# Geometry of Classical Nambu-Goldstone Fields

Slobodan M. Radošević  0000-0002-3211-1392<EMAIL_ADDRESS>Department of
Physics, Faculty of Sciences, University of Novi Sad, Trg Dositeja Obradovića
4, Novi Sad, Serbia
Department of Mathematics, University of Jaén, Jaén, Spain

###### Abstract

A coordinate-free formulation of first order effective field theory, in which
Nambu-Goldstone fields are described as sections on associated bundle, is
presented. This construction, which is based only on symmetry considerations,
allows for a direct derivation of number and types of Nambu-Goldstone fields
in a classical field theory without any reference to effective Lagrangian. A
central role in classification is shown to be played by Lorentz-symmetry
breaking order parameter which induces symplectic structure in the field space
of the theory.

Effective Field Theories, Global Symmetries, Sigma models

###### Contents

  1. I Introduction
  2. II Noether symmetries and momentum mapping
  3. III Symmetry breaking and coadjoint representation
  4. IV Nambu-Goldstone fields on associated bundle
    1. IV.1 Associated bundles and maps
    2. IV.2 Type B Nambu-Goldstone fields
    3. IV.3 Type A Nambu-Goldstone fields
    4. IV.4 Dynamical type A and B NG fields
    5. IV.5 SO(3) ferromagnet and antiferromagnet
  5. V Coordinate expressions
    1. V.1 Coordinates on homogeneous space $G/H$
    2. V.2 Coordinates on bundle $G/H$
  6. VI Explicit symmetry breaking
    1. VI.1 $\widetilde{\mathrm{Stab}(\bm{\Sigma})}={\mathrm{Stab}(\bm{\Sigma})}$
    2. VI.2 $\widetilde{\mathrm{Stab}(\bm{\Sigma})}\subset{\mathrm{Stab}(\bm{\Sigma})}$
    3. VI.3 Examples
      1. VI.3.1 Heisenberg antiferromagnet
      2. VI.3.2 Relativistic Bose-Einstein condensation
      3. VI.3.3 The $\mathrm{SO}(2N)$ vector model at fixed charge
  7. VII Sumarry

##  I Introduction

Effective field theory (EFT) describes low energy sector of physical systems
which undergo spontaneous symmetry breakdown. Being model independent, EFT
found applications across many areas of physics [1, 2, 3, 4, 5, 6] where it
not only provides a powerful tool for high-order perturbative calculations [7,
8, 9, 10] but also allows for a deeper insight into physical problems [11] or
other conventional methods used to deal with it [12]. The main dynamical
degrees of freedom in EFT for internal symmetries are spin-zero Nambu-
Goldstone (NG) bosons and, in the case of Lorentz-invariant systems, they
share many common features – they are gapless excitations whose fields take
values in the coset space $G/H$ and total number of which is
$\mathrm{dim}(G)-\mathrm{dim}(H)$, where $G$ represents the symmetry group of
the action (or Hamiltonian) and $H$ is the symmetry group of the ground state.
Also, their dispersion relation is $\omega(\bm{k})\propto|\bm{k}|$. On the
other hand, situation becomes more involved once the constrains imposed by the
Lorentz symmetry are dropped – neither the number of NG particles, nor their
dispersion relation, are universal. For example, $\mathrm{O}(3)$ ferromagnets
and antiferromagneets share the same symmetry breaking pattern, yet the
spectrum of ferromagnets contains one NG boson with dispersion
$\omega(\bm{k})\propto\bm{k}^{2}$ while there are two NG bosons with
$\omega(\bm{k})\propto|\bm{k}|$ in the case of antiferromagnets. Also, spinor
bose condensates support excitations with $\omega(\bm{k})\propto\bm{k}^{2}$
and $\omega(\bm{k})\propto|\bm{k}|$ at the same time, while explicit symmetry
breaking by finite charge density opens a gap for some of NG bosons. Giving
the obvious importance of these particles, efforts were made to understand the
mechanism responsible for such a diversity of NG bosons and their properties.

The original classification [13] of bosonic excitations which accompany
symmetry breakdown is based on the form of their dispersion relations:
excitations with $\omega\propto|\bm{k}|^{2n+1}$ are labeled as type I, while
those with $\omega\propto|\bm{k}|^{2n}$ as type II. If $N_{\rm{I}}$ and
$N_{\rm{II}}$ denote the numbers of different flavors of type I and II bosons,
respectively, the following inequality holds:
$N_{\rm{I}}+2N_{\rm{II}}\geq\mathrm{dim}(G/H)$. This statement goes by the
name of Nielsen-Chada theorem (NCT). Examples of type I NGB include pions in
QCD, antiferromagnetic magnons as well as phonons in solids and superfluids,
while type II NGB appear in ferromagnets and Bose-Einstein condensates [14,
15]. Even though NCT leaves $n$ as an arbitrary integer, in the most cases of
physical interest $n=0$ for type I and $n=1$ for type II NG bosons.

Subsequent development of nonrelativistic EFT came in the form of Lagrangian
field theory for NG fields [16]. By considering the symmetry conditions
imposed on the effective Lagrangian, together with the commutation relations
and vacuum-expectation values of Noether charges [17, 18], Watanabe and
Murayama have traced the origin of two types of excitations to the structure
of time derivatives in the leading order term of the effective Lagrangian [19,
20]. The proof presented by Watanabe and Murayama is a perturbative one. That
is, they have obtained a general solutions of Leutwyler’s equations in local
coordinate system near the origin of $G/H$. On the basis of this solution, the
effective Lagrangian up to quadratic order in NG fields is shown to, in
general, contain coupled canonically conjugate fields and their corresponding
first time derivatives. Accordingly, the new nomenclature for NG fields is
proposed: NG fields which are canonically paired in the effective Lagrangian
are denoted as type B, while remaining ones are denoted as type A. Also,
Watanabe, Murayama and Brauner [19, 20, 21] showed that the numbers of type B
($N_{\rm{B}}$) and type A ($N_{\rm{A}}$) NG boson flavors are given by
$N_{\rm{A}}={\rm{dim}}(G/H)-\rm{rank}\rho$,
$N_{\rm{B}}=\frac{1}{2}\rm{rank}\rho$, where $\mathrm{i}V\rho_{ab}=\langle
0|[Q_{a},Q_{b}]|0\rangle$, $V$ is the volume of system and $\\{Q_{a}\\}$ is
the set of broken generators. Since the authors of [20, 21] consider quantum
theory and discuss fluctuations and stability of the ground state, their proof
also implicitly makes use of a special basis of $\mathfrak{g}$ chosen in such
a way that only operators corresponding to Cartan generators of $\mathfrak{g}$
have nonzero vacuum expectation values. Another important aspect of the work
by Watanabe and Murayama is the geometric picture underlining their
classification. As it is discussed in [19, 20], the form of time derivatives
in the leading order terms essentially determines the presymplectic structure
on $G/H$. Therefore, the coset space $G/H$ is partially a phase space in which
canonically conugate NG fields (naimelly, type B bosons) live. The
identification of presymplectic structure on $G/H$ in these papers makes use
of the Borel’s theorem which applies to compact semisimple Lie groups and this
puts some restrictions on generality of their proof. However, the extension to
noncompact group $G$ may be of interest when considering Higgs effective field
theory, spontaneous braking of Lorentz symmetry or supergravity [22, 23, 24,
25, 26]. The classification was latter refined to include massive Nambu-
Goldstone bosons whose appearance accompanies explicit symmetry breakdown [27,
28, 29].

