Gauge-Invariant Observables and Asymptotic Amplitudes

The previous pages taught a useful lesson from anomalies: a quantity that looks local and innocent can depend on how the theory is regulated. Gauge symmetry and diffeomorphism symmetry sharpen this lesson. In a theory with redundancy, not every field that appears in the action is an observable. The photon field $A_\mu$, the metric perturbation $h_{ab}$, and a charged fermion field $\psi$ are excellent variables for calculations, but their raw correlation functions depend on gauge choices.

The goal of this page is to separate three layers that are often blurred together:

1. fields used to describe the theory, such as $A_\mu$, $h_{ab}$, and $\psi$;
2. gauge-invariant observables, such as field strengths, Wilson lines, stress-tensor responses, and properly dressed matter operators;
3. asymptotic amplitudes, where local gauge-dependent fields are converted into physical particle data by residues at on-shell poles.

The same logic becomes more severe in gravity. In an ordinary gauge theory, spacetime points are still labels with external meaning. In quantum gravity, even the location of an operator is gauge data, because diffeomorphisms move points. One must either use asymptotic boundary data or formulate observables relationally: not “the field at coordinate $x$,” but “the correlation of two scalar insertions separated by intrinsic geodesic distance $L$.” This is the conceptual bridge to random lattices and random surfaces.

Gauge-invariant content in QED

Abelian charge and Wilson-line orientation

For Abelian gauge theory in this page we use

$$\psi(x)\mapsto e^{iq\alpha(x)}\psi(x), \qquad A_\mu(x)\mapsto A_\mu(x)+\partial_\mu\alpha(x), \qquad D_\mu=\partial_\mu-iqA_\mu.$$

The Wilson line along a path $\Gamma$ from $y$ to $x$ is

$$U_\Gamma(x,y)=\exp\!\left(iq\int_y^x A_\mu dz^\mu\right).$$

With this convention $\bar\psi(x)U_\Gamma(x,y)\psi(y)$ is gauge-invariant. For gravity we use Euclidean functional-integral notation when discussing fluctuating metrics, but the asymptotic scattering discussion is Lorentzian.

A charged field is not itself gauge-invariant. Under the Abelian gauge transformation above,

$$\langle \bar\psi(x)\psi(y)\rangle \mapsto e^{-iq\alpha(x)}e^{iq\alpha(y)}\langle \bar\psi(x)\psi(y)\rangle .$$

Unless $x=y$, this is not invariant. This does not mean charged particles are unphysical. It means the bare local field $\psi(x)$ is not a complete observable. It creates charge without also specifying the electromagnetic field lines required by Gauss’s law.

There are two standard ways to make gauge-invariant quantities. The first is to use neutral local operators such as

$$F_{\mu\nu}, \qquad \bar\psi\psi, \qquad \bar\psi\gamma^\mu D_\mu\psi,$$

or products of such operators. The second is to join charged endpoints by a Wilson line:

$$\mathcal O_\Gamma(x,y)=\bar\psi(x)U_\Gamma(x,y)\psi(y).$$

Indeed,

$$U_\Gamma(x,y) \mapsto \exp\!\left(iq\int_y^x (A+d\alpha)\right) =e^{iq\alpha(x)}U_\Gamma(x,y)e^{-iq\alpha(y)},$$

so the phases from $\bar\psi(x)$ and $\psi(y)$ cancel. The price is that the operator is not only a function of its endpoints. It also depends on the chosen path $\Gamma$, which describes a particular electromagnetic dressing. The displayed bilocal has zero total charge. A gauge-invariant operator that creates a single charged excitation must instead carry its dressing to an asymptotic boundary (or use an equivalent nonlocal Coulomb dressing); Gauss’s law forbids a compactly supported local charged observable.

The bilocal product of bare charged fields is gauge-dependent. A Wilson-line dressing supplies the electromagnetic phase needed to make the neutral bilocal gauge-invariant, at the cost of introducing path dependence.

