Maxwell-Proca theory: Definition and construction

Abstract

We present a systematic construction of the most general first order Lagrangian describing an arbitrary number of interacting Maxwell and Proca fields on Minkowski spacetime. To this aim, we first formalize the notion of a Proca field, in analogy to the well-known Maxwell field. Our definition allows for a nonlinear realization of the Proca mass, in the form of derivative self-interactions. Consequently, we consider so-called generalized Proca/vector Galileons. We explicitly demonstrate the ghost-freedom of this complete Maxwell-Proca theory by obtaining its constraint algebra. We find that, when multiple Proca fields are present, their interactions must fulfill nontrivial differential relations in order to ensure the propagation of the correct number of degrees of freedom. These relations had so far been overlooked, which means previous multi-Proca proposals generically contain ghosts. This is a companion paper to the paper by Diez et al. [arXiv:1905.06967]. It puts on a solid footing the theory there introduced.

Full text

Maxwell-Proca theory: Definition and construction
Verónica Errasti Díez ,*Brage Gording,†Julio A. M´endez-Zavaleta,‡and Angnis Schmidt-May§
Max-Planck-Institut für Physik (Werner-Heisenberg-Institut) Föhringer Ring 6, 80805 Munich, Germany
(Received 3 June 2019; accepted 30 August 2019; published 13 February 2020)
We present a systematic construction of the most general first order Lagrangian describing an arbitrary
number of interacting Maxwell and Proca fields on Minkowski spacetime. To this aim, we first formalize
the notion of a Proca field, in analogy to the well-known Maxwell field. Our definition allows for a
nonlinear realization of the Proca mass, in the form of derivative self-interactions. Consequently, we
consider so-called generalized Proca/vector Galileons. We explicitly demonstrate the ghost-freedom of this
complete Maxwell-Proca theory by obtaining its constraint algebra. We find that, when multiple Proca
fields are present, their interactions must fulfill nontrivial differential relations in order to ensure the
propagation of the correct number of degrees of freedom. These relations had so far been overlooked,
which means previous multi-Proca proposals generically contain ghosts. This is a companion paper to the
paper by Diez et al. [arXiv:1905.06967]. It puts on a solid footing the theory there introduced.
DOI: 10.1103/PhysRevD.101.045009
I. INTRODUCTION: A BRIEF HISTORY
OF BEYOND ELECTRODYNAMICS
In the second half of the 19th century, the foundations of
classical electromagnetism were laid. The Lorentz force
law, together with Maxwell’s equations, provided a unified
framework to explain the origin and propagation of electric
and magnetic fields. Light was then understood to be an
electromagnetic phenomenon. This body of work has since
become one of the cornerstones of theoretical physics. It is
therefore not surprising that ongoing efforts are made so as
to generalize Maxwell’s theory. In the following, we
discuss the subset of extensions that provides the relevant
ingredients to our new proposal: the Maxwell-Proca theory.
(If the reader arrives from [1], we now elaborate on the
spin-one void noted in that introduction.)
Early extensions. As a field theory, Maxwell electrody-
namics describes an Abelian massless vector field and its
linear interactions with sources. During the first half of the
20th century, extensions to this setup were introduced in
order to formalize new observations. For instance, consider
nuclear phenomena. As pointed out by Yukawa [2], the
forces there implicated do not obey a Coulomb law, but
show a faster decay over distance. This insight was in good
agreement with the theory of a massive vector field intro-
duced by Proca [3]. In his paper, Proca proposed a relativistic
massive wave equation for a vector field (analogous to the
Klein-Gordon equation [4]). Subsequent work [5] refined the
implications of his proposal: it was to be understood as a
model of a massive spin-one force carrier, i.e., a massive
photon. Of course, the phenomenologies of the Maxwell and
Proca fields are different. For example, the Proca field
explicitly breaks the gauge symmetry and thus propagates
one more degree of freedom (d.o.f.) than the Maxwell field.
Alternatively, this helicity-zero mode can be excited
without spoiling gauge invariance. Shortly after Proca’s
work, Bopp [6] and Podolsky [7] presented a model of a
Uð1Þ-invariant Abelian vector field with a modified kinetic
structure. Such novel electrodynamics (and generalizations
[8]) was introduced to amend some of the theoretical
concerns at the time. Specifically, this was a consistent
proposal describing short distance interactions and thus
resolving the self-energy problem—a neat review of which
can be found in [9]. Even though the proposed Lagrangians
contain higher order self-derivative interactions, the equa-
tions of motion remain second order.1However, some other
peculiar characteristics appear, such as nonlocality.
Many other theories of relativistic vector fields emerged.
A prominent representative is the theory in which the
Maxwell kinetic term is promoted to an arbitrary smooth
function of itself. Concretely, an early proposal of this kind
was Born-Infeld electrodynamics [11]. In this model, the
*veroerdi@mppmu.mpg.de
†brageg@mppmu.mpg.de
‡julioamz@mpp.mpg.de
§angnissm@mpp.mpg.de
Published by the American Physical Society under the terms of
the Creative Commons Attribution 4.0 International license.
Further distribution of this work must maintain attribution to
the author(s) and the published article’s title, journal citation,
and DOI. Funded by SCOAP3.
1Usually, higher order field equations are avoided by restrict-
ing to first order Lagrangians. This straightforwardly prevents the
undesired Ostrogradsky instability [10].
PHYSICAL REVIEW D 101, 045009 (2020)
2470-0010=2020=101(4)=045009(17) 045009-1 Published by the American Physical Society

gauge symmetry is preserved and the equations of motion
are again second order. Besides, Born-Infeld theory (and
closely related constructions like the Heisenberg-Euler
model [12]) manages to resolve the infinities in the self-
energy of the electron. This is achieved at a price, though:
the field equations become nonlinear. In fact, all theories
sharing such properties are commonly labeled nonlinear
electrodynamics [13]. We stress that mass, derivative self-
interactions and nonlinearity are constituents of the
Maxwell-Proca theory here constructed.
Modern proposals. In recent times, attempts have been
made in pursuit of the most general theory of a massive
vector field whose self-interactions extend the standard
Proca mass term. Such theories are collectively known as
generalized Proca or vector Galileon—anamethatwillbe
clarified later. This picture was first suggested in [14].The
approach is based on the inclusion of derivative self-
interactions, carefully chosen so as to not excite further
modes than the three expected in four spacetime dimensions.
Throughout the paper, we refer to any further d.o.f. carried
by the action as a ghost. The motivation behind these models
goes beyond extending previous electromagneticlike theo-
ries: when considered over a cosmological background,
generalized Proca fields can give rise to self-acceleration
and fit data from late-time observations [15]. Evidently, a
coupling to gravity is required for such applications. Even
though this coupling is already used in the initial references
[14], a rigorous counting of d.o.f. is not carried out there.
This issue has been further investigated [16,17], but still
much is to be done in this direction.2
In closer detail, the above mentioned theories are defined
by the Lagrangian density of an Abelian spin-one field that
lives on Minkowski spacetime. Schematically, the
Lagrangian can be split into two parts,
L¼Lð0Þþb
L:ð1Þ
The first piece Lð0Þis an arbitrary smooth function built
from the field itself and its (dual) field strength. As such, it
generalizes both nonlinear electrodynamics and the original
Proca theory. On the other hand, the second piece b
Ladmits
a closed form and contains self-interactions composed of
contractions between the field and its derivatives. Contrary
to Lð0Þ, the derivatives within b
Ldo not come as field
strengths exclusively, but they are such that the Lagrangian
remains first order. In fact, it has been the work of a couple
of years to track down the complete set of interactions
comprising the so-claimed most general b
L[14,19,20]. The
interested reader can find thorough reviews on these
theories in [21].
The generalized Proca theories introduced so far share a
characteristic that has been exploited very actively. When
inspecting the longitudinal mode’s dynamics in the decou-
pling limit, one finds that it matches the acclaimed scalar
Galileon. Based on this feature, generalized Proca theories
are interchangeably labeled vector Galileons. First intro-
duced in [22], Galileon fields also arise in the context of
modified gravity, for instance in models that give an
effective mass to the graviton [23]. Both scalar and vector
Galileons are close relatives of Horndeski’s construction.
The latter gives the most general theory of a self-gravitating
scalar [24] or vector (gauge) field [25]. Indeed, when
Galileon-like theories are promoted to evolve in the presence
of gravity, they must couple to the curvature according to
Horndeski’s prescription. Conversely, Galileons can be
understood as the zero-curvature limit of a Horndeski theory.
Finally, theories of multiple spin-one fields have also
been investigated, albeit they have not received that much
attention. Interacting Maxwell fields were alluded to in the
search of one-form Galileons in [26].In[27,28], theories
that comprise several copies of generalized Proca fields
were introduced. There, a global rotational symmetry was
imposed on the field space, which drastically reduced the
allowed interactions. The motivation behind this restriction
was to provide an innate source capable of supporting
isotropic cosmologies. To our knowledge, there are no
models incorporating couplings between massless and mas-
sive vector fields.
Summary of results. In this work, we focus on ghost-free
interactions of multiple real Abelian fields on Minkowski
spacetime. Loosely speaking, these can be either massless
(Maxwell) or massive (Proca). We formalize both notions
by providing their canonical definitions in terms of
Hamiltonian constraints. Then, we present a bottom-up
construction of the most general first order Lagrangian
involving such fields. Our result, the Maxwell-Proca
theory, is axiomatically complete. It naturally contains
Maxwell electromagnetism, nonlinear electrodynamics
and Proca’s theory, together with its modern generaliza-
tions, as particular subcases. The general structure of the
Maxwell-Proca interactions is similar to the generalized
Proca action in (1), but it is subject to highly restrictive
differential conditions on the form of b
Lwhen multiple
Proca fields are involved. The origin of these conditionslies
at the secondary level of the Lagrangian constraint analysis,
which we explain in detail.