Important constrains on the dynamics of NG fields, resulting from the symmetry
breakdown, can be studied within classical field theory – a nonlinear
$\sigma$-model with the target space $G/H$. The dynamical degrees of freedom
in classical theory are sections on a properly chosen bundle and many of their
features, such as number and types of NG fields or their dispersion relation,
are shared between classical and lowest order quantum field theory. Also,
classical field theory allows for a clear geometric formulation which makes a
sharp distinction between physical degrees of freedom (sections on a bundle)
and local coordinates on $G/H$, which are devoid of any physical significance
[30, 31]. This is an important point since, giving the generality EFT usually
aims at [1], one should be in a position of deducing the properties of NG
fields, such as their number and type, not from the effective Lagrangian
(which, by definition, is a function of coordinates on the coset space), but
from the properties of the system itself. The main result of present paper is
that this can be done, at least on the level of classical description. Also,
the geometric arguments presented here allow for a unified description of
massive and massless NG fields.

The classical theory of NG fields presented here is build around the
assumption of spontaneous breakdown of symmetry according to $G\to H$. We
assume that $G$ is connected, simply connected Lie group and $H$, a symmetry
group of ground-state field configuration, a compact subgroup of $G$. To
discuss two types of NG fields, we need to consider the existence of the
Lorentz-symmetry breaking order parameter and its symmetry group to whom, in
general, $H$ is subgroup. As it turns out, Lorentz-symmetry breaking order
parameter is an element of $\mathfrak{g}^{*}$, a vector space dual to the lie
algebra of $G$, which carries the coadjoint representations of Lie group $G$
and its Lie algebra, $\mathfrak{g}$. The role of $\mathfrak{g}^{*}$ turns out
to be crucial in present construction, as a distinguished element of
$\mathfrak{g}^{*}$ (the order parameter) defines the symplectic structure on
space of its orbits under the action of $G$ and this introduces two types of
NG fields. The three groups, which describe symmetries of the theory, are most
conveniently taken into account starting from a principal $G$ bundle and
building the field space $\mathrm{E}$ as an associated bundle. Even though, in
many applications, the field space is a trivial bundle
($\mathrm{E}=(G/H)\times\mathrm{M}$, where $\mathrm{M}$ is spacetime), the
space $G/H$, as a bundle, is not, and the general theory of associated bundles
is well suited for this problem. Since the objective of present paper concerns
with types and number of NG fields, the field theory is presented using 1-jet
formalism. Finally, the associated bundle machinery will allow us to formulate
the theory of classical NG fields in a coordinate-free manner and to derive
classification of NG fields starting from the symmetry breaking pattern and
properties of the order parameter, without any reference to the effective
Lagrangian or particular basis of $\mathfrak{g}$. We shall assume that are no
redundancies due to equivalent space-time symmetries [32, 33].

The Section II contains a brief review of classical field theory in the
language of jets, together with the Noether theorem and momentum mapping,
while the symmetry breaking pattern, the Lorentz-symmetry breaking order
parameter as an element of $\mathfrak{g}^{*}$ and fibration of $G/H$ are
discussed in Section III. A coordinate-free definition of type A and type B
Nambu-Goldstone fields as sections on associated bundle build upon three
symmetry groups characterizing the system are presented in Section IV. Two
types of coordinate systems on $G/H$, one of which allows for a decoupling of
type A and type B NG fields from each other at the level of free field theory
are described in Section V. Finally, the case of explicit symmetry breaking,
triggered by the chemical potential coupled to the Noether charge in
Hamiltonian, is discussed Section VI.

##  II Noether symmetries and momentum mapping

Let $(\mathrm{E},\pi_{\mathrm{E}},\mathrm{M})$ represent a fiber bundle, where
$\mathrm{M}$ denotes $m+1$ dimensional spacetime,
$\mathrm{F}=\mathrm{E}_{\mathrm{p}}$, with $\mathrm{p}\in\mathrm{M}$, is $n$
dimensional typical fiber, $\mathrm{E}$ is $n+m+1$ dimensional total space and
$\pi_{\mathrm{E}}:\mathrm{E}\to\mathrm{M}$ is the projection. In this setting,
we identify classical fields $s$ with local sections of $\pi_{\mathrm{E}}$,
$s:\mathrm{M}\to\mathrm{E}$. Further, let
$(J^{1}\pi_{\mathrm{E}},\pi_{1},\mathrm{M})$ denote the corresponding first
jet bundle and $L\in C^{\infty}(J^{1}\pi_{\mathrm{E}})$ be the Lagrangian
density. If $\mathrm{Vol}\mathrm{M}$ is the volume form on $\mathrm{M}$, the
action functional for this classical field theory is

$S[s]=\int_{\mathrm{U}}(j^{1}s)^{*}\mathcal{L},$ (1)

where $\mathcal{L}=L\;\mathrm{Vol}\mathrm{M}\in\bigwedge_{0}^{m+1}\pi_{1}$ is
the Lagrangian form, $\mathrm{U}\subset\mathrm{M}$ is a compact $m+1$
dimensional submanifold of $\mathrm{M}$ and $j^{1}s$ is the first prolongation
of $s$. The classical field configurations are extremals of the action (1),
i.e. those sections of $\pi_{\mathrm{E}}$ which satisfy corresponding Euler-
Lagrange equations [34, 35].