This path dependence is physical, not a flaw. Different dressings represent different ways of arranging the long-range gauge field. In a theory with a mass gap for the gauge field, such differences can become short-ranged. In QED, however, the massless photon makes the dressing visible at arbitrarily large distances. This is the beginning of the infrared subtlety of charged scattering: strictly speaking, charged asymptotic states are not bare Fock states of electrons alone, but charged particles accompanied by soft electromagnetic clouds.

For many formal discussions one still writes gauge-fixed correlators of $A_\mu$ or $\psi$. That is fine, provided one remembers their status. Gauge-fixed Green functions are computational intermediates. Physical answers must be gauge-independent after combining diagrams, counterterms, and external-state prescriptions.

On-shell amplitudes from gauge-dependent fields

The S-matrix is a particularly important gauge-invariant object. In ordinary relativistic QFT with a mass gap and asymptotically free particle states, the scattering amplitude is extracted from time-ordered Green functions by the LSZ procedure. For a scalar field with overlap

$$\langle0|\phi(0)|p\rangle=\sqrt Z,$$

the exact two-point function has a one-particle pole

$$G(p)\sim {iZ\over p^2-m^2+i\epsilon}+\text{less singular terms}.$$

An $n$-point Green function has a product of such external poles when its external momenta approach the mass shell. Multiplying by $p_i^2-m_i^2$, taking the on-shell limit, and dividing by $\sqrt{Z_i}$ gives the scattering amplitude.

An isolated one-particle pole permits LSZ reduction. Its residue depends on the normalization of the interpolating field; after amputation and external-state normalization, the on-shell amplitude between physical states is the observable quantity.

For gauge theories, the raw Green functions and the wavefunction residues of gauge-variant interpolating fields can depend on gauge fixing. Physical pole positions, and the fully LSZ-reduced amplitude between physical states after contraction with physical polarizations, do not. The Ward identities are what make this possible. For example, in QED the photon self-energy must be transverse,

$$q^\mu\Pi_{\mu\nu}(q)=0,$$

so unphysical longitudinal polarizations decouple from conserved currents. A regulator or cutoff that violates this transversality will manufacture unphysical photon mass terms, such as a term proportional to $\Lambda^2\delta_{\mu\nu}$ in the polarization tensor. Gauge-invariant regularization schemes, or gauge-restoring counterterms when allowed, are not optional housekeeping; they are what keep the computed amplitude tied to the correct observable.

There is a useful slogan here:

$$\text{off-shell Green functions are gauge-scheme dependent,} \qquad \text{on-shell physical amplitudes are not.}$$

The slogan has two caveats. First, one must use physical external states. Second, in theories with massless gauge bosons, the S-matrix between naive charged Fock states is infrared ill-defined. In QED the exact charged sector has infraparticle behavior rather than the isolated one-particle pole assumed by elementary LSZ. Physically meaningful formulations use inclusive probabilities or appropriately dressed asymptotic states. The gauge-invariant content survives, but the dictionary from fields to particles is more delicate.

Gravity: coordinates are also gauge data

Gravity pushes the previous discussion one level deeper. In gauge theory, $A_\mu$ changes under gauge transformations, but the spacetime point $x$ remains part of the background. In gravity, diffeomorphisms act on the metric and on the coordinate labels themselves. A scalar field transforms as

$$\phi(x)\mapsto \phi'(x')=\phi(x), \qquad x' = f(x).$$

Thus a formal expression such as

$$\langle\phi(x_1)\phi(x_2)\rangle$$

is not an observable in a theory where the metric is integrated over and diffeomorphisms are gauge redundancies. The coordinate labels $x_1,x_2$ have no invariant meaning by themselves. Under a passive diffeomorphism, the same physical configuration obeys the exact covariance relation

$$\big\langle\phi'(f(x_1))\phi'(f(x_2))\big\rangle =\big\langle\phi(x_1)\phi(x_2)\big\rangle.$$

This covariance statement does not make either fixed-coordinate expression a diffeomorphism-invariant observable; it relates two gauge descriptions.

Likewise, the graviton two-point function

$$\langle h_{ab}(x)h_{cd}(y)\rangle$$

depends on the gauge used to fix linearized diffeomorphisms,

$$h_{ab}\mapsto h_{ab}+\nabla_a\xi_b+\nabla_b\xi_a.$$

It is a useful object for perturbation theory, but it is not itself an observable.