Organization of the paper. In Sec. II, we introduce the
mathematical formalism behind our setup. All information
here presented is based on the Lagrangian-Hamiltonian
constraint analysis in the Appendix. In Sec. III, we system-
atically generate the exhaustive set of ghost-free interactions.
2There is a wider collection of papers related to vector
Galileons in the presence of gravity. Their aim is not to prove
ghost-freedom. Instead, submodels are taken as a source to tackle
specific goals, such as finding black hole or star configurations
[18]. While the models that deal with a single vector field and
yield second order equations of motion are automatically ghost-
free, we highlight both the pertinence and nontriviality of the
pending proof of ghost-freedom in the remaining proposals.
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We continue in Sec. IV, discussing direct strategies to
enhance the generality of the Maxwell-Proca theory. At last,
in Sec. V, we summarize the key ideas behind the technical
details of our construction.
Conventions. We work on d-dimensional Minkowski
spacetime with the mostly positive metric signature
ð−;þ;;þÞ. Spacetime indices are denoted by the
Greek letters ðμ;ν;…Þand raised/lowered with ημν ¼
diagð−1;1;…;1Þand its inverse ημν. Latin indices
ði; j; …Þdesignate spatial coordinates and are trivially
raised/lowered. The alphabets ðα;β;…Þlabel massive fields,
while their barred counterparts ð¯
α;¯
β;…Þlabel massless
fields. Capital alphabets ðA; B; …Þand ðA1;A
2;…Þenu-
merate Abelian fields, massive and massless alike. All field
labels are trivially raised/lowered. Einstein summation con-
vention is to be understood throughout.
II. AXIOMATIZATION
The Maxwell and Proca actions are the elemental theories
of Abelian vector fields: a massless one in the former case and
a massive one in the latter. In d-dimensional Minkowski
spacetime and free of sources, the Lagrangian densities of
these theories have the form,
LM¼−
1
4AμνAμν ;LP¼−
1
4BμνBμν −
1
2m2BμBμ;ð2Þ
respectively. We take both fields to be real. Here, Aμν ¼
∂μAν−∂νAμis the field strength of the Maxwell field.
Similarly, Bμν is that of the Proca field and the constant m≠0
parametrizes its mass. We begin by convening the essentials
of these theories, since they serve as building blocks for the
more elaborate Maxwell-Proca theory we propose.
Maxwell electrodynamics is manifestly invariant under
the Uð1Þgauge transformation of the second kind,
Aμ→Aμþ∂μΦ;ð3Þ
where Φis an arbitrary real scalar field. This means that Aμ
transforms as a connection, in the affine representation of
the gauge group. We highlight that both the Lagrangian (off
shell) and the field equations (on shell) are invariant under
(3). The explicit mass term that tells apart both Lagrangians
in (2) has the crucial effect of spoiling such gauge
symmetry for the Proca field.
One of the central points throughout the paper is the
count of the d.o.f. propagated by a given Lagrangian, as a
means to ensure the absence of ghosts. It is well known that
LMpropagates d−2whereas LPpropagates d−1d.o.f. In
four spacetime dimensions, Maxwell’s theory accounts for
two transversal polarization states, while the Proca field
incorporates an additional longitudinal mode.
Both LMand LPare singular. In other words, the
determinant of their primary Hessians,
WðMÞ
μν ≔∂2LM
∂_
Aμ∂_
Aν;W
ðPÞ
μν ≔∂2LP
∂_
Bμ∂_
Bν;ð4Þ
vanishes. Here, _
Aμ≔∂0Aμstands for the velocity of the
massless field while _
Bμcorresponds to that of the massive
field. As a consequence, some of the Euler-Lagrange
equations following from (2) are first order. Namely, they
are not equations of motion—in the sense that they do not
involve accelerations of the fields—but relations between
the field variables and their velocities. These relations are
the primary Lagrangian constraints3of the system (if
tautologically true, they can be further distinguished as
identities). They are responsible for the mismatch between
the number of a priori d.o.f. and those that are physically
meaningful.
Given any singular Lagrangian, the existence of primary
constraints φ¼0that are not identically zero starts a
renowned iterative procedure to obtain the remaining
constraints. This algorithm is suitably referenced and
self-consistently reviewed in the Appendix Sec. I. The gist
of it is as follows. The demand that the primary constraints
be stable in time _
φ¼
!0yields the so-called secondary Euler-
Lagrange equations. Those which are independent of
accelerations, if any, are the secondary Lagrangian con-
straints ϕ¼0. When extant, the stability of those secon-
dary constraints that are not identically vanishing is not
guaranteed. Rather, requiring _
ϕ¼
!0leads to the third
iteration. The procedure terminates when no Lagrangian
constraints appear or when they are all identically zero,
since then their stability need not be assured.
Going back to the Maxwell and Proca theories in (2), the
just described constraint analysis tells us that Maxwell
electrodynamics is endowed with a single primary
(Bianchi) identity, while its massive counterpart contains
two Lagrangian constraints: one primary and one secon-
dary. Furthermore, the generator of the gauge symmetry
inherent to Maxwell’s theory can be regarded as an addi-
tional primary identity.
Knowledge of the identities and Lagrangian constraints
in a singular theory can be used to determine the d.o.f. it
propagates. Most often, this is done by means of the
Hamiltonian formula proposed by Dirac long ago,
ndof ¼M0−N1−N2
2;ð5Þ
where M0is the number of field variables and ðN1;N
2Þ
denote the number of first and second class constraints,
respectively. Recall that a first class constraint is one
that yields a vanishing Poisson bracket with all of the
3This discussion applies to field theories with irreducible
constraints. All LM,LPand the later introduced Maxwell-Proca
theory pertain to this class. The case involving reducible con-
straints is briefly reviewed in the Appendix Sec. I.
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constraints in the theory, whereas a second class constraint
does not. It is worth noting that in [29] an alternative
expression to (5) was developed for particle systems, which
depends solely on Lagrangian quantities, including the off
shell gauge generators. The result was later on adapted to
field theories in [30].
Indeed, Appendix Sec. II is devoted to the exposition of a
long established systematic procedure to calculate N1and
N2from the Lagrangian constraints. As is widely known,
the Bianchi and gauge identities in Maxwell’s theory
generate two first class constraints. On the Proca side,
the Lagrangian constraints translate to two second class
constraints. A quick inspection of (5) then yields the
anticipated number of d.o.f.
Considering all the above exposed, we introduce the
following two definitions:
Definition 1. A Maxwell field is any real Abelian vector
field whose Lagrangian is associated with two first class
constraints.
Definition 2. A Proca field is any real Abelian vector
field whose Lagrangian is associated with two second class
constraints.
From this point of view, the distinction between a Proca
field [3,5] and a generalized Proca field [14,19,20] is
insubstantial. Subsequently, we do not discern between
the two.
Notice that, in the above definitions, the Poisson bracket
is the relevant operation: it distinguishes between first and
second class constraints and thus defines the constraint
algebra. Bilinearity of the Poisson bracket implies linearity
of the constraint algebra. Consequently, a direct sum of
whatever number of copies of the Maxwell and Proca
Lagrangians will preserve the constraint patterns in
Definitions 1 and 2. This is generically not true when
there is an explicit interaction involving (some of) the Proca
fields. Specifically, we prove in the Appendix Sec. Ithat
coupling massive vectors alters the stability of their primary
Lagrangian constraints. What happens is that, at the
secondary level, the constraints ϕ¼0are absent.
Instead, we get independent Euler-Lagrange equations
responsible for exciting unwanted ghostlike d.o.f.
One of our key results lies in showing that one can
always find befitting interactions that prevent the afore-
mentioned deviation from the Proca constraint scheme.
This is uniquely achieved by setting the secondary Hessian
to zero. In more detail, consider some Lagrangian density
L0encoding the interaction of a set of MProca fields fBμg.
The antisymmetric matrix,
e
Wαβ ≔∂2L0
∂_
BðαÞ
0∂BðβÞ
0
−∂2L0
∂_
BðβÞ
0∂BðαÞ
0
;ð6Þ
conforms the nonvanishing part of the secondary Hessian.
Enforcing
e
Wαβ¼
!0∀α;β¼1;…;M; ð7Þ
induces the appearance of the secondary constraints that
restrain the ghost. We stress that only when (7) is fulfilled,
can the vectors fBμgcomposing L0be identified as Proca
fields in the sense of Definition 2. This prescription has
been irreconcilably overlooked in the existing literature
[27,28]. As a consequence, the models there submitted are
ghost-full.
The preceding observations motivate the proposition of
our main result:
Definition 3. The Maxwell-Proca theory is the most
general Lagrangian of Nnumber of Maxwell fields and M
number of Proca fields, such that
(i) The theory is defined over four-dimensional Min-
kowski spacetime.
(ii) The Lagrangian is at most first order.
By a first order Lagrangian density we mean
L¼LðAμ;B
μ;∂μAν;∂μBνÞ;ð8Þ
up to boundary terms. Then, the variational principle will
produce at most second order field equations. This is a
sufficient condition to safeguard our theory from
Ostrogradsky-like instabilities, as already noted in footnote
1. When a single Proca field is considered (i.e., M¼1),
this is the necessary and sufficient condition.
