Spectral representations and unitarity

Insert a complete set of energy-momentum eigenstates between two scalar operators. Translation invariance supplies their spacetime dependence, while each coefficient contains an absolute square of a vacuum-to-state matrix element. Positivity follows from the positive inner product on the physical Hilbert space. It is a property of physical operators and states; it should not be imposed unchanged on a gauge-fixed gluon propagator living in an enlarged space with unphysical components.

Lorentz invariance organizes these states by invariant mass squared $s$. A stable particle that the operator creates contributes a delta function at its mass, whereas multiparticle states usually generate a continuous distribution above a threshold. A resonance embedded in that continuum need not be a delta function on the physical spectral axis. The spectral density is an operator-dependent record of accessible states, rather than a list of classical particles passing through a detector.

For a sufficiently well-behaved scalar two-point function, Euclidean momentum replaces the oscillatory propagator denominator by $Q^2+s$. The result is a weighted superposition of free massive denominators. Differentiating a convergent representation at positive $Q^2$ gives a nonpositive derivative, and successive derivatives alternate in sign. These inequalities are useful consistency tests, although a few successful tests do not prove that an arbitrary proposed correlator belongs to a complete local quantum field theory.

Ultraviolet growth can require subtractions, especially for composite operators. Local contact terms appear as polynomials in momentum and must be separated before applying a simple monotonicity argument. Likewise, rescaling an operator rescales its entire spectral density. There is no universal sum rule saying every renormalized or composite operator has total spectral weight one. A canonical normalization with a suitable regulator may provide an additional sum rule; it must be stated rather than silently borrowed.

Recovering a continuous spectral density from finitely many noisy Euclidean data points is an inverse problem. The integral smooths sharp features, so quite different positive densities can fit the same observations within their errors. Positivity, threshold information, sum rules when available, and independently justified asymptotics constrain the reconstruction. A narrow peak produced by an algorithm is not automatically evidence for a sharply measured stable state; its dependence on those assumptions must be assessed.

The numerical exercise deliberately replaces the continuum by two positive discrete contributions. This makes the signs inspectable without pretending to solve the inverse problem. Change one weight to a negative number and the model loses its positive-measure interpretation, even if a particular momentum value still gives a positive correlator. Unitarity constrains a whole representation, not merely the sign of one plotted point. Determining spectral structure in strongly interacting theories remains an active computational challenge.