In gravity, fixed coordinate labels are part of the gauge description. Diffeomorphism-invariant observables must either be anchored to asymptotic boundary data or defined relationally using geometric information inside the state.

This is why local observables in quantum gravity are subtle. A local curvature scalar $R(x)$ is a scalar under coordinate changes, but the phrase “at $x$” is not invariant if $x$ is just a coordinate label. To turn it into an observable, one must say how the point is identified physically: by boundary anchoring, by matter clocks and rods, by geodesic distances, or by integrating over all points with a diffeomorphism-invariant measure.

Asymptotic observables

The cleanest observables in gauge theory and gravity often live at infinity. In asymptotically flat spacetime, the S-matrix is defined by preparing incoming particles at past infinity and measuring outgoing particles at future infinity. The interior fields are gauge-dependent variables, but the asymptotic particle data are tied to representations of the asymptotic symmetry group and to on-shell poles.

For a massive particle in flat space, the relevant large-time behavior of the two-point function is controlled by

$$G(p)\sim {iZ\over p^2-m^2+i\epsilon}+\text{regular terms}.$$

For massless particles, one works with null infinity and helicity states. Gauge redundancy removes longitudinal polarizations; only the transverse physical polarizations reach the asymptotic Hilbert space.

In curved spacetimes, the situation depends on the asymptotics:

• In asymptotically flat spacetime, an S-matrix is the natural candidate, though massless fields and gravitational memory introduce infrared structure.
• In anti-de Sitter space, timelike infinity is replaced by a conformal boundary. Boundary sources and boundary correlators are the natural gauge-invariant data. A conventional flat-space S-matrix is recovered only in suitable large-radius or local limits.
• In de Sitter space, there is no ordinary in/out S-matrix of the same kind. One instead studies wavefunction coefficients, late-time correlators, or in-in expectation values, again subject to gauge and diffeomorphism constraints.

Asymptotic flatness, AdS boundary conditions, and de Sitter expansion lead to different observable dictionaries. In all cases the same principle applies: gauge-dependent bulk fields are not the final observable data.

This point is easy to underestimate. The expression “the graviton propagator” is not wrong, but it is like “the vector potential propagator” in QED: useful after gauge fixing, not directly measurable. The measured content is encoded in gauge-invariant amplitudes, boundary correlators, fluxes, or relational quantities.

Relational correlators in fluctuating geometry

A diffeomorphism-invariant alternative to asymptotic data is to average over positions while imposing an invariant geometric condition. Let $A(x)$ and $B(y)$ be scalar operators (not scalar densities, because the invariant measures are written explicitly). Define

$$G_{AB}(L) = {1\over Z}\int {\mathcal D g\,\mathcal D\Phi\over \operatorname{Diff}} e^{-S[g,\Phi]} \int d^dx\sqrt g\,d^dy\sqrt g\, A(x)B(y)\,\delta\!\left(L-\ell_g(x,y)\right),$$

where $\ell_g(x,y)$ is the geodesic distance between $x$ and $y$ in the metric $g$. This object does not ask for the correlation between coordinate points. It asks for the correlation between two insertions at intrinsic separation $L$.

A useful Laplace transform is

$$G_{AB}(\mu)=\int_0^\infty dL\,e^{-\mu L}G_{AB}(L).$$

Equivalently,

$$G_{AB}(\mu) = \left\langle \int d^dx\sqrt g\,d^dy\sqrt g\, A(x)B(y)e^{-\mu\ell_g(x,y)} \right\rangle.$$

The kernel $e^{-\mu\ell_g(x,y)}$ is the semiclassical large-mass limit of a heavy-particle propagator in the background geometry. More precisely, a scalar propagator with mass $\mu$ admits a worldline representation, and at large $\mu$ its leading saddle is

$$G_\mu(x,y;g)\sim \exp[-\mu\ell_g(x,y)]\times\text{fluctuation determinant}.$$

Thus relational correlators can be viewed as correlators of matter insertions connected by a heavy probe worldline.