The most concise form of the Maxwell-Proca theory is
LMP ¼Lkin þLint;ð9Þ
where the kinetic part can always be expressed as
Lkin ¼−
1
4Að¯
αÞ
μν Aμν
ð¯
αÞ−
1
4BðαÞ
μν Bμν
ðαÞ;ð10Þ
and the interaction part encompasses every possible Lorentz
scalar built out of the fields and their first derivatives, while
respecting the defining axioms. We identify three sectors:
massless-massless, massless-massive and the massive-
massive. Explicitly,
Lint ¼LðAAÞþLðABÞþLðBBÞ:ð11Þ
We reassert that the Maxwell-Proca theory, in its final form of
Sec. III C,isonlyghost-freewhenregardedasawhole
together with the condition that its secondary Hessian
vanishes. Whilst the massless interactions LðAAÞtrivially
fulfill this condition, LðABÞand LðBBÞdo not: they are severely
restricted by it.
III. CONSTRUCTION OF THE GHOST-FREE
INTERACTIONS
In this section, we systematically construct the inter-
actions (11) building on the definitions given in the
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previous section. We first review the necessary and suffi-
cient conditions to ensure the existence of the suitable
constraints at the primary level. Albeit slightly more
involved, this construction is very similar to that for a
single Proca field [14]. It relies on the antisymmetric
structures first recognized in [26] as a necessary and
exhaustive tool for developing healthy Lorentz invariant
theories. After implementing consistency at the primary
level, we present the highly nontrivial conditions that arise
at the secondary level of the constraint algebra. We discuss
how these conditions notably restrict multi-Proca inter-
actions and apply them to generate ghost-free interactions
in several examples.
A. Primary constraints
For our present purposes, it is useful to introduce a
collective notation for all Abelian vector fields XðAÞ
μ, where
Aranges from 1 to MþN. This means that, for the
moment, we set aside the issue of gauge invariance.
As already stated, healthiness of the Maxwell-Proca
theory necessarily relies on a suitable rank reduction of its
primary Hessian. This is uniquely4achieved by
∂2LMP
∂_
XðAÞ
0∂_
XðBÞ
μ
¼
!0;∀A; B: ð12Þ
Notice that (12) must hold for all μ. This is true also for the
single Proca limit (N¼0and M¼1), as described, for
instance, at the very beginning of p. 3 in the review paper
[20]. We refer to the above as the primary constraint
enforcing relations, which are not to be confused with the
primary constraints themselves. Together with the demand
for Lorentz invariance, they will serve as a guiding
principle to forge the interactions. They are explicitly
derived in the Appendix. Specifically, they are equivalent
to (A7).
Next, recall that the Maxwell-Proca theory is of the form
(9). The kinetic part Lkin given in (10) automatically
satisfies the relations in (12). We thus focus on the
interactions Lint. In order to fulfill (12), they can have at
most a linear dependence on all the velocities associated
with temporal components _
XðAÞ
0. It is then natural to split
Lint into terms Lð0Þwhich do not depend on _
XðAÞ
0and terms
b
Lint which carry a linear dependence on these velocities,
Lint ¼Lð0Þþb
Lint:ð13Þ
By definition, Lð0Þis an arbitrary real smooth function of
the fields and their field strengths,
Lð0Þ¼Lð0ÞðXðAÞ
μ;X
ðAÞ
μν Þ:ð14Þ
Due to these loose restrictions, it contains infinitely
many terms.
We now turn to the composition of b
Lint. For a first order
Lagrangian and under the assumptions of smoothness and
reality, b
Lint can be formally expanded as
b
Lint ¼X
∞
n¼1
LðnÞ;ð15Þ
with
LðnÞ¼Tν1νnρ1ρn
A1An
∂ν1XðA1Þ
ρ1∂νnXðAnÞ
ρn:ð16Þ
Here, ncounts the number of derivatives and ðA1;A
2;…Þ
label all Abelian fields, as ðA; B; …Þbefore. The expansion
coefficients Tν1…ρn
A1…Anhave an arbitrary dependence on the
fields but contain no derivatives. Note that, if all derivatives
appear only in field strengths, the corresponding term
belongs in Lð0Þby definition. Observe that the exchange
of any triple ðAi;νi;ρiÞ↔ðAj;νj;ρjÞleaves (16) invariant.
Hence, without loss of generality, we take the expansion
coefficients to be symmetric under this replacement.
In order for b
Lint to satisfy (12), it is necessary and
sufficient that
∂b
Lint
∂_
XðAÞ
0
¼X
∞
n¼1
∂LðnÞ
∂_
XðAÞ
0
ð17Þ
does not contain any time derivatives. For Lð1Þthis is
automatically true, while for
∂LðnÞ
∂_
XðAÞ
0
¼nT0ν2…νn0ρ2…ρn
AA2…An
∂ν2XðA2Þ
ρ2…∂νnXðAnÞ
ρnð18Þ
with n>1, it must be enforced. The only Lorentz covariant
way to do so is to entertain three distinct antisymmetriza-
tions in (16):
(i) Antisymmetrization of all nderivatives with
each other.
(ii) Antisymmetrization of all but one derivatives with
each other and with all vector field indices. Then, the
respective n−1fields appear only inside field
strengths, while the derivative on the nth field can
be contracted in an arbitrary manner.
(iii) Antisymmetrization of k<nderivatives with l<n
fields, together with the antisymmetrization of the
remaining n−kderivatives with the left-over n−l
fields.
All these options restrict but do not fully determine the
coefficients Tν1…ρn
A1…An. Hence, the interactions contain free
tensorial functions.
On four-dimensional Minkowski spacetime, the anti-
symmetrizations are uniquely accomplished using the
4For a first order theory, the rank of the Hessian can only be
reduced by setting to zero entire rows. Requiring linear combi-
nations of rows to vanish amounts to a field redefinition.
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totally antisymmetric Levi-Civita tensor εμνρσ. The above
considerations show that all Lðn>1Þmust contain at least
one Levi-Civita tensor. There is another rather drastic
consequence: since we always need to antisymmetrize at
least nindices with each other, the in principle infinite
series expansion (15) truncates at n¼4in four spacetime
dimensions. Thus, the full interaction Lagrangian reads
Lint ¼Lð0ÞþX
4
n¼1
Tν1νnρ1ρn
A1An
∂ν1XðA1Þ
ρ1∂νnXðAnÞ
ρn:ð19Þ
For concreteness, we explicitly write out all independent
Lorentz contractions that respect the primary constraint
enforcing relation.
As stated earlier, for Lð1Þ, there is no condition on the
expansion coefficients,
Tνρ
A¼FAνρ;ð20Þ
where henceforth the spacetime tensors fFgare arbitrary
smooth real functions5. Generically, they do not possess
any symmetry properties, and they are built from the fields
XðAÞ
μand the Lorentz invariant objects ημν and εμνρσ. The
allowed coefficients in Lð2Þare of the form,
Tν1ν2ρ1ρ2
A1A2¼εν1ν2μ1μ2FA1A2
μ1μ2ρ1ρ2þερ1ρ2ν2μFð1Þ
A1A2
μν1
þεν1ρ2μ1μ2εν2ρ1σ1σ2Fð2Þ
A1A2
μ1μ2σ1σ2;ð21Þ
where each line corresponds to the cases i–iii, respectively.
For Lð3Þwe have
Tν1ν2ν3ρ1ρ2ρ3
A1A2A3¼εν1ν2ν3μFA1A2A3
μρ1ρ2ρ3
þεν1ν3μ1μ2εν2ρ1ρ2ρ3Fð1Þ
A1A2A3
μ1μ2
þεν1ν3ρ1μεν2ρ2ρ3σFð2Þ
A1A2A3
μσ
þεν1ν3ρ2μεν2ρ1ρ3σFð3Þ
A1A2A3
μσ
þεν1ν3ρ1ρ2εν2ρ3σ1σ2Fð4Þ
A1A2A3
σ1σ2:ð22Þ
Here, the first term implements case i. Case ii cannot be
realized for n>2in four dimensions. The remaining terms
all belong to and exhaust case iii. Finally, in Lð4Þ,
Tν1ν2ν3ν4ρ1ρ2ρ3ρ4
A1A2A3A4¼εν1ν2ν3ν4FA1A2A3A4
ρ1ρ2ρ3ρ4
þεν1ν3ν4ρ1εν2ρ2ρ3ρ4Fð1Þ
A1A2A3A4
þεν1ν3ν4ρ2εν2ρ1ρ3ρ4Fð2Þ
A1A2A3A4
þεν1ν3ρ1ρ2εν2ν4ρ3ρ4Fð3Þ
A1A2A3A4
þεν1ν3ρ2ρ4εν2ν4ρ1ρ3Fð4Þ
A1A2A3A4;ð23Þ
where the first term is associated to case i while the
remaining lines fully cover case iii.
To summarize, (19)–(23) define the complete set of
interactions that satisfy the primary constraint enforcing
relations.
B. Secondary constraints
Our most crucial result, arising from the constraint
analysis outlined in the Appendix Sec. I, is the unveiling
of a set of secondary constraint enforcing relations.
Namely, the axiomatically consistent closure of the con-
straint algebra requires the vanishing of the secondary
Hessian,
e
WAB ≔∂2Lint
∂_
XðAÞ
0∂XðBÞ
0
−∂2Lint
∂_
XðBÞ
0∂XðAÞ
0
¼
!0∀A; B; ð24Þ
with Lint as in (13). Together with (12), the above gives the
necessary and sufficient conditions to avoid the propaga-
tion of ghosts. Then, the algebra closes automatically at the
tertiary level.
The differential relations (24) further restrict the internal
structure of the interactions constructed in the previous
section, which will be shortly exemplified.
