematic have different status. The equal-source branch is the universal generating identity behind the $a$-field selection rules; the KMS box additionally requires a thermal state and equilibrium dynamics.

For normalized unitary evolution, the full generating functional gives $Z[J,J]=1$, every connected all-$a$ correlator vanishes, and the largest-time identity removes a correlator whose latest insertion is an $a$ field in this convention. These statements survive without thermal equilibrium. The quadratic-inversion branch pertains to the two-point propagators, while interaction vertices come from rotating the action separately. KMS applies only when the initial state is thermal and the dynamics preserves that equilibrium. The diagram is schematic and not to scale.

The sections Three exact identities, Largest-time identity, and Open and approximate dynamics give the text and equation equivalent of the normalization, causal, and scope conditions.

Checked two-point normalization

The branch components obey

$$G^{++}+G^{--}-G^{+-}-G^{-+}=0.$$

This is exactly $-i\langle\phi_a\phi_a\rangle=0$. Insert the Wightman decompositions: the coefficient of $G^>$ is $\theta_{xy}+\theta_{yx}-1=0$, and likewise for $G^<$, if a common equal-time prescription is used.

For a quartic bulk interaction,

$$S_{\mathrm{int}}[\phi_+]-S_{\mathrm{int}}[\phi_-] =-\int d^{d+1}x\left( \frac{\lambda}{6}\phi_a\phi_r^3 +\frac{\lambda}{24}\phi_a^3\phi_r \right),$$

which vanishes at $\phi_a=0$ and contains no all-$r$ vertex. This direct algebraic check should precede diagram generation.

Open and approximate dynamics

For a factorized normalized environment preparation—or after correlated initial boundary data have been separated—a trace-preserving influence action satisfies $S_{\mathrm{IF}}[\phi_r,0]=0$. It may contain imaginary $a a$ terms describing noise. Positivity or complete positivity imposes additional inequalities; it does not follow from normalization alone. Likewise, a finite 2PI or perturbative truncation may preserve $Z[J,J]=1$ yet violate spectral positivity or a gauge Ward identity.

Test an implementation by setting arbitrary equal branch sources, verifying the largest-time zero on time-ordered test data, and checking conjugation under branch exchange. These tests should be repeated after discretization and counterterm insertion.

Exercise

Why does $I_{\mathrm{dyn}}[\phi_r,0]=0$ forbid a local bulk term proportional to $\phi_r^2$ in a trace-preserving dynamical action but allow $\phi_a\phi_r$ and $i\phi_a^2$? Why can an initial Gaussian boundary kernel nevertheless contain $i\phi_rK_{rr}\phi_r/2$?

Solution

A pure bulk $\phi_r^2$ term remains nonzero when the two dynamical histories coincide, contradicting cancellation of identical forward and backward evolution. Both $\phi_a\phi_r$ and $i\phi_a^2$ vanish at $\phi_a=0$. The latter is possible in an influence action or effective description, but its sign is constrained by positivity conditions beyond trace normalization. The initial kernel has a different job: its pure-$r$ term supplies the nonconstant diagonal probability $\rho_0(\phi_r,\phi_r)$. Only its diagonal integral, not the boundary action pointwise, is fixed to one.

Continue

Rotate the branch matrix on retarded, advanced, and Keldysh bases, then use the largest-time rule to eliminate forbidden Keldysh diagrams.

References

• Crossley, M., Glorioso, P., and Liu, H. (2017). “Effective Field Theory of Dissipative Fluids.” Journal of High Energy Physics 2017(09), 095. arXiv:1511.03646; DOI.
• Veltman, M. (1963). “Unitarity and Causality in a Renormalizable Field Theory with Unstable Particles.” Physica 29, 186–207. DOI.