A gravitationally meaningful two-point function can be defined by integrating over insertion points and conditioning on the geodesic separation $\ell_g(x,y)=L$. Its Laplace transform is naturally represented by a massive worldline propagator in the fluctuating metric.

This construction is not always the most practical observable, but it is conceptually clean. It makes explicit what a diffeomorphism-invariant question must do: either refer to infinity, or define positions by relations among dynamical fields and geometric data. As a quantum composite, $G_{AB}(L)$ still needs ultraviolet renormalization, and geodesic-distance degeneracies or boundaries may require additional prescriptions; diffeomorphism invariance alone does not remove those analytic subtleties.

Sources, deformations, and effective actions

It is useful to package observables by coupling sources to gauge-invariant operators. In an ordinary QFT,

$$e^{-W[J]}=\int\mathcal D\Phi\,\exp\left[-S[\Phi]-\int d^dx\,J_i(x)\mathcal O_i(x)\right].$$

The derivatives of $W[J]$ generate connected correlators of the operators $\mathcal O_i$. In a gauge theory, the operators inserted this way should be gauge-invariant, or else the source itself must transform in a way that restores gauge invariance. In gravity, the source term must be written with the invariant measure,

$$\int d^dx\sqrt g\,J_i(x)\mathcal O_i(x),$$

and a fixed coordinate-dependent source $J_i(x)$ should be interpreted as part of the background structure. If the metric is dynamical and no boundary has been fixed, truly invariant deformations are typically integrated scalar operators such as

$$\lambda_i\int d^dx\sqrt g\,\mathcal O_i(x).$$

The response to such deformations is encoded in

$$W(\lambda_1,\ldots,\lambda_n) = -\log\int {\mathcal D g\,\mathcal D\Phi\over \operatorname{Diff}} \exp\left[-S[g,\Phi]-\sum_i\lambda_i\int d^dx\sqrt g\,\mathcal O_i\right].$$

Near a scale-invariant point, dimensional analysis and anomalous dimensions determine the scaling of $W$ with the couplings. The crucial point for the present page is not the exact exponent, but the type of question being asked: a deformation by an integrated scalar is a diffeomorphism-invariant question about the whole geometry, whereas a coordinate-space Green function at fixed $x_i$ is not.

This is also where local cutoff intuition can betray us. In a nongauge scalar theory on a fixed flat background, a condition like

$$|k|<\Lambda$$

is a familiar regulator, although it sacrifices some symmetries. Applied directly to a gauge loop, however, even this cutoff generally violates the momentum-shift identities behind the Ward identity. Trying to repair it pointwise by imposing

$$|k-qA(x)|<\Lambda$$

does not define a gauge-invariant regulator either. Likewise, a coordinate-momentum cutoff is not a diffeomorphism-invariant way to regulate gravity. A bad regulator may generate terms forbidden by the symmetry, such as a photon mass term or noncovariant metric terms. The remedy is not to abandon fields, but to use covariant regulators—or symmetry-restoring counterterms when justified—and enforce the Ward identities.

Why random lattices enter

A fixed lattice is an excellent regulator for many QFTs because it makes the path integral finite and concrete. But a fixed square lattice has a preferred coordinate structure and does not itself sum over geometry. In a theory where diffeomorphisms are gauge redundancies, recovering the desired continuum symmetry then requires care.

The idea of a random lattice is to avoid putting the geometry on a fixed coordinate grid. Instead one sums over combinatorial geometries. In two dimensions, this becomes especially powerful because triangulated surfaces and ribbon graphs can be generated by matrix integrals. The diagrams of a matrix model are not merely Feynman graphs in a pre-existing space; they are discretized worldsheets. Their connectivity defines the geometry being summed over.

A fixed lattice supplies a coordinate cutoff. A random lattice instead sums over graphs whose connectivity approximates geometry. This is the entry point to matrix models and random surfaces.

This is not yet string theory, but it points in that direction. A particle worldline is a sum over paths. A string worldsheet is a sum over surfaces. A random lattice is a discretized way to make “sum over surfaces” precise. The next page develops this idea through matrix models and planar diagrams.