Let now a Lie Group $G$ act on $\pi_{\mathrm{E}}$ via bundle automorphism
$(\eta,\bar{\eta})$ where $\eta:\mathrm{E}\to\mathrm{E}$ and
$\bar{\eta}:\mathrm{M}\to\mathrm{M}$ is a diffeomorphism such that
$\pi_{\mathrm{E}}\circ\eta=\bar{\eta}\circ\pi_{\mathrm{E}}$ [36]. Under this
action of $G$ the sections of $\pi_{\mathrm{E}}$ transform as

$G:s\to s^{\prime}\equiv\eta\cdot s:=\eta\circ s\circ\bar{\eta}^{-1}.$ (2)

Also, the first jet prolongation of $(\eta,\bar{\eta})$, to be denoted as
$j^{1}(\eta,\bar{\eta})\equiv j^{1}\eta:J^{1}\pi\to J^{1}\pi$, is defined to
act on $j^{1}s$ as

$j^{1}s\to j^{1}s^{\prime}=j^{1}\left(\eta\circ
s\circ\bar{\eta}^{-1}\right):=j^{1}\eta\circ j^{1}s\circ\bar{\eta}^{-1}.$ (3)

The actions of $(\eta,\bar{\eta})$ and $j^{1}(\eta,\bar{\eta})$ are
conveniently represented in a commutative diagram

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}\pgfsys<EMAIL_ADDRESS>(4)

Here $(J^{1}\pi_{\mathrm{E}},\pi_{1,0},\mathrm{E})$ is an affine bundle and
$(j^{1}\eta,\eta)$ as well as $(j^{1}\eta,\bar{\eta})$ are bundle
automorphisms. If $X\in\chi(\mathrm{E})$ is a projectable vector field with
the flow $\eta_{\epsilon}$, whose compact support lies within
$\pi_{\mathrm{E}}^{-1}(\mathrm{U})$

$X=\frac{\mathrm{d}}{\mathrm{d}\epsilon}\bigg{|}_{0}\eta_{\epsilon},\hskip
28.45274ptX\bigg{|}_{\pi_{\mathrm{E}}^{-1}(\partial\mathrm{U})}=0,$ (5)

then the pair $(\eta_{\epsilon},\bar{\eta}_{\epsilon})$ defines a bundle
morphism for each $\epsilon$ and $\bar{\eta}_{\epsilon}$ denotes the flow of
$\bar{X}$ defined by $\bar{X}\circ\pi_{\mathrm{E}}=\pi_{\mathrm{E}*}\circ
X\in\chi(\mathrm{M})$. We also note that the (infinite dimensional) Lie
algebra of projectable vector fields on $\mathrm{E}$ is homomorphic to
$\mathfrak{g}$, the Lie algebra of group $G$. That is, induced vector fields
represent the Lie algebra $\mathfrak{g}$ and describe infinitesimal action of
$G$ on various geometric objects on $\mathrm{E}$. When it becomes important,
we shall denote by $X_{\bm{A}}$ the vector field on $\mathrm{E}$ induced by
the action of $\mathrm{exp}\epsilon\bm{A}\in G$, with $\bm{A}\in\mathfrak{g}$.

The variation of a section $s$, induced by the action of $G$, is a family of
sections $s_{\epsilon}:=\eta_{\epsilon}\circ
s\circ(\bar{\eta}_{\epsilon})^{-1}$. Since the field $X$ is assumed to vanish
at $\partial\mathrm{U}$, the variation $s_{\epsilon}$ coincides with $s$ at
all points of $\partial\mathrm{U}$. The invariance of action (1) under the
bundle automorphimsms may now be expressed as [34]

$0=\frac{\mathrm{d}}{\mathrm{d}\epsilon}\bigg{|}_{0}\int_{\bar{\eta}_{\epsilon}(\mathrm{U})}(j^{1}s_{\epsilon})^{*}\mathcal{L}=\int_{\mathrm{U}}(j^{1}s)^{*}\mathrm{d}_{X^{1}}\mathcal{L},$
(6)

where we have used (3) and properties of the pull-back operation. Here
$\mathrm{d}_{X^{1}}$ denotes the Lie derivative with respect to the 1-jet
prolongation of $X$,

$X^{1}=\frac{\mathrm{d}}{\mathrm{d}\epsilon}\bigg{|}_{0}j^{1}\eta_{\epsilon}$
(7)

and the condition $\mathrm{d}_{X^{1}}\mathcal{L}=0$ simply states that
invariance of the Lagrangian form under transformations induced by the vector
field $X$ always implies the invariance of action (1). This constrain on
$\mathcal{L}$, which guarantees the invariance of classical equations of
motion, may in fact be relaxed. If $x^{\mu}$ are (local) coordinates on
$U\subset\mathrm{M}$ ($\mu=0,1,2,\dots,m$) and
$\mathrm{Vol_{\mu}M}=\mathrm{i}_{\partial_{\mu}}\mathrm{Vol}\mathrm{M}$, where
$\mathrm{i}$ denotes the interior product, the classical field configurations
will remain the same if

$\mathrm{d}_{X^{1}}\mathcal{L}=\mathrm{d}\left[\Delta^{\mu}\mathrm{Vol}_{\mu}\mathrm{M}\right]\equiv\mathrm{d}\Delta,$
(8)

where $\Delta^{\mu}$ are functions on $\mathrm{E}$ and $\mathrm{d}$ is the
exterior derivative on $\mathrm{E}$. Indeed, a direct application of the
Stokes theorem gives

$\int_{\mathrm{U}}(j^{1}s)^{*}\mathrm{d}\left[\Delta^{\mu}\mathrm{Vol}_{\mu}\mathrm{M}\right]=\int_{\mathrm{U}}\mathrm{d}\left[(j^{1}s)^{*}\Delta^{\mu}\mathrm{Vol}_{\mu}\mathrm{M}\right]=\int_{\partial\mathrm{U}}(j^{1}s)^{*}\Delta^{\mu}\mathrm{Vol}_{\mu}\mathrm{M}$
(9)

which obviously does not affect the field configurations in the interior of
$\mathrm{U}$.

