Abstract
We derive conditions which are sufficient for theories consisting of multiple vector fields, which could also couple to non-dynamical external fields, to have the required structure of constraints of multi-field generalised Proca theories, so that the number of degrees of freedom is correct. The Faddeev–Jackiw constraint analysis is used and is cross-checked by Lagrangian constraint analysis. To ensure the theory is constraint, we impose a standard special form of Hessian matrix. The derivation benefits from the realisation that the theories are diffeomorphism invariance. The sufficient conditions obtained include a refinement of secondary-constraint enforcing relations derived previously in literature, as well as a condition which ensures that the iteration process of constraint analysis terminates. Some examples of theories are analysed to show whether they satisfy the sufficient conditions. Most notably, due to the obtained refinement on some of the conditions, some theories which are previously interpreted as being undesirable are in fact legitimate, and vice versa. This in turn affects the previous interpretations of cosmological implications which should later be reinvestigated.
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Notes
In this paper, the external fields are non-dynamical in the sense to be described in Sect. 2. The consideration of dynamics of the external fields especially gravity is not within the scope of this paper.
It is understood that LHS of Eq. (103) is actually the pullback of \(\tilde{\Omega }_\alpha \) to tangent bundle. Throughout this paper, we do not use different notations to distinguish the functions from their pullbacks as it should be clear from the context.
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Acknowledgements
We are grateful to Sheng-Lan Ko for interests and discussions. We would also like to thank Claudia de Rham for interests, discussions, and comments. Furthermore, we are grateful to anonymous referees for useful comments and suggestions which improve our manuscript. S.J. is supported by the Research Professional Development Project Under the Science Achievement Scholarship of Thailand (SAST).
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Appendices
Appendix A: Conditions from diffeomorphism invariance
In this appendix, we consider a class of theories described in Sect. 2. Since these theories are diffeomorphism invariant, their Lagrangians would satisfy the conditions to be presented in this appendix.
Under diffeomorphism \(x^\mu \mapsto x^\mu - \epsilon ^\mu (x),\) the vector fields transform as
and the external fields \(\{K\}\) transform under standard diffeomorphism. The Lagrangian density transforms as
Demanding the expression \(\delta _\epsilon \mathcal{L}- \partial _\mu (\epsilon ^\mu \mathcal{L})\) to vanish will give rise to useful conditions. In order to evaluate this expression, we begin by recall that \(\mathcal{L}= T + U_\alpha \dot{A}^\alpha _0.\) Then we consider
Next, let us consider
Combining the two expressions, we obtain
Let us also compute
and
So we have
Combining Eq. (A5) with Eq. (A8), we obtain the expression for \(\delta _\epsilon \mathcal{L}- \partial _\mu (\epsilon ^\mu \mathcal{L}).\)
We are now ready to obtain useful conditions. Let us note that the expression \(\delta _\epsilon \mathcal{L}- \partial _\mu (\epsilon ^\mu \mathcal{L})\) is a polynomial in \(\dot{A}_0^\alpha \) up to degree two. Consider the term containing \(\dot{A}_0^\alpha \dot{A}_0^\beta \) in \(\delta _\epsilon \mathcal{L}- \partial _\mu (\epsilon ^\mu \mathcal{L})\). It can easily be seen that there is only one term which is
Demanding this expression to vanish gives
Let us next turn to the coefficients of \(\dot{A}_0^\beta .\) For this, it would be convenient to consider Eqs. (A5) and (A8) and collect the terms proportional to \(\dot{A}_0^\beta .\) We have
and
Therefore, we have
Although the above equation looks complicated especially due to the explicit presence of external fields, we will only extract some parts of this equation to obtain the conditions that we will need. These conditions will look much more simple. For example, the dependence on the external fields and their derivatives are only through T and \(U_\beta .\) We may derive these conditions as follows. Taking derivative of Eq. (A13) with respect to \(\dot{A}_j^\alpha \) gives
Let us take derivative of Eq. (A13) with respect to \(\partial _j A_0^\alpha \), then swap the indices \(\alpha \) and \(\beta \), add it to the original equation, and use Eq. (A10), we obtain
Expressing in phase space, the conditions Eqs. (A14)–(A15) become
and
By substituting Eq. (A16) into Eq. (A17), we obtain
Appendix B: Expressions of \(\partial \phi _\alpha /\partial \partial _{\mathcal{I}}\dot{Q}^\beta \) in phase space
In this appendix, we outline necessary steps to express \(\partial \phi _\alpha /\partial \partial _{\mathcal{I}}\dot{Q}^\beta \) in phase space. We use the same set-up and notations as those given in Sects. 2–3. For convenient, let us denote \(P_A\) and \(\Lambda ^A\) as collective for \(\pi _\alpha ^i\) and \(\Lambda ^\alpha _i,\) respectively.