Summary

Gauge symmetry means that some variables in the action are redundant. In QED, $A_\mu$ and $\psi$ are useful fields, but gauge-invariant observables are built from field strengths, neutral composites, Wilson lines, or asymptotic charged states with appropriate dressing. The S-matrix is extracted from on-shell pole data, not from arbitrary off-shell Green functions.

Gravity adds the fact that spacetime coordinates themselves are gauge labels. A fixed-coordinate bulk correlator is therefore not a gravitational observable unless the coordinates have been physically anchored. Good observables are asymptotic, boundary-defined, integrated, or relational. Relational correlators condition on invariant geometric data such as geodesic distance.

Random lattices arise naturally from this logic. If geometry is dynamical, a single fixed coordinate lattice does not represent the gravitational path integral. Summing over lattices or graphs gives a discrete version of summing over geometries, which becomes the matrix-model and random-surface technology of the next lecture.

Common pitfalls

A gauge-fixed propagator is not “wrong.” It is a gauge-dependent ingredient. The mistake is to interpret it directly as an observable.

A gauge-invariant charged operator is usually nonlocal. This is not an aesthetic inconvenience but a consequence of Gauss’s law.

In gravity, scalar fields are not enough to make local observables. A scalar evaluated at a coordinate point is still tied to a coordinate choice.

A cutoff that looks rotationally invariant in one coordinate system need not respect gauge symmetry or diffeomorphism symmetry. Regulators must be judged by the Ward identities they preserve.

Exercises

Exercise 1: Wilson-line cancellation of endpoint phases

Using the gauge-transformation convention of this page, show that

$$\mathcal O_\Gamma(x,y)=\bar\psi(x)U_\Gamma(x,y)\psi(y), \qquad U_\Gamma(x,y)=\exp\!\left(iq\int_y^x A\right),$$

is gauge-invariant.

Solution

Under

$$\psi(x)\mapsto e^{iq\alpha(x)}\psi(x), \qquad \bar\psi(x)\mapsto \bar\psi(x)e^{-iq\alpha(x)}, \qquad A\mapsto A+d\alpha,$$

the Wilson line transforms as

$$U_\Gamma(x,y) \mapsto \exp\!\left(iq\int_y^x A+iq\int_y^x d\alpha\right) =e^{iq\alpha(x)}U_\Gamma(x,y)e^{-iq\alpha(y)}.$$

Therefore

$$\bar\psi(x)U_\Gamma(x,y)\psi(y) \mapsto \bar\psi(x)e^{-iq\alpha(x)} \left(e^{iq\alpha(x)}U_\Gamma(x,y)e^{-iq\alpha(y)}\right) e^{iq\alpha(y)}\psi(y),$$

and all phases cancel. Thus $\mathcal O_\Gamma(x,y)$ is gauge-invariant.

Exercise 2: Amputation and external-state normalization in LSZ

Suppose a scalar exact two-point function has the pole

$$G(p)={iZ\over p^2-m^2+i\epsilon}+\text{regular}.$$

Explain why the factor $Z^{-1/2}(p^2-m^2)$ appears for each external scalar leg in LSZ reduction.

Solution

In a two-point function the isolated pole has residue $Z$ because there is an overlap $\sqrt Z$ at each end. In a general time-ordered $n$-point function, the factor associated with one external leg is instead schematically

$${i\sqrt Z\over p^2-m^2+i\epsilon},$$

times the matrix element for the remaining process. The pole reflects propagation of an asymptotic one-particle state created by the field. Multiplying by $p^2-m^2$ removes the propagator denominator, and multiplying by $Z^{-1/2}$ removes the overlap of the interpolating field with the normalized external state, up to the standard phases and factors of $i$.

The field does not create a normalized one-particle state with unit amplitude; it creates it with overlap $\sqrt Z$:

$$\langle0|\phi(0)|p\rangle=\sqrt Z.$$

Therefore the combination $Z^{-1/2}(p^2-m^2)$ is precisely the operation of amputating the pole and normalizing the external state. This derivation assumes an isolated stable-particle pole; the charged infraparticle sector of massless QED requires a dressed or inclusive treatment instead.