To discuss further implications of symmetry in a classical field theory, let
us recall the definition of Cartan form [37, 36, 34]. In the case of first-
order Lagrangian, the Cartan form is unique and is given by

$\Theta_{\mathcal{L}}=\mathcal{L}+\mathrm{d}_{S}(L)$ (10)

where $S$ denotes the vertical endomorphism associated to $\rm{VolM}$, a
vector-valued $m+1$ form on $J^{1}\pi$ and $\mathrm{d}_{S}$ is the
corresponding derivative operator [34]. If $(x^{\mu},u^{a})$ are adapted
coordinates on $\mathrm{E}$ ($a=1,2\dots,n$) and $u^{a}_{\mu}$ are associated
derivative coordinates, then

$\Theta_{\mathcal{L}}=\mathcal{L}+\frac{\partial L}{\partial
u^{a}_{\mu}}\theta^{a}\wedge\mathrm{Vol}_{\mu}\mathrm{M}$ (11)

where $\theta^{a}$ are basis contact forms, defined by
$(j^{1}s)^{*}\theta^{a}=0$ and $s$ is an arbitrary section on
$\pi_{\mathrm{E}}$. In local coordinates

$\theta^{a}=\mathrm{d}u^{a}-u^{a}_{\mu}\mathrm{d}x^{\mu}.$ (12)

Therefore, $(j^{1}s)^{*}\mathcal{L}=(j^{1}s)^{*}\Theta_{\mathcal{L}}$. A
section $s:\mathrm{M}\to\mathrm{E}$ is a solution of the Euler-Lagrange
equations iff [35]

$(j^{1}s)^{*}\left[\mathrm{i}_{Y}\mathrm{d}\Theta_{\mathcal{L}}\right]=0$ (13)

for every vector field $Y$ on $J^{1}\pi$. In particular, these equation remain
the same if [38]

$\mathrm{d}_{X^{1}}\Theta_{\mathcal{L}}=\mathrm{d}\Delta,$ (14)

with $\Delta$ as in (8). A vector field $X$, whose 1-jet extension $X^{1}$
generates the flow $j^{1}\eta_{\epsilon}$ which changes the Cartan form by a
total divergence, is said to be a Noether symmetry [37]. Given the Noether
symmetry condition on $\Theta_{\mathcal{L}}$, we introduce the covariant
momentum map in the Lagrangian representation

$\bm{J}^{\mathcal{L}}:J^{1}\pi_{\mathrm{E}}\to\mathfrak{g}^{*}\otimes\wedge^{n}J^{1}\pi_{\mathrm{E}},$
(15)

by defining

$J^{\mathcal{L}}(\bm{A}):=\Delta_{\bm{A}}-\mathrm{i}_{X_{\bm{A}}^{1}}\Theta_{\mathcal{L}},\hskip
28.45274pt\mathrm{with}\;\;\;\;\bm{A}\in\mathfrak{g},$ (16)

to be a $n$ form on $J^{1}\pi_{\mathrm{E}}$ given by
$\langle\bm{J}^{\mathcal{L}}(\mathrm{f}),\bm{A}\rangle:=J^{\mathcal{L}}(\bm{A})|_{\mathrm{f}}$
for $\mathrm{f}\in J^{1}\pi_{\mathrm{E}}$. Above, $\mathfrak{g}^{*}$ denotes
the dual of the Lie algebra $\mathfrak{g}$, $X_{\bm{A}}^{1}$ is the first jet
prolongation of the vector field $X_{\bm{A}}$, $\langle\;\;,\;\;\rangle$
stands for the natural pairing between the elements of $\mathfrak{g}$ and
$\mathfrak{g}^{*}$ while $\Delta_{\bm{A}}$ is an $n$ form on $\mathrm{M}$
linear in $\bm{A}$. If $\\{\bm{T}_{i}\\}$ with $i=1,2\dots,\mathrm{dim}(G)$,
denotes a basis of $\mathfrak{g}$ and $\\{\bm{T}^{*i}\\}$ is the corresponding
dual basis, we have

$\bm{J}^{\mathcal{L}}=\bm{T}^{*i}\otimes J^{\mathcal{L}}_{i},\hskip
28.45274pt\mathrm{with}\;\;\;\;J^{\mathcal{L}}_{i}:=J^{\mathcal{L}}(\bm{T}_{i})=\Delta_{i}-\mathrm{i}_{X_{i}^{1}}\Theta_{\mathcal{L}}$
(17)

where

$X_{i}:=X_{\bm{T}_{i}}$ (18)

are induced vector fields corresponding to the basis elements of
$\mathfrak{g}$. By using definition (16) and the Noether condition (14), one
easily shows that the quantity $\mathrm{d}J^{\mathcal{L}}(\bm{A})$ is
conserved on configurations which satisfy EL equations

$(j^{1}s)^{*}\left[\mathrm{d}J^{\mathcal{L}}(\bm{A})\right]=0,$ (19)

which is just the Noether’s first theorem. To summarize, if the action of the
theory enjoys symmetry described by the Lie Group $G$, which acts on fields by
the bundle automorphism, the corresponding conservation law may be expressed
in terms of momentum map and elements of $\mathfrak{g}$ (19). The importance
of this construction, at least for the purpose of the present paper, lies in
the clear geometric formulation which reveals the significance of
$\mathfrak{g}^{*}$ for underlying effective field theory.

##  III Symmetry breaking and coadjoint representation

Let us suppose now that the classical field theory we are interested in
describes a system which undergoes spontaneous symmetry breakdown according to
the scheme $G\to H$, where $G$ is connected, simply connected Lie group and
$H$ closed subgroup of $G$. The low energy sector of such a theory is
dominated by Nambu-Goldstone fields which take their values in the homogeneous
space $\mathrm{E}_{\mathrm{p}}=G/H$. Each point of $\mathrm{a}\in G/H$ is of
the form $gH=\\{gh|h\in H\\}$ and corresponds to a field configuration. This
homogeneous space is equipped with a left transitive action $G\times G/H\to
G/H$, which we denote by $(g,\mathrm{a})\to l(g,\mathrm{a})$. It is convenient
to pick $eH\equiv O$, where $e$ denotes the unit element in $g$, as the origin
in $G/H$. The ground state field configuration corresponds to $O$ and it is
invariant under the action of $H$. Then, standard arguments [39] show that
$T_{O}(G/H)\cong\mathfrak{g}/\mathfrak{h}$. Further, any homogeneous space
admits a $G$ invariant metric and we may decompose $\mathfrak{g}$ into the
direct sum [39, 40, 41]

$\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{m}$ (20)

where $\mathfrak{m}$ is subspace of $\mathfrak{g}$, canonically isomorphic to
$T_{O}(G/H)$, invariant under adjoint action of $H$:
$\mathrm{Ad}(H)\mathfrak{m}\subset\mathfrak{m}$. Given a group $G$ and
homogeneous space $G/H$, we may introduce projection
$\widetilde{\pi}_{\mathrm{P}}:G\to G/H,g\mapsto gH$. In this manner, Lie group
$G$ can be viewed as a principal bundle with the base space $G/H$ and typical
fiber $H$.