C. Gauge invariance
We now identify each Abelian vector XðAÞwith either a
Maxwell or a Proca field. Then, the interactions (19)–(23)
can be classified as in (11):LðAAÞdescribing purely
massless interactions, LðBBÞfor purely massive ones and
LðABÞcontaining the interactions between massless and
massive fields. In doing so, we need to respect gauge
invariance.
All NMaxwell fields Að¯
αÞ
μtransform as connections
under their own Uð1Þgauge group, as in (3). It has long
been established that consequently the Maxwell vectors can
only show up inside their field strengths. As a result, the
purely massless interactions are a subset of the general
zeroth interactions: LðAAÞ⊆Lð0Þ.
The MProca fields BðαÞ
μ, on the other hand, are gauge
invariant by definition. Therefore, the purely massive sector
LðBBÞis straightforwardly obtained by replacing the fields
XðAÞwith Proca fields,
5They need not be tensors with respect to the field labels. For
example, FAνρ ¼ηνρ for all Ais a valid choice for (20).
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LðBBÞ¼X
4
n¼0
LðBBÞ
ðnÞ;LðBBÞ
ðnÞ≔LðnÞXðAÞ→BðαÞ:ð25Þ
The coupling LðABÞbetween massless and massive fields
is obtained in a similar way: replacing XðAÞwith either Að¯
αÞ
or BðαÞ, with the Maxwell fields appearing only inside field
strengths. In particular, this means that the functions F
depend exclusively on the Proca fields.
In terms of Maxwell and Proca fields, the secondary
Hessian takes the form,
e
WAB ¼e
W¯
α¯
βe
W¯
αβ
e
Wα¯
βe
Wαβ ≡00
0e
Wαβ ;ð26Þ
since Maxwell fields, which appear only inside field
strengths, automatically satisfy (24). For the same reason,
the interactions Lð0Þare not restricted at the secondary
level. In other words, (24) solely constrains the n>0terms
in LðABÞand LðBBÞby
∂2b
Lint
∂_
BðαÞ
0∂BðβÞ
0
−∂2b
Lint
∂_
BðβÞ
0∂BðαÞ
0
¼
!0∀α;β:ð27Þ
Notice that only when two or more Proca fields are
coupled, does (27) become nontrivial. The derivation of
the above relations can be found in the Appendix.
Concretely, (27) is a rewriting of (A20). The nature and
relevance of these relations can be grasped as follows.
Equation (12) enforces the existence of a set of suitable
(primary) constraints. Then, Eq. (27) guarantees their
stability. The (secondary) constraints that follow from
Eq. (27) hold true at all times; i.e., they are automatically
stable. Therefore, no further constraint enforcing rela-
tions apply.
The internal structure of the expansion coefficients in
(20)–(23) is so rich that implementing (27) in full generality
seems intractable, because these conditions form a set of
coupled non-linear partial differential equations. Hence, we
do not incorporate these relations into a building principle
like we did at the primary level. Instead, we provide a few
examples of how to use them to extract ghost-free
interactions.
D. Explicit examples
First, consider LðBBÞ
ð1Þ, which remained unconstrained at
the primary level. It can be written in the form,
LðBBÞ
ð1Þ¼Fαμν∂μBðαÞ
ν;ð28Þ
where the coefficient is an arbitrary function of the Proca
fields obeying Fαμν ¼Fανμ, so as to exclude terms that
belong in LðBBÞ
ð0Þ. Even this simple interaction is not
guaranteed to be free of ghosts in a multi-Proca setup
(i.e., M>1). The constraint enforcing relations (27)
reduce to
∂Fα
00
∂BðβÞ
0
−
∂Fβ
00
∂BðαÞ
0
¼
!0∀α;β:ð29Þ
There are two straightforward solutions to this equation.
The functions Fαμν can be constants or they can depend
exclusively on the field with label α:Fαμν ¼FαμνðBðαÞÞ
for all α. In the first case, the interaction becomes a
boundary term. In the second case, we are left with M
decoupled Proca theories. While it is difficult to find the
most general form of Fαμν, one can always construct
particular solutions. A nontrivial solution fulfilling (29) is
given in [1].
Next, we take an example from [27]. In our notation,
their term reads
LðBBÞ
ð2-iÞ¼ðερ1ρ2μ1μ2ησ1σ2þεσ1σ2μ1μ2ηρ1ρ2ÞBðαÞ
σ1BðβÞ
σ2
×εμ1μ2
μ3μ4∂μ3BðαÞ
ρ1∂μ4BðβÞ
ρ2ð30Þ
and belongs in LðBBÞ
ð2Þ. For this interaction (which passes the
primary level by completely antisymmetrizing the deriva-
tives), the secondary level (27) requires
∂iðB½ðαÞ
0BðβÞ
iÞ¼
!0∀α;β:ð31Þ
Clearly, this condition is not satisfied for generic field
configurations. The interaction term spoils the closure of
the constraint algebra and propagates additional d.o.f. We
conclude that (30) and various other interactions proposed
in [27,28] carry ghosts.
We finish by providing a few examples of ghost-free
interactions. Any Lagrangian compatible with the primary
constraint enforcing relations (12) that contains a single
Proca field (M¼1) immediately passes the secondary
level as well. This is true for any number Nof Maxwell
fields. Accordingly, a large number of healthy terms can
directly be obtained from (19)–(23). A purely multi-Proca
example (M>1,N¼0)is
LðBBÞ
ð2-iiÞ¼Fαβεμ1μ2μ3μ4ερ1ρ2μ3μ4∂μ1BðαÞ
ρ1∂μ2BðβÞ
ρ2;ð32Þ
which belongs in LðBBÞ
ð2Þand has
Fαβ ¼δαβBðγÞ·BðγÞþ2BðαÞ·BðβÞ:ð33Þ
An LðABÞ
ð2Þexample with M,N>1is given by
LðABÞ
ð2-iÞ¼Fα¯
βμ1μ2ερ1ρ2ν2σεσμ1μ2
ν1∂ν1BðαÞ
ρ1Að¯
βÞ
ν2ρ2;ð34Þ
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where the expansion coefficients are set to
Fα¯
βμ1μ2¼K½α1α2
¯
βBμ1
ðα1ÞBμ2
ðα2Þfα:ð35Þ
Here, the K½α1α2
¯
βare arbitrary constants and
fα¼BðγÞ·BðγÞþ2BðαÞ·X
M
γ¼1
BðγÞ∀α:ð36Þ
It is easy to check that the interactions (32) and (34) satisfy
(27) and thus do not propagate any ghostlike d.o.f. We
emphasize that this holds true because (33) and (36) have
been tailored to be in agreement with the secondary
constraint enforcing relation.
In closing this section we recall that, for interactions
involving more than one Proca field, one must always
ensure that (27) is satisfied. This imposes highly nontrivial
restrictions on the interactions.
IV. RELAXING THE AXIOMS: BEYOND
MAXWELL-PROCA
The completeness of the just proposed Maxwell-Proca
theory rests on the four axioms encoded in Definitions 1–3
of Sec. II:
(i) All vector fields are Abelian.
(ii) All vector fields are taken to be real.
(iii) The Lagrangian is at most first order.
(iv) We work on Minkowski spacetime.
To conclude, we comment on the scenarios that unfold
upon concrete relaxations of the assumed postulates. In all
cases, the materialization of our subsequent proposals into
an exhaustive Lagrangian is nontrivial and lies beyond the
scope of this paper.
Generalization to non-Abelian fields. As is widely
known, classical Yang-Mills theory [31] can be regarded
as an extension of Maxwell electrodynamics to non-
Abelian vector fields. Conversely, electromagnetism can
be understood as a Yang-Mills theory with gauge group
Uð1Þ. This relationship has already been exploited to
construct generalized Yang-Mills theories starting from
specific nonlinear electrodynamics [32]. On the other hand,
non-Abelian extensions of the Proca field are not possible.
Indeed and as already noted in [21,27], attempts at
constructing a massive version of the Yang-Mills theory
boil down to the consideration of multiple interacting
Abelian Proca fields in the presence of a global rotational
symmetry in the field space. Putting together both obser-
vations, the non-Abelian analogue to the Maxwell-Proca
theory could be dubbed Yang-Mills-Proca. Notice that the
constraint enforcing relations (12) and (27), as well as the
purely massive interactions LðBBÞin (11), are the same as in
the Maxwell-Proca theory.
Generalization to complex fields. While the extension of
the massless sector to the complex field case is straightfor-
ward, the study of a complexified massive sector is not. For
example, allowing for Proca fields to be complex makes it
possible to charge them under the fundamental representa-
tion of the Uð1Þgauge group of any of the Maxwell fields,
Bμ→eieΦBμ;B

μ→B
μe−ieΦ;ð37Þ
with ea coupling constant and Φthe same real scalar field
that appeared in (3) before. Then, one can generate new
interaction terms by replacing partial derivatives ∂μacting
on the Proca fields by covariant derivatives Dμ≔∂μþ
ieAμ, as long as the resulting term is invariant under the
simultaneous transformations (3) and (37). Observe that
such interactions imply the Maxwell fields are no longer
restricted to appear as part of a field strength. This
generalization does not affect the form of the constraint
enforcing relations (12) and (27).