Exercise 3: Why a scalar at a coordinate point is not observable

Let $\phi$ be a scalar field in a dynamical geometry. Why is $\phi(x)$ not by itself a diffeomorphism-invariant observable, even though $\phi$ is a scalar?

Solution

A scalar field has the transformation law

$$\phi'(x')=\phi(x)$$

under a diffeomorphism $x'=f(x)$. This means the value of the field at the physical point is invariant, but the coordinate label of that point changes.

The expression $\phi(x)$ with fixed coordinate label $x$ assumes that $x$ has physical meaning. In a theory of dynamical geometry, diffeomorphisms are gauge redundancies, so two configurations related by moving the coordinate label represent the same physical configuration. Therefore a local insertion at a bare coordinate point is gauge-dependent.

To make an observable, one must identify the point relationally, anchor it to boundary data, or integrate over positions with an invariant measure.

Exercise 4: Diffeomorphism invariance of a geodesic-distance correlator

Show that the relational two-point function

$$G_{AB}(L) = \left\langle \int d^dx\sqrt g\,d^dy\sqrt g\, A(x)B(y)\delta(L-\ell_g(x,y)) \right\rangle$$

is invariant under diffeomorphisms, assuming $A$ and $B$ are scalar operators.

Solution

Under a diffeomorphism, the volume element is invariant as a density:

$$d^dx\sqrt g \mapsto d^dx'\sqrt{g'}.$$

Scalar operators satisfy

$$A'(x')=A(x), \qquad B'(y')=B(y).$$

The geodesic distance is an intrinsic quantity, so

$$\ell_{g'}(x',y')=\ell_g(x,y).$$

Therefore every factor in the integrand transforms covariantly, and after changing integration variables from $(x,y)$ to $(x',y')$, the value of the integral is unchanged. The delta function imposes a condition on an invariant distance, not on coordinate separation.

Exercise 5: Detecting a gauge-violating photon mass term

A naive hard cutoff in QED gives a vacuum polarization tensor of the schematic form

$$\Pi_{\mu\nu}(q)=\Lambda^2\delta_{\mu\nu}+(q^2\delta_{\mu\nu}-q_\mu q_\nu)\Pi(q^2)+\cdots .$$

Why is the first term incompatible with gauge invariance?

Solution

Gauge invariance implies the Ward identity

$$q^\mu\Pi_{\mu\nu}(q)=0.$$

The transverse structure satisfies this identity:

$$q^\mu(q^2\delta_{\mu\nu}-q_\mu q_\nu)=q^2q_\nu-q^2q_\nu=0.$$

The cutoff term gives

$$q^\mu\Lambda^2\delta_{\mu\nu}=\Lambda^2q_\nu,$$

which is nonzero. It is equivalent to generating a photon mass term

$$\Lambda^2 A_\mu A^\mu,$$

which is forbidden by gauge invariance. A gauge-invariant regulator avoids such a term, or the term must be subtracted by enforcing the Ward identity.

References and further reading

A. M. Polyakov, Gauge Fields and Strings, Chapters 7, 9, and 10, develops loop observables, random paths, random surfaces, and the relation between gauge fields and strings.

M. Srednicki, Quantum Field Theory, Chapters 5, 67–68, and 82, gives compact treatments of LSZ reduction, Ward identities, Wilson loops, lattice gauge theory, and confinement.

S. Weinberg, The Quantum Theory of Fields, Volume I, Chapters 3 and 10, and Volume II, Chapters 15–18, gives a systematic account of scattering theory, gauge theories, effective actions, and renormalization-group methods.

M. D. Schwartz, Quantum Field Theory and the Standard Model, Chapters 8, 14, 24, 25, and 34, is useful for gauge invariance, path-integral Ward identities, unitarity, Wilson lines, and background-field effective actions.

J. Polchinski, String Theory, Volume 1, Chapters 1–3, gives the standard treatment of worldline and worldsheet path integrals, gauge fixing, conformal gauge, and boundary observables in string perturbation theory.

Further reading

This lesson follows the manuscript’s progression from gauge-dependent propagators to invariant scattering and relational gravitational probes. For maintained reference accounts, see BRST cohomology and physical observables, Gauge-invariant dressed observables, and Relational, boundary, and asymptotic observables.