Of special importance is a situation in which the conserved quantity from the
Noether theorem plays the role of the order parameter. Asides from breaking
the Lorentz symmetry on a fundamental level, such an occurrence puts further
restrictions on the total space $\mathrm{E}$ and, consequently, classical
field configurations. According to (19), a Lorentz-symmetry breaking order
parameter induces a distinguished element of $\mathfrak{g}^{*}$, which we
formally write as

$\bm{\Sigma}:=\bm{T}^{*i}\left\langle\frac{1}{V}\;Q_{i}\right\rangle_{\rm{g.s.}}=\;\bm{T}^{*i}\left\langle\frac{1}{V}\int_{\bar{\mathrm{U}}}(j^{1}s)^{*}\left[\Delta_{i}^{0}-X^{0}_{i}L-X^{a}_{i}\frac{\partial
L}{\partial u^{a}_{0}}+X^{0}_{i}\frac{\partial L}{\partial
u^{a}_{\nu}}u^{a}_{\nu}\right]\mathrm{Vol_{0}M}\right\rangle_{\rm{g.s.}}\equiv\Sigma_{i}\bm{T}^{*i}$
(21)

where $Q_{i}$ is a Noether charge, $X^{\mu}_{i},X^{a}_{i}$ are the components
of $X_{i}$ in a local chart on $\pi_{\mathrm{E}}$,
$\langle\dots\rangle_{\rm{g.s.}}$ denotes the ground state average value of a
corresponding quantum (statistical) theory, $\bar{\mathrm{U}}$ is a spacelike
region of $\mathrm{M}$ and $V$ is the volume of $\bar{\mathrm{U}}$. While it
is obvious that this average can not be calculated within classical theory,
this is of no concern to us and we shall simply assume that $\Sigma_{i}\neq 0$
break the Lorentz invariance. The order parameter $\bm{\Sigma}$ transforms
with the coadjoint representation $\mathrm{Ad}^{*}$, defined by
$\langle\mathrm{Ad}^{*}(g)\bm{\Sigma},\bm{A}\rangle=\langle\bm{\Sigma},\mathrm{Ad}(g^{-1})\bm{A}\rangle$,
for $\bm{A}\in\mathfrak{g}$ [42]. Further, let us introduce the coadjoint
orbit as the set ${\mathcal{O}}=\\{\mathrm{Ad}^{*}(g)\bm{\Sigma}|g\in G$ }.
Since all the points in ${\mathcal{O}}$ are generated by the action of $G$,
this set may be identified with the homogeneous space
$G/\mathrm{Stab}(\bm{\Sigma})$, where $\mathrm{Stab}(\bm{\Sigma})$ denotes the
isotropy group of $\bm{\Sigma}$

$\mathrm{Stab}(\bm{\Sigma})=\\{g\in
G|\mathrm{Ad}^{*}(g)\bm{\Sigma}=\bm{\Sigma}\\}.$ (22)

Therefore, the group $G$ may also be viewed as the principal fiber bundle with
projection $\pi_{\tilde{\mathrm{P}}}$, base space
$G/\mathrm{Stab}(\bm{\Sigma})$ and typical fiber $\mathrm{Stab}(\bm{\Sigma})$.
In complete analogy with (20), we may decompose $\mathfrak{g}$ also as

$\mathfrak{g}=\mathfrak{stab}(\bm{\Sigma})\oplus\mathfrak{n}$ (23)

where $\mathfrak{n}\cong\mathfrak{g}/\mathfrak{stab}(\bm{\Sigma})\cong
T_{\tilde{O}}(G/\mathrm{Stab}(\bm{\Sigma}))$, $\tilde{O}$ is the origin of
$\mathcal{O}$ and

$\mathfrak{stab}(\bm{\Sigma})=\\{\bm{B}\in\mathfrak{g}|\mathrm{ad}^{*}(\bm{B})\bm{\Sigma}=0\\},\hskip
28.45274pt\langle\mathrm{ad}^{*}(\bm{B})\bm{\Sigma},\bm{A}\rangle=\langle\bm{\Sigma},-\mathrm{ad}(\bm{B})\bm{A}\rangle$
(24)

is the Lie algebra of $\mathrm{Stab}(\bm{\Sigma})$ which acts on the order
parameter by corresponding coadjoint representation $\mathrm{ad}^{*}$. The
order parameter $\bm{\Sigma}$ must also be $H$ invariant and, therefore, in
general $H\subseteq\mathrm{Stab}(\bm{\Sigma})$. We shall assume that $H$ as
well as $\mathrm{Stab}(\bm{\Sigma})$ are compact subgroups of $G$.

When $H\subset\mathrm{Stab}(\bm{\Sigma})$, and both $H$ and
$\mathrm{Stab}(\bm{\Sigma})$ are compact subgroups of $G$, previously
introduced fibrations induce yet another one. With the natural projection
$f:G/H\to G/\mathrm{Stab}(\bm{\Sigma})$, given by $gH\mapsto
g\mathrm{Stab}(\bm{\Sigma})$, homogeneous space $G/H$ becomes a bundle itself
[43]: the base space is $G/\mathrm{Stab}(\bm{\Sigma})$ and the typical fiber
is $\mathrm{Stab}(\bm{\Sigma})/H$. The pattern of spontaneous symmetry
breaking (SSB), together with the structure of order parameter configuration,
completely determines three fibrations summarized in the diagram:

${G}$${G/H}$${G/\mathrm{Stab}(\bm{\Sigma}).}$$\textstyle{\widetilde{\pi}_{\mathrm{P}}}$$\textstyle{\pi_{\tilde{\mathrm{P}}}}$$\textstyle{f}$

Of particular interest in the following is the fact that
${\mathcal{O}}=G/\mathrm{Stab}(\bm{\Sigma})$ is a symplectitc manifold
equipped with the Kirillov-Kostant symplectic form [42, 44, 45]. Thus, a $G$
invariant Lagrangian, together with $\mathrm{Stab}(\bm{\Sigma})\subset G$
invariant Lorentz-symmetry breaking order parameter and
$H\subset\mathrm{Stab}(\bm{\Sigma})\subset G$ invariant ground state field
configuration, naturally lead to two types of NG fields: the ones who take
values in a symplectic manifold $G/\mathrm{Stab}(\bm{\Sigma})$ and the rest
whose values belong to the space $\mathrm{Stab}(\bm{\Sigma})/H$. Indeed, these
are type B and type A Nambu-Goldstone fields introduced in [19]. If
$H=\mathrm{Stab}(\bm{\Sigma})$, one takes symmetry breaking pattern to be
$G\to\mathrm{Stab}(\bm{\Sigma})$ and constructs a theory of type B NG fields
[See IV.5 for an example] whenever $G$ is non-Abelian [16].