Generalization to higher order field theory. If the
Maxwell-Proca theory is promoted to depend on arbitrarily
high order time derivatives of the fields, i.e.,
LMP →LMPðAμ;B
μ;_
Aμ;_
Bμ;
̈
Aμ;
̈
Bμ;…Þ;ð38Þ
the Lagrangian and Hamiltonian formalisms underlying the
Appendix no longer constitute the suitable framework to
study the constraint algebra. Indeed, the development of an
appropriate formalism has been investigated for a long time
and continues to be pursued; see e.g., [33]. The foundations
of a consistent approach amenable to geometrization were
laid in [34], based on the notions put forward in [35] for
second order theories. The main idea consists on introduc-
ing so-called intermediate spaces that connect the tangent
bundles T2k−1C, natural to the Lagrangian analysis, to the
cotangent bundles TðTk−1CÞ, which appear on the
Hamiltonian side. Here, k¼1;…;n for a nth order
Lagrangian and Cis the configuration space. The inter-
mediate spaces sustain a systematic inspection of the higher
velocities and a step-by-step inference of the corresponding
momenta. Moreover, iterative algorithms that ensure the
stability of the constraints in the theory are naturally
defined in these spaces, under mild requirements on the
form of (38). Obviously, the constraint enforcing relations
(12) and (27) do not apply to higher order cases.
Generalization to curved backgrounds. Relaxing our
last Axiom iv means allowing for general background
configurations, such as (anti–)de Sitter or Friedmann-
Robertson-Walker spacetimes. The complete set of permit-
ted metrics can be obtained by coupling the Maxwell-Proca
theory to gravity. As discussed elsewhere [1], this leads to
diverse physical applications.
Regarding free vector fields with a standard quadratic
mass, a consistent coupling to general relativity is obtained
by direct covariantization. For instance, the Proca mass
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term in Minkowski space m2ημνBμBνis promoted to
m2gμνBμBν, where gμν is (the inverse of) the dynamical
spacetime metric. For terms containing two or more
derivatives, this procedure becomes ambiguous due to
the presence of the derivatives ∂μ, which do not commute
anymore when replaced by covariant derivatives ∇μ.
When covariantizing the Maxwell-Proca action one
needs to be particularly careful to not destroy the under-
lying constraint algebra. This algebra should contain the
constraints for the massless spin-two together with those of
the Maxwell-Proca theory. In other words, we expect the
constraint enforcing relations (12) and (27) to be supple-
mented by additional conditions coming from the gravita-
tional sector.
In fact, already for the single Proca theory in [14],
nontrivial curvature couplings appear in the action. The
precise form of these terms was motivated by focusing on
the longitudinal mode of the vector field and requiring that
its couplings to gravity be the same as in Horndeski’s
theory, which is known to be ghost-free. This is a necessary
but not sufficient condition for a (first order) Lagrangian to
propagate the correct number of d.o.f.
The authors of [16] took a more rigorous approach and
initiated an ADM analysis [36] of the interactions proposed
in [14]. Their results show that the nonminimal curvature
couplings are precisely such that both the B0component of
the Proca field and the lapse function in the metric appear
without time derivatives in the action. This demonstrates
that the Lagrangian in [14] satisfies another necessary
condition for consistency. A full constraint analysis of the
theory that provides a solid proof of ghost freedom,
however, is still pending. Based on these partial results
for the single Proca field, it is natural to expect that the
ghost-free covariant version of our Maxwell-Proca theory
will also contain non-minimal curvature couplings.
We finish by pointing out two recent developments
pertinent to the beyond Maxwell-Proca scenarios above
described. First, it is interesting to note that a particular
example of a consistent theory that simultaneously relaxes
Axioms iii and iv arises in [37]. One of the decoupling
limits of nonlinear massive gravity [38] there considered
distributes the theory’s five propagating modes into 2 plus
3 d.o.f., corresponding to a massless graviton and a Proca
field, respectively. The resulting vector field Lagrangian
contains higher order terms and is defined on four-dimen-
sional anti–de Sitter spacetime, meanwhile it is ghost-free
by construction, as proven in [39]. Second, when relaxing
Axiom iv and coupling the Maxwell-Proca theory to
gravity, one should be careful not to miss the kind of
terms unveiled shortly after this work, in [40]. Such terms
reduce to total derivatives in flat spacetime.
V. CONCLUDING REMARKS
The key ideas behind the results here obtained are as
follows. We begin by formalizing the notion of a Proca field
in Definition 2, putting it on the same footing as that of the
renowned Maxwell field (see Definition 1). Based on these
two concepts, we construct the Maxwell-Proca Lagrangian:
the most general first-order theory in Minkowski spacetime
for an arbitrary number of interacting Maxwell and Proca
fields propagating the correct number of d.o.f.
In more detail, the Maxwell-Proca Lagrangian is most
conveniently expressed as the sum of kinetic plus inter-
action terms (9). The kinetic piece is (10). The interactions
are obtained by systematically converting the uniquely
defined primary constraint enforcing relations (12)—
derived in the Appendix (A7)—into as definite a
Lagrangian as possible. We distinguish two types of
interactions in (19): those that trivially satisfy (12) and
those that do not. The first set can succinctly be written as
(14), while the coefficients of the second set are made
explicit in (20)–(23).
Additionally, when multiple interacting Proca fields are
considered, this Lagrangian must be forced to satisfy the
uniquely defined secondary constraint enforcing relations
in (27)—derived in the Appendix (A20)—to avoid ghosts.
No further conditions on the Lagrangian apply because,
once Eq. (27) is fulfilled, the constraint algebra automati-
cally closes at the tertiary level. Notice that, in full
generality, (27) is a set of coupled nonlinear partial
differential equations whose solution is not known. This
explains our inability to turn (27) into a Lagrangian
building principle, unlike (12) before.
The vast and diverse applicability of the Maxwell-Proca
theory is discussed and exemplified elsewhere [1].
ACKNOWLEDGMENTS
The authors thank Axel Kleinschmidt for a fruitful
discussion, which gave us the idea for this work. We are
grateful to Chrysoula Markou for enlightening conversa-
tions in the early stages of the project. V. E. D. thanks
Nicolás Coca López for lucid explanations regarding
superconductors. J. A. M. Z. is thankful to E. A. Ayón-
Beato, and D. F. Higuita-Borja for the computing power
granted at ZymboLab (CINVESTAV). The work of
J. A. M. Z. is partially supported by the “Convocatoria
Max-Planck-CONACyT 2017 para estancias postdoctor-
ales”fellowship. This work is supported by a grant from the
Max Planck Society.
APPENDIX: DEGREE OF FREEDOM COUNT
In this appendix, we provide the explicit count of the
d.o.f. propagated by the Maxwell-Proca theory. We thus
demonstrate the absence of ghosts claimed around Eq. (11)
in Sec. II. Namely, given the Lagrangian constructed in
Sec. III, this count ensures that Definitions 1–3 in Sec. II
hold true. Indeed, in Secs. A1and A2we obtain the results
summarized in Tables Iand II, respectively.
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Our starting point is the Maxwell-Proca action con-
structed in Sec. III, Eqs. (19)–(23). For our present
purposes, it suffices to consider this action in the schematic
form,
S¼ZR×R3
d3þ1xL;L¼LðQI;_
QIÞ:ðA1Þ
Here, the fQg’s are the M0¼4ðNþMÞfield variables
that label the points in the configuration space Cof the
system,
QI¼ðAð1Þμ;…;A
ðNÞμ;B
ð1Þμ;…;B
ðMÞμÞðA2Þ
with I¼1;…;M
0, and the dot denotes derivation with
respect to time _
QI≔∂0QI. The velocities f_
Qgspan the
tangent space TC of the configuration space C. In all
generality, we take Cto be a differentiable Banach mani-
fold. The Lagrangian Ldepends on the spatial derivatives
of the field variables as well, L¼Lð∂iQIÞ. However, such
dependencies are not relevant to the upcoming d.o.f. count,
and hence we omit them. Note that our Ldoes not depend on
time explicitly. This simplifies a bit the subsequent algebra,
but it does not introduce nor remove any conceptual feature
compared to the generic scenario L¼Lðt; QI;_
QIÞ.
Our end point will be the well-known classical
Hamiltonian formula that enumerates propagating d.o.f.,
ndof ¼M0−N1−N2
2;ðA3Þ
where N1and N2stand for the number of first and second
class constraints, respectively. Recall that, for physically
sensible theories, N2is always an even integer. We shall
find N1¼2Nby construction and restrict the form of the
Lagrangian through the demand N2¼
!2M, in order to obtain
the desired count of d.o.f.: 2 per Maxwell field and 3 per
Proca field.
Needless to say, the formalism underlying this Appendix
is not new. It has been long established that any field theory,
such as the Maxwell-Proca theory here studied, is always
described by a singular Lagrangian—for example, see [41].
The equivalence between the Lagrangian and Hamiltonian
pictures [42] implies that any field theory is always a
constrained Hamiltonian system. Then, the famous Dirac-
Bergmann (Hamiltonian) theory of constraints [43,44]—
which was geometrized in [45]—provides the canonical
formalism to study such theories. In particular, it gives rise
to our end point formula (A3).
Our Appendix describes the fastest and easiest way to
obtain the results of interest. We predominantly make use
of the Lagrangian formalism for two fundamental reasons.
First and foremost, this allows for the construction of the
Maxwell-Proca theory. We anticipate that the uniquely
defined primary constraint enforcing relations in (A7) are
rewritten as Eq. (12), which is then systematically
converted into as definite a Lagrangian as possible in
Sec. III A. Then, the also uniquely defined secondary
constraint enforcing relations in (A20) are rewritten as
Eq. (27). This latter set of equations is left as an additional
requirement on the Lagrangian that must be satisfied to
ensure ghost-freedom. Second, the applications of the
Maxwell-Proca theory we propose in [1], as well as the
cosmological applications that motivated the very notion of
a generalized Proca field [14] use the Lagrangian (and not
the Hamiltonian) of the theory as their starting point. We
choose to make use of the Hamiltonian formalism only to
explicitly count the d.o.f. propagated by the Maxwell-Proca
theory, thereby showing it is free of ghosts. The
Hamiltonian itself is not worked out because this is not
required for the counting process.
