The Fierz–Pauli mass term and the Boulware–Deser ghost

The Fierz–Pauli mass term is the only mass term that gives a free spin-2 field a mass without a ghost, a mode whose energy has the wrong sign. In matrix notation it is proportional to Tr(h²) − (Tr h)², where h is the field; any other relative weight of the two terms lets an extra mode propagate, and that mode is a ghost. When the theory is made nonlinear, a generic completion brings the extra mode back, the Boulware–Deser ghost, and keeping it out takes the special de Rham–Gabadadze–Tolley (dRGT) and Hassan–Rosen potentials. A related standard fact, the signature obstruction, says that a ghost-free two-derivative kinetic term for a symmetric tensor field must have the gauge symmetry of the massless graviton; the Fierz–Pauli theory keeps that kinetic term and breaks the symmetry only through its mass term. The program uses all three facts to decide what its second field, the momentum graviton X, can be. Its tuning test checks X’s mass term against the Fierz–Pauli form, and the version built from metrics fails that check.

Status: Established — [T] — textbook massive-gravity theory, with the signature obstruction tagged “[T — standard]” in #6 §2. The program’s documents cite no source for the Fierz–Pauli term or the Boulware–Deser ghost; the references below are added from model memory (†). The program’s comparison of its own mass term with the Fierz–Pauli form is machine-checked as battery test T4 [V], which passes in the current gate run (the verification battery).

A mass for a spin-2 field [T]#

The starting point is the free massless graviton of general relativity: a symmetric tensor field h_μν on flat spacetime, with the two-derivative Lagrangian L_kin[h] obtained by expanding the Einstein–Hilbert action to second order. L_kin is invariant under the linearized diffeomorphisms δh_μν = ∂_μξ_ν + ∂_νξ_μ, and that gauge symmetry is what keeps the theory free of ghosts.

A mass term is a term quadratic in h with no derivatives. Lorentz invariance allows only the scalars h_μν h^μν and h², where h = h^μ_μ is the trace. Treating h as a matrix (one index raised with the flat metric), these are Tr(h²) and (Tr h)². A general mass term is a combination of them with some relative weight. Any such term breaks the gauge symmetry, because it changes when a pure-gauge perturbation ∂_μξ_ν + ∂_νξ_μ is added to h.

The Fierz–Pauli weight [T]#

Fierz and Pauli (1939†) wrote down the Lagrangian of a free massive spin-2 field. Its mass term has one particular relative weight, and with any other weight the field carries a ghost (van Nieuwenhuizen 1973†):

L_FP = L_kin[h] + L_mass,        L_mass ∝ −m² (h_μν h^μν − h²) = −m² (Tr(h²) − (Tr h)²)

What the weight does. With this weight, taking the divergence and then the trace of the source-free field equations yields the constraints ∂^μ h_μν = 0 and h = 0. The field is then transverse and traceless, and each component obeys the massive wave equation, the Klein–Gordon equation with mass m. What propagates is a massive spin-2 particle with five polarizations, the count calc note #6 §3 gives for X: “a massive spin-2 with its five polarizations”.

Why only this weight. The mechanism shows up in the time components. In h_μν h^μν and in h², the term quadratic in h₀₀ appears with the same coefficient, so with the Fierz–Pauli weight it cancels, and h₀₀ enters the action only linearly. A field that appears linearly and without time derivatives is a Lagrange multiplier: its equation is a constraint, and that constraint removes the extra mode a symmetric tensor would otherwise carry. With any other weight h₀₀ appears quadratically, the constraint is lost, and the extra mode propagates. It is a scalar with a wrong-sign kinetic term.

What a ghost does. A field whose kinetic term has the wrong sign carries negative energy. Coupled to ordinary fields, it leaves the energy unbounded below: the vacuum can decay by producing ghosts together with ordinary particles, and energy conservation does not stop it. In the quantum theory the choice is between negative-energy states and negative-norm states, an unstable vacuum or a loss of unitarity. A theory with a ghost coupled to ordinary matter is not considered physical.

The signature obstruction#

The Fierz–Pauli construction keeps the gauge-invariant kinetic term L_kin even though its mass term breaks the gauge symmetry. The reason is the signature obstruction, tagged “[T — standard]” in calc note #6 §2 and named in calc note #7 §5: without the gauge invariance, no two-derivative kinetic term for a Lorentz-covariant symmetric tensor is free of ghosts, because in Lorentzian signature the mixed time–space components h₀ᵢ enter with the wrong sign. L_kin contains no time derivatives of h₀₀ and h₀ᵢ, which are constrained rather than propagated (general relativity and the graviton), and the Fierz–Pauli weight keeps h₀₀ a Lagrange multiplier when the mass term is added. The checkpoint states the program’s use as “ungauged X by Lorentz signature” (#11 §7.1). Calc note #6 §3 adds that the chain beneath that use was derived for positive-definite metrics; its extension to spacetime signature is open (Lorentzian dressing (OP3)).