##  IV Nambu-Goldstone fields on associated bundle

As we saw in previous section, a Lorentz-symmetry breaking order parameter
naturally induces the fibration of $G/H$ so that Nambu-Goldstone fields come
into two types. In this section we shall explicitly construct the bundle
$(\mathrm{E},\pi_{\mathrm{E}},\mathrm{M})$, with $\mathrm{E}_{\mathrm{p}}=G/H$
and show that both types of NG fields may be defined in a coordinate-free
manner, in a relatively general case. To do so, we first recall some basic
facts about associated bundles [41] and then proceed with the exposition.

###  IV.1 Associated bundles and maps

To construct the fiber bundle $(\mathrm{E},\pi_{\mathrm{E}},\mathrm{M})$,
where $\mathrm{M}$ is spacetime and typical fiber is $G/H$, we start from the
principal $G$ bundle $(\mathrm{P},\pi_{\mathrm{P}},M)$ equipped with right
principal action $\mathrm{P}\ni\mathrm{u}\mapsto\mathrm{u}\cdot g$ and pick a
local section $\sigma:\mathrm{M}\to\mathrm{P}$ to provide a local
trivialization. If we consider a reduction of symmetry group $G\to H$, then
there is a well defined associated bundle $\mathrm{P}_{G}[G/H;l]$ with the
fiber $G/H$ and the left action $l:G\times(G/H)\to G/H$. The total space of
associated bundle $\pi_{\mathrm{E}}$ is build using projection
$q:\mathrm{P}\times(G/H)\to\mathrm{E}$, which is defined as an equivalence
class with respect to orbits of the right action
$R:(\mathrm{P}\times(G/H))\times G\to\mathrm{P}\times(G/H)$, defined by
$R\left((\mathrm{u},\mathrm{a}),g\right)=\bigl{(}\mathrm{u}\cdot
g,l(g^{-1},\mathrm{a})\bigr{)}$ where dot denotes the right principal action.
If we fix $\mathrm{u}\in\mathrm{P}$, then
$q_{\mathrm{u}}:\\{\mathrm{u}\\}\times(G/H)\to\mathrm{E}_{\pi_{\mathrm{P}}(\mathrm{u})}$
is a diffeomorphism for each $\mathrm{u}\in\mathrm{P}$. The second important
map defined for an associated bundle $\pi_{\mathrm{E}}$ is
$\tau:\mathrm{P}\times_{\mathrm{M}}\mathrm{E}\to G/H$, where
$\tau(\mathrm{u}_{\mathrm{p}},\mathrm{a}_{\mathrm{p}}):=q^{-1}_{\mathrm{u}_{\mathrm{p}}}(\mathrm{a}_{\mathrm{p}})$
with
$\pi(\mathrm{a}_{\mathrm{p}})=\pi(\mathrm{u}_{\mathrm{p}})=\mathrm{p}\in\mathrm{M}$.
The maps $q$ and $\tau$ can be used to express a local section $s$ of
$\pi_{\mathrm{E}}$ in terms of the frame form $\bar{s}:\mathrm{P}\to G/H$ and
a local section of $\pi_{\mathrm{P}}$ as

$s=q\circ(\mathrm{i}\mathrm{d}_{\mathrm{P}}\times\bar{s}\bigr{)}\circ\sigma=q\bigl{(}\sigma,\bar{s}\circ\sigma\bigr{)}$
(25)

where

$\bar{s}=\tau\bigl{(}\mathrm{i}\mathrm{d}_{\mathrm{P}}\times
s\circ\pi_{\mathrm{P}}\bigr{)}.$ (26)

Various maps introduced here are conveniently represented in a commutative
diagram

${G/H}$${\mathrm{P}\times_{\mathrm{M}}\mathrm{E}}$${\mathrm{P}}$${\mathrm{P}\times(G/H)}$${\mathrm{E}}$${\mathrm{M}}$${\mathrm{M}}$${\mathrm{M}}$$\textstyle{\tau}$$\textstyle{\mathrm{i}\mathrm{d}_{\mathrm{P}}\times
s\circ\pi_{\mathrm{P}}}$$\textstyle{\bar{s}}$$\textstyle{\mathrm{i}\mathrm{d}_{\mathrm{P}}\times\bar{s}}$$\textstyle{\pi_{\mathrm{P}}}$$\textstyle{q}$$\textstyle{\pi_{\mathrm{E}}}$$\textstyle{\phi}$$\textstyle{\mathrm{i}\mathrm{d}_{\mathrm{M}}}$$\textstyle{\mathrm{i}\mathrm{d}_{\mathrm{M}}}$$\textstyle{\sigma}$$\textstyle{s}$
(27)

where $\mathrm{i}\mathrm{d}_{\mathrm{M}}$ denote the identity map on
$\mathrm{M}$. The formalism of first-order jets outlined Section II can now be
used to construct the classical NG fields (see recent papers [46, 47]). The
map $\phi:\mathrm{M}\to G/H$ defined as

$\phi=\bar{s}\circ\sigma=\tau\bigl{(}\mathrm{i}\mathrm{d}_{\mathrm{P}}\times
s\circ\pi_{\mathrm{P}}\bigr{)}\circ\sigma=\tau\bigl{(}\sigma,s\bigr{)}$ (28)

can be understood as a local representative of a section $s$ and $\phi$ is
usually the object which is considered as a classical Nambu-Goldstone field
[19, 20, 48]. Given $\phi$, we can reconstruct the section $s$ using $q$
projection

$s=q\bigl{(}\sigma,\phi\bigr{)}.$ (29)

The local representative $\phi$ has ${\rm{dim}}(G)-{\rm{dim}}(H)$ components,
and they are split into physical degrees of freedom carried by NG fields of
type A and type B.

###  IV.2 Type B Nambu-Goldstone fields

Suppose now that the ground state of system is such that the Noether charge
develops vacuum expectation value which we identify with the order parameter
$\bm{\Sigma}\in\mathfrak{g}^{*}$. The coset space $G/H$ may now be viewed as a
bundle $\bigl{(}G/H,f,G/\mathrm{Stab}(\bm{\Sigma})\bigr{)}$, where $f$ denotes
the canonical projection $f:G/H\to G/\mathrm{Stab}(\bm{\Sigma})$, $gH\mapsto
g\mathrm{Stab}(\bm{\Sigma})$. The typical fiber of $f$ is
$\mathrm{Stab}(\bm{\Sigma})/H$ and we shall focus on this space in next
subsection.