The fact that the Maxwell-Proca theory is a gauge
theory introduces subtleties on both the Lagrangian and
TABLE I. Summary of the Lagrangian composition of the Maxwell-Proca theory with NMaxwell and MProca fields. The table
shows the number gof gauge identities present, together with the independent Euler-Lagrange equations of motion, LBianchi identities
and KLagrangian constraints. When possible, these are further classified into primary, secondary and tertiary types. The explicit form of
all the tabulated quantities can be found in the quoted formulas.
On shell Off shell
Primary Secondary Tertiary
Euler-Lagrange Eqs. M0−M1¼3ðNþMÞ(A12) K1−M2¼
!0K2−M3¼M(A25) ∅
Identities ∅L2¼0L3¼0L1¼N(A14)
g¼N(A29)
Lagrangian constraints K1¼M(A15) K2¼M(A24) K3¼0∅
TABLE II. Number of Hamiltonian constraints present in the
Maxwell-Proca theory with NMaxwell and MProca fields,
according to their type. Their explicit form can be found in the
quoted formulas. The nonsingular part of the constraint algebra is
given by the Poisson bracket matrix (A46).
First class Second class
Primaries (A33) Nb
Ψin (A40) Me
Ψin (A40)
Secondaries (A36) Nb
χin (A43) Me
χin (A43)
N1¼2NN
2¼2M
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Hamiltonian side of the story. We will discuss and reference
this important point where appropriate. For the benefit of
the reader, the Appendix is written as an up-to-date and
self-contained partial review of the theory of singular
Lagrangians and constrained Hamiltonians in the presence
of gauge symmetries.
1. Lagrangian constraint analysis
As already anticipated, we begin the study of the
constraint structure of the Maxwell-Proca theory in the
Lagrangian formalism. In this section, we obtain the results
conveyed in Table I. Namely, we count the number of
independent Euler-Lagrange equations of motion, Bianchi
identities and Lagrangian constraints present in the theory
and classify them into primary, secondary and tertiary. We
show that no tertiary Lagrangian constraints appear, and
hence no quaternary level arises. We also find the gauge
identities in the theory.
What follows is not just a counting exercise, though. In
the process, we will find the necessary and sufficient
conditions to make our count match that of the auxiliary
theory,
Laux ≔−
1
4Að¯αÞ
μν Aμν
ð¯αÞ−
1
4BðαÞ
μν Bμν
ðαÞ−m2
2BðαÞ
μBμ
ðαÞ:ðA4Þ
This is the theory of NMaxwell and MProca free fields,
where for simplicity we have taken the mass m≠0of all
the Proca fields to be the same. The aim is to show that the
(self-)interactions between all NþMfields that we pro-
posed in (19) do not introduce further unphysical d.o.f.,
compared to Laux. We will see this does not happen auto-
matically: for M>1, the constraint enforcing relation (7)
must be fulfilled.
Because we adopt a coordinate-dependent approach, the
subsequent analysis depends on the form of the Lagrangian:
the number of identities and Lagrangian constraints varies
with the field basis (unlike the number of d.o.f., which is
basis-independent). The results here reported correspond to
the mass eigenbasis formulation of the Maxwell-Proca
theory, which was introduced in Sec. III C.
Our analysis is largely based on the iterative algorithm
presented in [46]. However, there exists a vast literature on
the topic that could be consulted equivalently: [47],to
mention but a few of the standard references.
Iterative analysis. Given the classical field theory
Lagrangian density (A1) of interest, the principle of sta-
tionary action leads to the following Euler-Lagrange
equations of motion:
EJ≔
̈
QIWIJ þαJ¼0∀J; ðA5Þ
where we have defined
WIJ ≔∂2L
∂_
QI∂_
QJ;αJ≔_
QI∂2L
∂QI∂_
QJ−∂L
∂QJ:ðA6Þ
In (A5), the right-hand side is an on shell statement, while
WIJ is a M0×M0matrix known as the Hessian.
By construction,6our Lis such that
WIJ ≡0ðA7Þ
for all Jand for I¼4z−3, with z¼1;…;NþM. The
above is the necessary and sufficient condition to ensure
ghost-freedom of the theory at the primary level. In the
language of the main text, this condition reads (12).Itwas
until now believed [21,27] that (A7) is the only condition
needed to avoid ghosts. We will show that this expectation
is too naive and that, for M>1,(A7) must be supple-
mented by an additional condition (arising at the secondary
level) in order to truly remove ghosts.
Note that (A7) makes the rank of the Hessian be smaller
than its dimension and so its determinant vanishes. This
implies that the Lagrangian density under consideration is
singular, as we said it should for a field theory. What
is more, (A7) yields a Hessian of rank 3ðNþMÞ. This is
what is needed to reproduce the primary level of (A4).The
desired rank reduction of the Hessian can only happen by
setting entire columns (or, equivalently, rows) of this matrix
to zero for any first order Lagrangian, i.e., as in (A7).Rank
reductions in the form of NþMcolumns being linear
combinations of the remaining set of linearly independent
3ðNþMÞcolumns can be absorbed into a field redefinition.
Given (A7), it is clear that not all the equations in (A5)
are independent second order differential equations; only
M0−M1number of them are. The remaining M1equations
do not involve the accelerations f̈
Qg. Instead, they are
relations between the generalized field variables fQgand
their time derivatives f_
Qg. Here,
M1≔dimðWIJÞ−RankðWIJÞ¼NþM: ðA8Þ
Among these M1relations, we will further distinguish
between Bianchi identities and primary Lagrangian con-
straints, whose number we denote by L1and K1, respectively.
In order to split (A5) into the above mentioned three
subsets of equations, one must first obtain the M1linearly
independent zero vectors fγIgassociated to the Hessian,
γaIWIJ ¼0;a¼1;2;…;M
1:ðA9Þ
6At this point, we explicitly verify that the Maxwell-Proca
interaction terms in Eq. (19) satisfy the requirement of ghost-
freedom at the primary level. Recall that these interactions were
generated in Sec. III A, using a systematic procedure based on
pairwise antisymmetrization of the derivatives ∂μQI⊂Lusing
Levi-Civita symbols.
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These null vectors must be chosen so that they satisfy
γaIγIb¼δab:ðA10Þ
Additionally, one must also compute an auxiliary M0×M0
matrix MIJ. This matrix always exists and is uniquely
determined from the relations,
WIJMJK −δIKþγIaγaK¼0;M
IJγJa¼0:ðA11Þ
Then, the M0−M1independent second order (primary)
Euler-Lagrange equations are the nonzero subset in
ð̈
QJþαIMIJÞWJK ¼0;ðA12Þ
while the M1relations between the Q’s and their velocities
are
φa≔γaIαI¼0∀a: ðA13Þ
Notice that the right-hand side is, in principle, an on shell
equality. For the Maxwell-Proca theory, Nnumber of the
φ’s are identically zero,
φ¯
α≡0∀¯
α¼1;…;N: ðA14Þ
These are the L1¼NBianchi identities associated to the
Maxwell fields, which will be later related to first class
constraints. It is important to realize that (A14) holds true
off shell: it does not require the right-hand side of (A13).In
general, any identity at the primary level is off shell by
definition. On the other hand, the imposition that the
remaining φ’s vanish,
φα≈0∀α¼1;…;M; ðA15Þ
is an on shell statement that gives rise to K1¼Mprimary
Lagrangian constraints on the Proca fields.7These will be
later connected with second class constraints. It is worth
pointing out that, in agreement with the standard notation,
“≈”stands for a weak equality. In more detail, the
requirement that those φ’s which are not identities vanish
defines a subspace C0⊆Cin the configuration space,
generally referred to as the constraint surface. A weak
equality is one which only holds true in C0, not necessarily
in the entire of C. We here finish the study of the primary
level of the Maxwell-Proca theory.
A significant remark on nomenclature follows.
Equation (A14) is referred to as Bianchi identities because
these are the natural generalization of the familiar Bianchi
identity in classical electromagnetism to the full Maxwell-
Proca theory. In the language of differential forms, one can
succinctly write these full Maxwell-Proca Bianchi iden-
tities as dF −J¼0, where ddenotes exterior derivative,
Fis the field strength of the usual Maxwell field and Jis
the most general possible source term within the defining
axioms of the theory. That is, Jconsists of a set of sources
coming from the nonlinearities of the usual Maxwell field,
from additional Maxwell fields and from all the Proca
fields. In the well-known Maxwell electrodynamics limit
(N¼1and M¼0with no interaction terms), J¼0and
the usual Bianchi identity dF ¼0is recovered.
Before proceeding to the secondary level, we will briefly
discuss the increased algorithm needed for reducible con-
strained systems. These are theories for which the on shell
relations (A13) are not functionally independent. In our
setup, we encounter this feature in two instances:
(i) If working with a mixed eigenstate basis, obtained
from the mass eigenbasis fA; Bgby simple linear
field redefinitions.
(ii) If studying the charged complex Proca proposal
outlined in Sec. IV.