Beyond quadratic order: the Boulware–Deser ghost [T]#

The Fierz–Pauli term fixes a massive spin-2 theory only to quadratic order in h. Coupling the field to gravity, or giving it self-interactions, requires a nonlinear potential: a function of the full metric g_μν and a second, reference metric f_μν (a fixed background in massive gravity, a dynamical field in bimetric gravity) whose expansion about g = f begins with the Fierz–Pauli term. Boulware and Deser (1972†) showed that a generic completion of this kind propagates the extra mode after all, beyond quadratic order, and that the mode is a ghost: the Boulware–Deser ghost.

In the Hamiltonian formulation the mechanism is the one described above, at full nonlinear order [T]. In general relativity the lapse, which carries g₀₀, appears linearly and enforces the Hamiltonian constraint. A generic potential makes the lapse appear nonlinearly; the constraint disappears, and with it the condition that removed the extra mode. The linear constraint from h₀₀ in the Fierz–Pauli theory is the quadratic-order form of that Hamiltonian constraint. In the literature the name Boulware–Deser ghost usually refers to the extra mode as it reappears beyond quadratic order; at quadratic order the same extra mode is what any non-Fierz–Pauli weight lets propagate, and the program’s notes use the name for both.

Avoiding the ghost at every order takes a special family of potentials, for which the constraint survives: dRGT massive gravity for one dynamical metric and Hassan–Rosen bimetric gravity for two. Calc note #6 §3 summarizes: “Massive spin-2 coupled to gravity has exactly one known consistent structure: the Fierz–Pauli tuning at quadratic order completed by a dRGT/Hassan–Rosen-class potential [T: HR 2011]”, and “a generic potential reintroduces the Boulware–Deser ghost”.

Frame variables [T]#

In the vielbein, or frame, formulation the ghost-free potentials are polynomials in two frames, and the Fierz–Pauli weight comes out of them without being imposed. Calc note #7 §3 calls these potentials “ghost-free by construction, Fierz–Pauli automatic, dRGT structure inherited rather than tuned” (the vielbein formulation and wedge potentials).

How the program uses it#

The ungauged option is closed#

X = Re Z is a symmetric tensor with its own gauge symmetry, the dual diffeomorphisms δX_μν = ∂₍μλ_ν₎ (#6 §2). Two routes would avoid giving it a mass, and both are closed (#11 §7.1, [D + T]): “Massless X is excluded by the multi-graviton no-go [8†]; ungauged X by Lorentz signature.” The second exclusion is the signature obstruction; the first is the multi-graviton no-go theorem. What remains is a mass that respects the Fierz–Pauli structure, generated when the vacuum’s choice of polarization breaks the dual diffeomorphisms (why X must be massive; the gravitational Higgs mechanism). The lab notebook’s addendum v0.4 states the same pair: massless X “excluded by the multi-graviton no-go”, and “the ghost-free no-gauge option is excluded by Lorentz signature”.

The metric tier misses the Fierz–Pauli weight#

The tuning test asks whether the potential that gives X its mass lands in the ghost-free family “or in the generic Boulware–Deser-ghost family” (#7 §1). Its object is the misalignment potential, built from the anticommutator {J₀, I[Z]} = [[S, A], [A, S]] with S = Y + XY⁻¹X − Y⁻¹ and A = [X, Y⁻¹], in the vacuum frame where the metric Y = Im Z of the modular field is the identity, on the Euclidean skeleton (#7 §2).

Calc note #7 §2 expands it about the vacuum [D — rigorous]: “With Y = 1 + h, X = x: S = 2h + x² + O(3), A = [h, x] + O(3). Any potential of the class Tr({J,I}²)-with-positive-norm gives, at quadratic order, m²Tr(h²) — a graviton mass without the −(Tr h)² companion. That violates Fierz–Pauli already at linear order: Boulware–Deser ghost, immediately.” Flag 4 in review/wiki-build-flags.md records a suspected missing −h² term in this expansion of S, found by executed checks of two agents (review/wiki-build-checks/s_expansion.py), and notes that the quadratic-order test, battery test T4, is unaffected, because it uses only the leading term 2h. A second natural invariant, built from [J, I] + 2K, gives X a mass “as Tr(x²) with no trace subtraction. Same disease.” #7’s conclusion: “Every metric-tier (second-order, H-coupled) misalignment invariant tested lands off the tuning”, a “theorem-grade negative”. The handoff records the first failure with its machine-check mark ✓ (HANDOFF §2); the second invariant has no battery test, so its tag stays [D]. See the metric tier fails twice.