The canonical projection $f$ is well defined and we may simply introduce type
B Nambu-Goldstone fields as local representatives of section
$s:\mathrm{M}\to\mathrm{E}$ which take values in
$G/\mathrm{Stab}(\bm{\Sigma})$. That is, for each
${\rm{a}}=\phi(\mathrm{p})\in G/H$, there is a map
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{\rm
B}}:\mathrm{M}\to G/\mathrm{Stab}(\bm{\Sigma})$

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{\mathrm{B}}}$
$\displaystyle=$ $\displaystyle f\circ\phi=f\circ\bar{s}\circ\sigma$ (30)
$\displaystyle=$ $\displaystyle
f\circ\tau\bigl{(}\mathrm{i}\mathrm{d}_{\mathrm{P}}\times
s\circ\pi_{\mathrm{P}}\bigr{)}\circ\sigma$

such that
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{\rm
B}}(\mathrm{p})=f\bigl{(}\phi(\mathrm{p})\bigr{)}$. This definition is given
in the following diagram

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}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{129.86124pt}{-7.74306pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\hbox{$\textstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{\mathrm{B}}}$}
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}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>(31)

As we remarked earlier, $G/\mathrm{Stab}(\bm{\Sigma})$ is a symplectic
manifold. Thus, type B NG fields come in pairs and the number of physical
degrees of freedom which they carry is

$N_{\mathrm{B}}=\frac{1}{2}\mathrm{dim}\biggl{(}G/\mathrm{Stab}(\bm{\Sigma})\biggr{)}$
(32)

Of course, this is exactly the result obtained in [19, 20, 49].

###  IV.3 Type A Nambu-Goldstone fields

To give a proper definition of type A Nambu-Goldstone fields, recall that the
bundle $\bigl{(}G/H,f,G/\mathrm{Stab}(\bm{\Sigma})\bigr{)}$ may be viewed as
an associated bundle to the principal $\mathrm{Stab}(\bm{\Sigma})$ bundle
$\bigl{(}G\equiv\tilde{\mathrm{P}},\pi_{\tilde{\mathrm{P}}},G/\mathrm{Stab}(\bm{\Sigma})\bigr{)}$
[43]. Thus

$\Bigl{(}G/H,f,G/\mathrm{Stab}(\bm{\Sigma})\Bigr{)}=\tilde{\mathrm{P}}_{\mathrm{Stab}(\bm{\Sigma})}\Bigl{[}\mathrm{Stab}(\bm{\Sigma})/H;\tilde{l}\Bigr{]}$
(33)

where $\tilde{l}$ denotes the left action of $\mathrm{Stab}(\bm{\Sigma})$ on
${\mathrm{Stab}(\bm{\Sigma})}/H$. The associated bundle
$\tilde{\mathrm{P}}_{\mathrm{Stab}(\bm{\Sigma})}\bigl{[}\mathrm{Stab}(\bm{\Sigma})/H;\tilde{l}\bigr{]}$
comes equipped with mappings $\tilde{q}$ and $\tilde{\tau}$ which are defined
analogously to $q$ and $\tau$ of $\mathrm{P}_{G}[G/H;l]$. We can use
$\tilde{q}$ to connect the total space of
$\tilde{\mathrm{P}}_{\mathrm{Stab}(\bm{\Sigma})}\bigl{[}\mathrm{Stab}(\bm{\Sigma})/H;\tilde{l}\bigr{]}$
with the fibers of $(\mathrm{E},\pi_{\mathrm{M}},\mathrm{M})$ and
$\tilde{\tau}$ to define NG fields which take values in
$\mathrm{Stab}(\bm{\Sigma})/H$.

Let $\tilde{\sigma}$ denote a local section on $\pi_{\tilde{\mathrm{P}}}$, and
$c$ is be local section on $\bigl{(}G/H,f,\tilde{\mathrm{M}}\equiv
G/\mathrm{Stab}(\bm{\Sigma})\bigr{)}$ determined by
$\phi=c\circ{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{\mathrm{B}}}$.
Since
$f\circ\phi={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{\mathrm{B}}}$,
we have $f\circ c=\rm{id}_{\tilde{M}}$, so that $c$ is indeed a section on
$\bigl{(}G/H,f,\tilde{\mathrm{M}}\equiv G/\mathrm{Stab}(\bm{\Sigma})\bigr{)}$.
If $\bar{c}$ denotes the frame form of $c$, we have the following diagram

${\mathrm{Stab}(\bm{\Sigma})/H}$${\tilde{\mathrm{P}}\times_{\tilde{\mathrm{M}}}\tilde{\mathrm{E}}}$${\tilde{\mathrm{P}}\equiv
G}$${\tilde{\mathrm{P}}\times(\mathrm{Stab}(\bm{\Sigma})/H)}$${\tilde{\mathrm{E}}\equiv
G/H}$${\tilde{\mathrm{M}}}$${\tilde{\mathrm{M}}\equiv
G/\mathrm{Stab}(\bm{\Sigma})}$${\tilde{\mathrm{M}}}$${\mathrm{M}}$${\mathrm{M}}$${\mathrm{M}}$$\textstyle{\tilde{\tau}}$$\textstyle{\mathrm{i}\mathrm{d}_{\tilde{\mathrm{P}}}\times
c\circ\pi_{\tilde{\mathrm{P}}}}$$\textstyle{\bar{c}}$$\textstyle{\mathrm{i}\mathrm{d}_{\mathrm{P}}\times\bar{c}}$$\textstyle{\pi_{\tilde{\mathrm{P}}}}$$\textstyle{\tilde{q}}$$\textstyle{f}$$\textstyle{\tilde{\phi}}$$\textstyle{\mathrm{i}\mathrm{d}_{\tilde{\mathrm{M}}}}$$\textstyle{\mathrm{i}\mathrm{d}_{\tilde{\mathrm{M}}}}$$\textstyle{\tilde{\sigma}}$$\textstyle{c}$$\textstyle{\mathrm{i}\mathrm{d}_{\mathrm{M}}}$$\textstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\phi_{\mathrm{B}}}}$$\textstyle{\definecolor[named]{.}{rgb}{0,0,1}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi_{\mathrm{A}}}$$\textstyle{\mathrm{i}\mathrm{d}_{\mathrm{M}}}$$\textstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\phi_{\mathrm{B}}}}$$\textstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\phi_{\mathrm{B}}}}$$\textstyle{\phi}$
(34)

and type A Nambu-Goldstone fields are defined as

$\displaystyle{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi_{\mathrm{A}}}$
$\displaystyle=$
$\displaystyle\tilde{\phi}\circ{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\phi_{\mathrm{B}}}}=\bar{c}\circ\tilde{\sigma}\circ{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\phi_{\mathrm{B}}}}$
(35) $\displaystyle=$
$\displaystyle\tilde{\tau}\bigl{(}\mathrm{i}\mathrm{d}_{\tilde{\mathrm{P}}}\times
c\circ\pi_{\tilde{\mathrm{P}}}\bigr{)}\circ\tilde{\sigma}\circ f\circ\phi$
$\displaystyle=$
$\displaystyle\tilde{\tau}\Bigl{(}\tilde{\sigma}\circ{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\phi_{\mathrm{B}}}},\phi\Bigr{)}.$