For any reducible theory, (A13) satisfy L1relations of the
form,
Z¯
αaφa≡0with Z¯
α∈TC ∀¯
α:ðA16Þ
The above are the analogues of the Bianchi identities (A14)
of the irreducible case. The case i is trivial, in the sense that
(A16) amount to linear relations between the reducible
constraints fφg. On the other hand, for the case ii, the Z’s
include differential operators and can be determined
iteratively order by order,8as explained in [30]. Once that
(A16) has been established, only the M1−L1irreducible
Lagrangian constraints fφganalogue to (A15) are to be
used to continue with the iterative algorithm.
Going back to the Maxwell-Proca theory, the demand
that (A15) holds true at all times (i.e., _
φα≈0for all α)
initiates the study of the secondary level,9which is an
innately on shell level. Specifically, it gives rise to K1¼M
secondary equations of motion,
e
Eβ≔
̈
QIγIαe
Wαβ þe
αβ¼0∀β;ðA17Þ
where we have defined
7Notice that in (A14)–(A15) we reuse the field labels ( ¯
α,α)
that in the main text enumerate Maxwell and Proca fields,
respectively. Throughout the Lagrangian analysis, these labels
enumerate Bianchi identities and primary Lagrangian constraints
instead. Since these are intrinsically associated to the Maxwell
and Proca fields, the slight abuse in the notation serves as a
reminder of their origin.
8Although we are not concerned with such involved scenarios,
in [30] it is noted that there exist theories where infinitely many
iterations would be required to obtain the Z’s.
9For clarity, we will use a notation where tilde quantities
correspond to the secondary level (or second iteration in the
constraint analysis algorithm in [46]) and hat quantities belong to
the tertiary level (third iteration).
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e
Wαβ ≔γαI∂φβ
∂_
QI;e
αα≔−αIMIJ ∂φα
∂_
QJþ_
QI∂φα
∂QI;
ðA18Þ
in close analogy to the primary quantities in (A6).As
happened with (A5) before, generally not all the above
secondary equations of motion are independent second
order differential equations. Instead, they can again be
divided into three subsets: K1−M2secondary Euler-
Lagrange equations, L2identities and K2secondary
Lagrangian constraints, with
M2≔dimðe
WαβÞ−Rankðe
WαβÞ:ðA19Þ
In order for the Maxwell-Proca theory to have the desired
Lagrangian conformation [i.e., that of the auxiliary theory
in (A4)], M2¼K1is needed, implying that there are no
independent secondary Euler-Lagrange equations. It is easy
to see that this is uniquely achieved by setting
e
Wαβ≡
!0∀α;β;ðA20Þ
which is the sufficient and necessary condition that ensures
ghost-freedom at the secondary level. Fulfillment of (A20)
implies the constraint enforcing relations (27) on the
Maxwell-Proca interaction terms. Written in this manner,
it is easy to see that the above identical vanishing happens
by construction for any number of Maxwell fields and
when a single Proca field is considered. In other words,
(A20) trivially holds true if M≤1. However, when two or
more Proca fields are present in the theory, (A20)—
equivalently, (27)—nontrivially restricts the proposed inter-
action terms. We highlight that this is precisely the
anticipated condition that was missed out in all [21,27]
and that supplements the known condition (A7). To avoid
the propagation of ghosts in any multi-Proca setup, both
(A7) and (A20) must hold true. As noted in the main text,
we regard this realization as our main result.
Given (A20), repetition of the logic exposed for
the primary level requires us to obtain the M2null vectors
feγαg10associated to e
Wαβ,
eγrαe
Wαβ ≈0;r¼1;2;…;M
2;ðA21Þ
in such a way that they satisfy the orthonormality relation
eγrαeγαs¼δrs. Further, it is easy to see that the always extant
M1×M1auxiliary matrix e
Mαβ, which is analogue to (A11)
at the secondary level and which is uniquely determined
from
e
Wαβ e
Mβγ −δαγþeγαreγrγ¼0;e
Mαβ eγβr¼0;ðA22Þ
identically vanishes in this case: e
Mαβ ≡0. Subsequently,
the M2secondary relations defined in C0among the
generalized coordinates and their time derivatives are
given by11
ϕr≔eγrαe
αα≈0∀r: ðA23Þ
We see that none of the ϕ’s vanish identically, implying
L2¼0. To obtain the K2¼M2secondary Lagrangian
constraints in the theory, one must indeed demand that they
are zero,
ϕr≈
10∀r: ðA24Þ
We will soon see that these relate to second class constraints
for the Proca fields. For clarity, we have here introduced the
symbol “≈
1”to indicate that the requirement that the ϕ’s
vanish defines a subspace C1⊆C0in the constraint surface.
As a result, a weak equality that only holds true in C1(and
not necessarily in C0or the entire of C) shall be denoted by
≈
1in the following.
One more iteration is required. Again, all statements at
this tertiary level hold on shell only. Demanding that (A24)
holds true at all times (namely _
ϕr≈0for all r), M2¼M
number of tertiary equations of motion are obtained
b
Es≔
̈
QIγIαeγαrb
Wrs þb
αs¼0;ðA25Þ
with b
Wrs and b
αr, the tertiary counterparts of (A18) before,
defined as
b
Wrs ≔eγrαγαI∂ϕs
∂_
QIðA26Þ
and
b
αr≔−
e
ααe
MαβγβI∂ϕr
∂_
QI−αIMIJ ∂ϕr
∂_
QJþ_
QI∂ϕr
∂QI;ðA27Þ
respectively. The M2equations (A25) can generically be
divided into M2−M3independent tertiary Euler-Lagrange
equations, L3ð≤M3Þidentities and M3−L3tertiary
Lagrangian constraints, where
M3≔dimðb
WrsÞ−Rankðb
WrsÞ:ðA28Þ
However, M3¼0for the Maxwell-Proca theory. This
implies that all equations in (A25) are independent tertiary
10Note that the eγ’s are defined in C0. In particular, they are only
determined modulo the φ’s.
11For reducible theories, the increased iterative algorithm
described around (A16) should be applied at the secondary level
as well.
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Euler-Lagrange equations. It also means L3¼0¼K3,
which signals the termination of the iterative procedure:
an on shell quaternary level would unfold from guarantee-
ing the stability of the K3tertiary Lagrangian constraints,
but there are none.
Gauge identity analysis. We turn our attention to gauge
identities, which are known to escape the above algorithm
and require separate consideration. We shall use gto denote
their number. The key point is to notice that the subset of N
number of α’s defined in (A6) stemming from the Maxwell
fields vanish identically,
α¯α≡0∀¯
α:ðA29Þ
This is obvious from their very definition, once one takes
into account that our theory depends on the Maxwell
fields only through their field strengths, as pointed out
in Sec. III C. This is clearly an off shell identity, since it is
independent of (A5). Matter of fact, (A29) are precisely the
g¼Ngauge identities we were looking for. We will soon
link them with first class constraints. Notice that the
Bianchies in (A14) are projections of the gauges in
(A29) to the kernel of the Hessian WIJ defined in (A6).
In this way, the Bianchi identities can be viewed as relics of
the gauge invariance of the theory. Accordingly, we will
shortly relate them to first class constraints as well.12
We have now fulfilled our initial promise and completed
the derivation of Table I.
2. Hamiltonian constraint analysis
Next, we focus on the study of the Hamiltonian con-
straints in the Maxwell-Proca theory. In this section, we
derive the results summarized in Table II. In more detail,
we first relate the identities and Lagrangian constraints we
found in the previous section to the primary and secondary
Hamiltonian constraints in the Dirac-Bergmann formalism
[43,44]. We loosely refer to these as Dirac primaries and
secondaries, respectively. Our analysis is primarily based
on [46] again. Then, we reclassify these Dirac constraints
as first or second class. To this aim, we stick to [48].We
restate, however, that an extensive bibliography exists
elaborating on this topic—for example, see [49] and
references therein. In this way, we obtain the number of
propagating d.o.f. in the theory, according to (A3).
As already mentioned, the Hamiltonian formalism is an
equivalent reformulation of the Lagrangian formalism in
Sec. A1. Its central object of study is the Hamiltonian
density H, whose explicit form we do not work out here
(as it is not required to count the propagating d.o.f.). The
interested reader can consult in e.g., [50] the procedure to
determine the Hfollowing from (A1). Formally, Hrelates
to the Lagrangian density Las
H≔_
QIPI−LðQI;_
QIÞ:ðA30Þ
Generically, His a function of three types of variables: the
generalized coordinates QIintroduced in (A2), their time
derivatives _
QIand the momenta PI. The fPg’s span the
cotangent space TCof the configuration space C.For
clarity, we note that PIencodes both the momenta Pð¯
αÞ
μ
associated to the Maxwell fields and the momenta ΠðαÞ
μof
the Proca fields,
PI¼ðPð1Þ
μ;…;PðNÞ
μ;Πð1Þ
μ;…;ΠðMÞ
μÞ:ðA31Þ
We denote as Rthe ð3M0Þ-dimensional space comprising
C,TC and TC.
The Hamiltonian density is independent of the _
Q’s when
restricted to the subspace R0⊆Rdefined by13
PI−∂L
∂_
QI≈0:ðA32Þ
In the case of the Maxwell-Proca theory of our interest,
only M1¼NþMof the above relations are independent.
These are the Dirac primary constraints Ψa. We shall only
be concerned with their determination modulo the soon
to be introduced Poisson bracket operation, so we define
them as
Ψa≔γaIPI−∂L
∂_
QI:ðA33Þ
The Dirac primaries relate to the Lagrangian quantities fφg
defined in (A13) as
_
Ψa≈φa;ðA34Þ
where the weak equality means this holds true in R0only.