Battery test T4 [V]#

T4 checks the first failure numerically. With the generator seeded at 0 and d = 4, verify/battery.py draws one random symmetric h and x, sets Y = 1 + εh and X = εx, computes S, and compares Tr(S²)/ε² with two forms: 4Tr(h²), and the Fierz–Pauli structure with the same normalization, 4(Tr(h²) − (Tr h)²). The normalization follows from the expansion: S begins with 2εh, the 2h of #7’s expansion scaled by ε, so Tr(S²)/ε² tends to 4Tr(h²). The current gate run prints:

      T4 eps=0.0001: 66.5772 vs 4Tr(h^2)=66.5770 vs FP=63.7221
      T4 eps=1e-05: 66.5770 vs 4Tr(h^2)=66.5770 vs FP=63.7221
PASS  T4 Tr(S^2)/eps^2 -> 4Tr(h^2) (rel 1e-5 at eps=1e-5)
PASS  T4 limit differs from Fierz-Pauli form (rel > 1%)
PASS  T4 parity: 66.58 vs 63.72 as quoted  [66.58 vs 63.72]

The limit is 4Tr(h²) = 66.5770, against the Fierz–Pauli form 63.7221. The checkpoint describes the test as “the linearized potential (Tr(S²)/ε² → 4Tr(h²) to six digits, visibly ≠ the Fierz–Pauli form)” (#11 §2) and quotes it in §7.2 as “the 66.58-vs-63.72 test of §2”. Those figures are not in #11 §2, and the script’s docstring repeats the pointer to §2; they are in the handoff, “66.58 vs 63.72 at seed 0” (HANDOFF §5), and in the script’s output, whose parity check confirms that the environment still reproduces them (conflicts between sources; see flag 36 in review/wiki-build-flags.md).

The figures belong to one seeded random h; another draw gives other numbers. What T4 confirms is the arithmetic of the expansion. That the metric-tier potential is the right object to test, and that missing the trace term means a ghost, rest on #7’s derivation and on the literature (verification checks arithmetic, not the model).

The frame tier, and X’s mass#

Built from frames instead of metrics, the misalignment potential is a wedge polynomial of the symplectic vielbein E (H = EᵀE) and the polarization frame E₀[J]. That is the Hinterbichler–Rosen class, “ghost-free by construction, Fierz–Pauli automatic” (#7 §3) — a structural claim [D — structural, resting on T] with no machine check (the frame tier). The verdict (#7 §3; #1 ‘Addendum (v0.5)’) is that the theory fails in metric variables and passes in frame variables (the verdict: a conditional pass).

X’s mass is then the Fierz–Pauli mass of the passing potential. Calc note #8 §3 [D + T] writes it after reciprocity has constrained the coefficients (the palindrome βₙ = β₄₋ₙ): “the Fierz–Pauli mass combination m_FP² ∝ β₁ + 2β₂ + β₃ = 2(β₁+β₂), i.e. M_X”. The checkpoint writes “M_X ∝ 2(β₁+β₂)” [D] (#11 §7.5): #8 states the proportionality for the squared mass, #11 for M_X, and flag 5 in review/wiki-build-flags.md suspects that #11 means M_X², a question of notation only. No battery test covers it (spectrum, parameters and reciprocity parity).

The lab notebook’s v0.1 text had claimed a different route to ghost freedom, “no second independent metric”, which addendum v0.4 reversed; that superseded argument is recorded with the v0.1 action.

References#

The program’s documents cite no source for the Fierz–Pauli term, the Boulware–Deser ghost or the signature obstruction. Added by the wiki from model memory (†):

  • M. Fierz, W. Pauli (1939)†: the mass term for free fields of spin 2 and higher.
  • D. G. Boulware, S. Deser (1972)†: the ghost of generic nonlinear massive gravity.
  • P. van Nieuwenhuizen (1973)†: ghost-free quadratic Lagrangians for a symmetric tensor.
  • K. Hinterbichler, “Theoretical aspects of massive gravity,” Rev. Mod. Phys. (2012)†; C. de Rham, “Massive gravity,” Living Rev. Relativ. (2014)†: reviews.

Used by#

Sources#

  • notes/debt1a-ii-X-classification.md §2, §3 — the signature obstruction; the five polarizations; the one known consistent structure and the Boulware–Deser criterion; the Lorentzian caveat.
  • notes/debt1a-iii-tuning-test.md §1, §2, §3, §5 — the test’s question; the missing trace term and the second invariant; Fierz–Pauli automatic in frame variables; the name “signature obstruction”.
  • papers/broken-born-checkpoint-v2.md §2, §7.1, §7.2, §7.5 — the T4 description; the exclusion of an ungauged X; the metric-tier failure and the “66.58-vs-63.72” pointer to §2, where the numbers do not appear; M_X.
  • verify/battery.py T4 — the printed comparison, 66.5770 against 63.7221, and the parity check.
  • HANDOFF.md §2, §5 — the failure marked ✓; “66.58 vs 63.72 at seed 0”.
  • notes/debt1a-iv-beta-coefficients.md §3 — the Fierz–Pauli mass combination after the palindrome.
  • notes/broken-born-theory.md ‘Addendum (v0.4)’, ‘Addendum (v0.5)’ — the two exclusions; the tuning test’s verdict as first recorded.
  • review/wiki-build-flags.md item 4 — the suspected missing −h² in #7’s expansion of S; item 5 — M_X versus M_X²; item 36 — where the figures 66.58 and 63.72 are printed. All await Paul’s ruling.
  • review/wiki-build-checks/s_expansion.py and review/wiki-build-checks/s_expansion.out — the expansion of S checked to second order.
Backward Ghosts and perturbative instabilities The signature obstruction Forward