The idea is to first express \(\partial \phi _\alpha /\partial \partial _{\mathcal{I}}\dot{Q}^\beta \) in terms of \(\alpha _M\). This can be achieved by recalling from Sect. 3 the Eq. (98). Recall also that diffeomorphism invariance requirements and demanding \(\dot{\alpha }_\alpha = 0\) to not introduce further dynamics on the vector fields imply that \(\partial \alpha _\alpha /\partial \dot{Q}^\beta = 0 = \partial \alpha _\alpha /\partial \partial _i\dot{Q}^\beta \). Furthermore, due to the form of the Lagrangian of interest, we also have \(\partial \alpha _\alpha /\partial \partial _{i_1}\partial _{i_2}\cdots \partial _{i_l}\dot{Q}^\beta = 0\) for \(l\ge 2.\) So
Then since \(\phi _\alpha = \dot{\alpha }_\alpha \) we have, from Eqs. (98) and (B1)
Due to Eq. (B1), it can be seen that \(\phi _\alpha \) depend on \(\partial _{\mathcal{I}}\dot{Q}^\beta \) only through the expressions \(\partial _\mathcal{I}\dot{Q}^M\) and \(\partial _\mathcal{I}(M^{BC}\alpha _C)\) which appear in the above equation. This gives
We then need to compute each expression on RHS of Eq. (B3). For this, let us directly express \(\alpha _A\) in terms of Lagrangian then transforming to phase space, but transform \(\alpha _\alpha \) to \(-\tilde{\Omega }_\alpha \) (cf. Equation (104)). Direct calculations can be given as follows. In order to evaluate \({\partial \partial _\mathcal{J}(M^{AB}\alpha _B)}/{\partial \partial _\mathcal{I}\dot{Q}^\beta }\) we note that \(\partial M^{AB}/\partial \partial _\mathcal{I}\dot{Q}^\beta = 0\) for \(|\mathcal{I}|\ge 0\) whereas \(\alpha _B\) depends on \(\dot{Q}^\beta \) and \(\partial _i\dot{Q}^\beta \) but not on \(\partial _\mathcal{I}\dot{Q}^\beta \) where \(|\mathcal{I}|\ge 2.\) By writing \(\partial _k\alpha _B\) using chain rule and taking derivative with respect to \(\partial _\mathcal{I}\dot{Q}^\beta ,\) we obtain
This gives
Next, let us express \(\partial \alpha _B/\partial \partial _\mathcal{I}\dot{Q}^\beta \) in terms of phase space variables. The calculations will involve \(\partial (\partial _j(\partial \mathcal{L}/\partial \partial _j Q^B))/\partial \partial _\mathcal{I}\dot{Q}^\beta ,\) which can be computed by first using the chain rule for \(\partial _j\) and then taking derivative with respect to \(\partial _\mathcal{I}\dot{Q}^\beta .\) The relevant results are
So
Next, let us express \(\partial \alpha _\alpha /\partial \partial _\mathcal{I}Q^\beta \) in phase space. For this, we first use Eq. (104) to transform \(\alpha _\alpha \) to \(-\tilde{\Omega }_\alpha .\) More precisely, this is
such that \(P_B = P_B(Q^M,\partial _i Q^M,\dot{Q}^B,\{K,\partial K,\partial \partial K,\ldots \})\), in which both sides of Eq. (B11) are both functions on the tangent bundle. So when taking derivative of \(\alpha _\alpha \) with respect to \(\partial _\mathcal{I}Q^\beta ,\) we need to also take into account that \(P_B\) and \(\partial _i P_B\) also depend on \(\partial _\mathcal{I}Q^\beta .\) As part of the intermediate calculations, we need to compute \(\partial \partial _k P_B/\partial \partial _\mathcal{I}Q^\beta ,\) which can be done by first writing \(\partial _k P_B\) using chain rule, then taking derivative with respect to \(\partial _\mathcal{I}Q^\beta .\) We have
Then we use Eq. (45), which is equivalent to \(P_B = \partial \mathcal{T}/\partial \Lambda ^B.\) Keeping these in mind, we have
Finally, let us compute \(\partial \alpha _\alpha /\partial \partial _\mathcal{I}Q^\beta \). For this, as intermediate steps we compute
Then we have
Then by substituting Eqs. (B5B6)–(B7), (B9)–(B10), (B13B14)–(B15), (B17)–(B18) into Eq. (B3), we obtain
By using diffeomorphism invariance requirements, Eq. (65) is realised. This simplifies Eq. (B19). Further simplifications are possible. For this, let us note that using Eqs. (B3)–(B9) and diffeomorphism invariance requirements, one obtains
Then by expressing \(\alpha _M\) in terms of Lagrangian and using diffeomorphism invariance and secondary-constraint enforcing relations, we obtain
which is equivalent to the phase space expression
Finally, this gives
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Janaun, S., Vanichchapongjaroen, P. On sufficient conditions for degrees of freedom counting of multi-field generalised Proca theories. Gen Relativ Gravit 56, 5 (2024). https://doi.org/10.1007/s10714-023-03191-8
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DOI: https://doi.org/10.1007/s10714-023-03191-8