Can You Determine Whether a Theory Is Unitary from Its Lagrangian?

A Lagrangian is a starting point for analyzing unitarity, not a complete yes-or-no test. It tells you the fields, interactions, and classical equations of motion. To decide whether the resulting quantum theory is unitary, you also need to know how it is quantized, which states are physical, and whether quantum effects spoil consistency.

In quantum mechanics, unitary time evolution preserves inner products and probabilities. In scattering theory, the corresponding condition is that the S-matrix obey

$$S^\dagger S = 1.$$

A real-looking Lagrangian is therefore not enough: reality is relevant, but unitarity is a statement about quantum evolution on a physical Hilbert space.

What to inspect in the Lagrangian

Start with the terms quadratic in the fields, often called the free or kinetic terms. They determine the propagating degrees of freedom and, after quantization, the poles and residues of propagators. A wrong-sign kinetic term can indicate a ghost: a state with negative norm, or, under a different quantization choice, an energy spectrum unbounded below. Either problem threatens a consistent unitary theory.

For a real scalar field in flat spacetime with metric signature $(+---)$, a conventional free Lagrangian is

$$\mathcal{L} = \frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - \frac{1}{2}m^2\phi^2.$$

Its time-derivative term has the standard positive sign. If you reverse the sign of the kinetic term, the field’s propagator has the opposite-sign residue relative to an ordinary scalar. This is a warning that the corresponding excitation cannot be treated as an ordinary positive-probability particle. The exact diagnosis depends on the quantization and conventions, but the sign should not be ignored.

Next, examine constraints and gauge symmetries. Some fields appearing in a Lagrangian are not independent physical particles. Gauge fields, for example, require constraints or gauge fixing; covariant gauge choices can introduce unphysical modes and Faddeev–Popov ghosts. Those intermediate objects do not automatically mean that the physical theory is nonunitary. The relevant question is whether the physical state space, after imposing the constraints (or using an equivalent method such as BRST quantization), has a positive inner product and unitary evolution.

Also check whether the action is Hermitian in the quantum theory. For real bosonic fields, this generally requires real coefficients and appropriate boundary conditions. For fermions and complex fields, the conjugation properties must be handled explicitly. A non-Hermitian interaction can make the Hamiltonian non-Hermitian, though specialized theories can have more subtle definitions of probability and unitarity. Do not infer the answer from the word “real” alone.

A practical test, step by step

1. Specify the theory. State the fields, spacetime dimension, metric convention, boundary conditions, and whether the Lagrangian is fundamental or an effective approximation.
2. Find the physical degrees of freedom. Identify constraints, gauge redundancies, and auxiliary fields. Do not count gauge-dependent modes as physical particles.
3. Check the quadratic theory. Derive the propagators and inspect their poles and residues. Look for negative-norm states, tachyonic instabilities, or modes with an ill-behaved energy spectrum. A tachyonic mass term often signals an unstable chosen vacuum; it does not, by itself, prove that the eventual theory is nonunitary.
4. Check interactions and quantum corrections. Confirm that the interactions respect Hermiticity and examine whether quantization introduces anomalies or other inconsistencies.
5. Test quantum evolution. In a scattering calculation, verify that amplitudes satisfy the unitarity relations, such as the optical theorem. A healthy classical Lagrangian does not replace this quantum check.

Why the Lagrangian alone may not decide the answer

A Lagrangian does not always uniquely specify a complete quantum theory. Quantization prescriptions, choices of physical state space, boundary conditions, and regularization can matter. A classical symmetry can also fail after quantization through an anomaly. Conversely, a gauge-fixed Lagrangian may contain unphysical fields even though the physical theory is unitary.

Effective field theories need another qualification. They are normally used only below a cutoff, and higher-derivative terms may be part of an approximation. If you treat a truncated higher-derivative Lagrangian as exact, its extra solutions can appear ghost-like. That observation does not automatically establish that the underlying theory is nonunitary; you must interpret the theory within its stated range of validity.

Common mistakes

• Equating a real Lagrangian with a unitary theory. Reality is a useful consistency check, not a proof of unitarity.
• Treating every minus sign as a ghost. Signs depend on metric and field conventions. Compare the full kinetic term and propagator with a stated convention.
• Counting gauge artifacts as physical states. Unphysical modes in a gauge-fixed calculation must be distinguished from states in the physical Hilbert space.
• Confusing stability with unitarity. A theory can have a unitary time evolution but an unstable vacuum, or have other problems despite appearing stable classically. These are separate tests.
• Checking only the classical equations. Quantum anomalies and violations of scattering unitarity can arise beyond the classical description.

So, the Lagrangian is where you begin: it can expose serious warning signs and help establish necessary conditions. A reliable conclusion about unitarity requires the quantized theory and its physical states, with quantum consistency checked in the setting where the theory is meant to apply.