# Stuckelberg Formalism for Spin-2 - Metric Signature and Ghost Fields

Caution: This question may be trivial to experts, since I am looking at the consequence of metric conventions on the nature of fields in the calculation. My aim is to spot an error in either my understanding or in the calculation. Also, I've looked at other question on this forum, but they are neither specific nor pertaining to my question.

Background: I'm working out the Stuckelberg formalism for massive spin-2 fields following this paper. This paper uses the mostly plus convention $\eta_{\mu\nu} = \text{diag} (-1,1,1,1)$.

The kinetic term for spin-2 field is (eq(2.1) in the above reference) :

$$\mathcal{K} = \frac{1}{2} h^{\mu\nu} (\Box - m^2) h_{\mu\nu} - \frac{1}{2} h'(\Box - m^2)h') + \text{other terms}$$

Applying the stuckelberg transformation ($h_{\mu\nu} \to h_{\mu\nu} + \frac{1}{m} \partial_\mu \xi_\nu + \frac{1}{m}\partial_\nu\xi_\mu$) in this term, will lead to the following kinetic term for the $\xi_\mu$ field:

$$\frac{-1}{2} F_{\mu\nu}F^{\mu\nu}$$

which comes with the right sign, i.e., $\xi_\mu$ is not a ghost field. Note, the above sign is strictly fixed by the sign of the fierz-pauli mass term.

Question: If the same thing as above is done in the mostly minus metric signature, i.e. $\eta_{\mu\nu} = \text{diag} (1,-1,-1,-1)$. The kinetic term will be given by:

$$\mathcal{K} = -\frac{1}{2} h^{\mu\nu} (\Box + m^2) h_{\mu\nu} +\frac{1}{2} h'(\Box + m^2)h') + \text{other terms}$$

Applying the same field transformation will again give the same kinetic term for the $\xi_\mu$ field, since the sign is again dictated by the mass term.

$$\frac{-1}{2} F_{\mu\nu}F^{\mu\nu}$$

Now, the $\xi_\mu$ field is behaving like a ghost field (in accordance with the metric signature).

I understand, that this should not be the case? A field which is not a ghost in one convention SHOULD NOT BECOME a ghost field in another convention.

Any suggestion is welcome.

• Nope. The correct version is given by: $\mathcal{K} =\color{red}{-} \frac{1}{2} h^{\mu\nu} (\Box + m^2) h_{\mu\nu} +\cdots$. I just looked up Tong's notes for a double-check. Apr 2, 2018 at 23:34
• So, the paper you've posted uses the mostly '+' convention (footnote on pg-2), and their kinetic term is in accordance with the one - I've posted here. Additionally, one simple way to check the sign of the mass term is from the following: $$\mathcal{H} = \pi \dot{\phi} - \mathcal{L}$$ Since, the mass term does not participate in the momenta fields, it's sign is fixed irrespective of the metric signature to NEGATIVE! Hence, the issue. Apr 3, 2018 at 9:05

Since, I had to continue on with my work I've convinced myself that the fallacy in my argument presented above lied in not realizing the following:

Irrespective of metric signature, the correct sign for the kinetic term for a spin-1 field will always be negative. $$\mathcal{L}_{kinetic \ spin-1} = -1 \ \text{(normalization factor)} \ F_{\mu\nu}F^{\mu\nu}$$

This can be seen from two equivalent ways:

1. The kinetic term for the real degrees of freedom in the vector field $\xi_{i}$, get a positive sign.

2. Due to two indices, there are two metric contractions involved - which ultimately cancel their relative signs.

I find this arrangement of field theory, rather elegant!

Suggestions and comments are, as before, very welcome.