The demand that the Dirac primaries be stable,
_
Ψa≈
10∀a; ðA35Þ
connects them with all the Bianchi identities in (A14), the
primary Lagrangian constraints in (A15) and the gauge
12At this point, the reader may benefit from a lightning review
of the familiar case of Maxwell electrodynamics. Calling ð
⃗
E;
⃗
BÞ
the electric and magnetic fields, the Bianchi identities read ⃗
∇×
⃗
Eþ_
B¼0and ⃗
∇·⃗
B¼0. They are identically solved by ⃗
E¼
−
⃗
∇ϕ−_
⃗
Aand ⃗
B¼
⃗
∇×⃗
A, for any ðϕ;
⃗
AÞ. The gauge symmetry
tells us that not only these off shell identities, but also the
Maxwell Lagrangian itself and the on shell electric and magnetic
Gauss’laws following from it are invariant under ⃗
A→
⃗
Aþ
⃗
∇λ
and ϕ→ϕ−_
λ, with λan arbitrary smooth function of spacetime.
13The reader should not confuse the weak equality of the
Lagrangian formalism, defining C0⊆C, with the weak equality of
the Hamiltonian formalism, which defines R0⊆R.
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identities in (A29). On the Hamiltonian side, (A35) defines
the subspace R1⊆R0.
The Dirac secondary constraints χaare defined as the
M1¼NþMindependent relations between the Q’s and
the P’s that vanish in R1and that are independent of the
velocities f_
Qgin R0. As long as the Poisson bracket is
concerned, it suffices for us to choose them in the rather
obvious way,
χa≔_
Ψa:ðA36Þ
A subset of MDirac secondaries relates to the Lagrangian
objects fϕgdefined in (A23) as
_
χr≈ϕr:ðA37Þ
The time derivative of the remaining NDirac secondaries is
weakly vanishing, which can be viewed as the Hamiltonian
reflection of the intrinsic stability of the L1¼NBianchi
identities in (A14). The requirement that the χ’s be stable
gives rise to the Lagrangian secondaries in (A24).
Having found the Dirac primary and secondary con-
straints in the theory, we proceed to their splitting into first
and second class. As a reminder, first class constraints are
those which have a weakly vanishing Poisson bracket with
all Dirac constraints. The constraints not having this
property are second class. Equivalently, one can think of
the first (second) class constraints as those which do (do
not) form a closed constraint algebra in Poisson brackets.
Recall as well that, in canonical (or Darboux) coordinates14
ðQI;P
IÞ, the Poisson bracket of two functions fand gin
R0takes the form,
ff;gg≈∂f
∂QI
∂g
∂PI
−∂f
∂PI
∂g
∂QI;∀f; g ∈R0:ðA38Þ
In order to divide the Dirac primaries into first and
second class, we need to consider the matrix of the Poisson
brackets of the constraints in the theory,
Δ≔
0
B
B
B
B
B
B
B
B
B
B
B
B
B
@
fΨ1;Ψ1gfΨ1;Ψ2g  fΨ1;ΨM1g
.
.
..
.
...
..
.
.
fΨM1;Ψ1gfΨM1;Ψ2g  fΨM1;ΨM1g
fχ1;Ψ1gfχ1;Ψ2g  fχ1;ΨM1g
.
.
..
.
...
..
.
.
fχM1;Ψ1gfχM1;Ψ2g  fχM1;ΨM1g
1
C
C
C
C
C
C
C
C
C
C
C
C
C
A
:
ðA39Þ
This is a 2M1×M1matrix and, for the Maxwell-Proca
theory, the rank equals the number of Proca fields under
consideration: R¼M. Note that R<M
1, which reflects
the singular nature of the theory in the Dirac-Bergmann
approach. Then, it is possible to define two sets of linear
combinations of the Dirac primaries fe
Ψ;b
Ψg, the first set
containing Relements and the second set containing S≔
M1−R¼Nelements,
e
Ψα≔e
vαaΨa;b
Ψ¯α≔b
v¯αaΨa;ðA40Þ
where we have chosen to reuse the indices introduced in
(A14) and (A15) for simplicity.15 Here, the parameters
e
vαa;b
vαa∈Rcan and should be chosen so that the newly
defined e
Ψ’s and b
Ψ’s are linearly independent and so that the
b
Ψ’s are first class (primary) constraints,
fΨa;b
Ψ¯
αg≈0≈fχr;b
Ψ¯
αg∀a; ¯
α;r: ðA41Þ
Clearly, the e
Ψ’s are second class (primary) constraints.
The procedure to distinguish between first and second
class constraints among the Dirac secondaries is similar,
albeit a little more cumbersome algebraically. First, we
define the following set of linear combinations of all the
Dirac secondaries and the second class Dirac primary
constraints:16
χ0
a≔JabχbþKaαe
Ψαwith Jab;K
aα∈R;ðA42Þ
where the parameters Jabcan and should be chosen so that
Jis a non-singular matrix: detðJÞ≠0. Then, in analogy to
(A40), it is possible to form two sets of linear combinations
of the χ0’s,
e
χα≔e
wαaχ0
a;b
χ¯α≔b
w¯αaχ0
a:ðA43Þ
Once again, the reuse of the ðα;¯
αÞis due to the criterion
explained in footnote 7. We denote by m2ð≤M2Þthe
number of b
χ’s. Here, e
wαa;b
w¯αa∈R−f0gshould be chosen
so that the b
χ’s are all first class,
fΨa;b
χ¯
αg≈0≈fχa;b
χ¯
αg∀a; ¯
α;ðA44Þ
for a maximal number m2. Then, the M2−m2Dirac
secondaries e
χ’s are second class constraints. For the
Maxwell-Proca theory of our interest, m2¼N. This
completes the derivation of Table II.
As a double-check, we verify that (A46), the matrix ¯
Δof
the Poisson brackets of all the constraints that we identified
14By definition, these are natural orthonormal coordinates
on R0.
15Again, their correspondence to the field labels in the main
text is not coincidental: ðe
Ψα;b
Ψ¯αÞare constraints on the (Proca,
Maxwell) fields.
16It is possible to trivially extend the construction to include all
Dirac primaries, but this is not necessary for our purposes.
MAXWELL-PROCA THEORY: DEFINITION AND CONSTRUCTION PHYS. REV. D 101, 045009 (2020)
045009-15

as second class, is nonsingular: det ¯
Δ≠0. Note that ¯
Δis
always an antisymmetric square matrix of dimension Rþ
M2−m2(¼2M, for the Maxwell-Proca theory).
For completeness, we indicate the relation between the
first and second class constraints and the Lagrangian
picture next. As shown in the original works [43,44],
linear combinations of the second class constraints ðe
Ψ;e
χÞ
generate the Lagrangian primary (A15) and secondary
(A24) constraints. This happens through their time deriv-
atives and within the suitable Hamiltonian subspaces, in
agreement with (A34)–(A37). Observe that these con-
straints are all associated to the Proca fields. Similarly,
linear combinations of the first class constraints ðb
Ψ;b
χÞ
generate the Bianchi identities (A14), all of which stem
from the Maxwell fields. Surprisingly enough, the con-
nection between first class constraints and gauge identities
has been a subject of great controversy for decades and
continues to receive attention. Although the correct relation
was long ago suggested [44], this was for a long time
believed to be a matter of interpretation with no physical
implication. The subject was again pondered over some
thirty years later; for instance see [51], but remained as a
majoritarily misconceived issue. A lucid review of the
origin and resolution of the main puzzlements to this
respect appeared in [52] (see also references within). At
last, the topic was shown to be of physical transcendence
for the case of classical electrodynamics in [53]. The
interested reader can consult this same reference for a
meticulous historical review. In conclusion, only a suitably
tuned linear combination between a primary and secondary
first class constraints is able to generate a gauge symmetry
consistently. For the Maxwell-Proca theory, this means
particular compositions of ðb
Ψ;b
χÞyield (A29).
Finally, adding the number of first (second) class con-
straints coming from the Dirac primaries and secondaries, we
see that N1¼2N(N2¼2M), as anticipated. Using (A3),
the number of propagating d.o.f. for the proposed Maxwell-
Proca theory is that of the auxiliary theory (A4),
ndof ¼4ðNþMÞ−2N−M¼2Nþ3M: ðA45Þ
Namely, each Maxwell field propagates 2 d.o.f. and each
Proca field propagates 3 d.o.f. In other words, the interaction
terms between the fields in Eq. (19) supplemented by the
condition (27) are well-behaved, in the sense that they do not
give rise to the propagation of ghostlike d.o.f.
As a final remark, note that we have worked in four
spacetime dimensions for definiteness. However, all the
results in this Appendix can be straightforwardly gener-
alized to arbitrary dimension d. The only change would
be in the dimension of the configuration space C:
M0¼dðNþMÞ. This then would yield a d.o.f. count
ndof ¼ðd−2ÞNþðd−1ÞM.
¯
Δ≔
0
B
B
B
B
B
B
B
B
B
B
B
B
B
@
fe
Ψ1;e
Ψ1gf
e
Ψ1;e
ΨMgf
e
Ψ1;b
χ1gf
e
Ψ1;b
χMg
.
.
...
..
.
..
.
...
..
.
.
fe
ΨM;e
Ψ1gf
e
ΨM;e
ΨMgf
e
ΨM;b
χ1gf
e
ΨM;b
χMg
fb
χ1;e
Ψ1gf
b
χ1;e
ΨMgf
b
χ1;b
χ1gf
b
χ1;b
χMg
.
.
...
..
.
..
.
...
..
.
.
fb
χM;e
Ψ1gf
b
χM;e
ΨMgf
b
χM;b
χ1gf
b
χM;b
χMg
1
C
C
C
C
C
C
C
C
C
C
C
C
C
A
;
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