Particle Spectrum for Any Tensor Lagrangian (PSALTer)

PSALTer logo.

Go to the GitHub repository for download.

PSALTer is designed to predict the propagating quantum particle states in any tensorial field theory, including (but not limited to) just about any theory of gravity. The free action \(S_{\text{F}}\) must have the structure \[S_{\text{F}}=\int\mathrm{d}^4x\ \zeta(x)^{\text{T}}\cdot\Big[\mathcal{O}(\partial)\cdot\zeta(x)-j(x)\Big],\] where the ingredients are:

  1. The dynamical fields \(\zeta\) are real tensors, which may be a collection of distinct fields, each field having some collection of spacetime indices, perhaps with some symmetry among the indices.
  2. The wave operator \(\mathcal{O}\) is a real, second-order (Ostrogradsky's theorem discourages higher-derivative operators, but even if it did not we note that the apparent order may always be lowered by the introduction of extra fields) differential operator constructed from the flat-space metric \(\eta\) and partial derivative \(\partial\) (but not the totally antisymmetric \(\epsilon\) tensor), linearly parameterised by a collection of coupling coefficients.
  3. The source currents \(j\) are conjugate to the fields \(\zeta\). They encode all external interactions to second order in fields, whilst keeping the external dynamics completely anonymous.

For theories of this form, the spin-projection operator (SPO) algorithm applies and the PSALTer package may be used. Of course, spectra can also be obtained for more exotic theories, but these require the algorithm to be modified beyond its minimal form.