In other words, given $\phi(\mathrm{p})\in G/H$ and
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{\rm
B}}(\mathrm{p})\in G/\mathrm{Stab}(\bm{\Sigma})$, there is a corresponding
point in $\mathrm{Stab}(\bm{\Sigma})/H$ given by
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi_{\rm
A}}(\mathrm{p})$. Since there is no natural symplectic structure on
$\mathrm{Stab}(\bm{\Sigma})/H$, the number of physical degrees freedom
described by type A Nambu-Goldstone fields is

$N_{\mathrm{A}}=\mathrm{dim}\Bigl{(}\mathrm{Stab}(\bm{\Sigma})/H\Bigr{)}.$
(36)

Note tat this definition of type A NG fields assumes the existence of Lorentz-
symmetry breaking order parameter as, given $\phi(\mathrm{p})$,
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi_{\mathrm{A}}}(\mathrm{p})$
depends on
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\phi_{\mathrm{B}}}}(\mathrm{p})$.
In the case when $\bm{\Sigma}=\bm{0}$, one proceeds in a usual way and
introduces local representatives of type A NG fields as maps
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi_{\mathrm{A}}}:\mathrm{M}\to
G/H$ [48] and

$N=\mathrm{dim}\Bigl{(}G/H\Bigr{)}.$ (37)

is the total number of physical degrees of freedom in this case.

###  IV.4 Dynamical type A and B NG fields

Results of IV.2 and IV.3 show that given section $s:\mathrm{M}\to\mathrm{E}$,
which is a solution of variational problem and whose a local representative is
$\phi:\mathrm{M}\to G/H$, each point $\phi(\mathrm{p})$ van be represented by
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{\rm
B}}(\mathrm{p})$ and
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi_{\rm
A}}(\mathrm{p})$ which are completely determined by $\phi(\mathrm{p})$ and the
structure of associated bundle
$\bigl{(}\mathrm{E},\pi_{\mathrm{E}},\mathrm{M}\bigr{)}$. If we want to treat
type A and type B fields as true dynamical degrees of freedom we need to
express the effective Lagrangian in terms of
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{\rm
B}}$ and
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi_{\rm
A}}$, where
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{\rm
B}}:\mathrm{M}\to G/\mathrm{Stab}(\bm{\Sigma})$ and
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi_{\rm
A}}:\mathrm{M}\to\mathrm{Stab}(\bm{\Sigma})/H$ are unknown functions, not
specified by $\phi(\rm p)$. To do so, we first need to reconstruct the section
$s:\mathrm{M}\to\mathrm{E}$. Let
$({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}z^{\alpha}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}v^{\bar{\alpha}}})$,
where
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\alpha}=1,2,\dots\mathrm{dim}\bigl{(}G/\mathrm{Stab}(\bm{\Sigma})\bigr{)}$
and
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bar{\alpha}}=1,2,\dots\mathrm{dim}\bigl{(}\mathrm{Stab}(\bm{\Sigma})/H\bigr{)}$,
be a local coordinate system on $G/H$ and let $(x^{\mu},w^{j})$,
$j=1,2,\dots\mathrm{dim}\bigl{(}G\bigr{)}$ be a local coordinate system on
$(\rm P,\pi_{\rm P},\rm M)$. Then, the map $\phi:{\rm M}\to G/H$ can be
expressed as
$\Bigl{(}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{\rm
B}^{\alpha}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi_{\rm
A}^{\bar{\alpha}}}\Bigr{)}$, where
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{\rm
B}^{\alpha}}={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}z^{\alpha}}\circ{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{\rm
B}}$ and
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi_{\rm
A}^{\bar{\alpha}}}={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}v^{\bar{\alpha}}}\circ{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi_{\rm
A}}$. Given $\phi(\rm p)$, we can obtain section $s$ using (29). If
$\sigma^{j}=w^{j}\circ\sigma$,
$\psi_{\sigma}:\pi_{\mathrm{P}}^{-1}(\mathrm{U})\to\mathrm{U}\times G$ is a
local trivialization of $\pi_{\mathrm{P}}$ and
$\psi_{\mathrm{E}}:f^{-1}({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{\mathrm{B}}}(\mathrm{U}))\to{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{\mathrm{B}}}(\mathrm{U})\times\bigl{(}\mathrm{Stab}(\bm{\Sigma})/H\bigr{)}$,
a local trivialization on $G/H$, for each
$\mathrm{p}\in\mathrm{U}\subset\mathrm{M}$

$s(\mathrm{p})=q\Bigl{(}\psi_{\sigma}^{-1}\bigl{(}x^{\mu}(\mathrm{p}),\sigma^{j}(\rm
p)\bigr{)},\psi_{\mathrm{E}}^{-1}\bigl{(}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\phi_{\rm
B}^{\alpha}}(\rm
p),{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi_{\rm
A}^{\bar{\alpha}}}(\rm p)\bigr{)}\Bigr{)}$ (38)

and effective Lagrangian can be obtained as discussed in Section II. Several
examples are given below.

###  IV.5 SO(3) ferromagnet and antiferromagnet

The construction from previous subsections enable us to compare two well known
cases of spontaneous symmetry breaking: SO(3) Heisenberg ferromagnet and
antiferromagnet. Spontaneous symmetry breaking occurs along the scheme
$\mathrm{SO}(3)\to\mathrm{SO}(2)$ in both of these cases, yet systems behave
rather differently al low energies. The mechanism which produces differences
is well understood – effective Lagrangians for both systems are derived in
[16] and subsequent studies [5, 10, 50, 51] exploited them to reach at
detailed description of free field theories, possible magnon-magnon
interactions, scattering processes and low temperature thermodynamics in both
cases [52, 53, 54]. In short, the effective Lagrangian for an antiferromagnets
takes a form of Lorentz-invariant nonlinear $\sigma$-model with the target
space $S^{2}$ (type A NG fields) and non-interacting theory is of a Klein-
Gordon type. In contrast, the effective Lagrangian for a ferromagnet contains
additional term with single time derivative of magnon fields, frequently
refereed to as a Wess-Zumino term, with fields also taking values in $S^{2}$
(ferromagnetic magnons are classified as type B NG fields). The presence of
this term, which makes the free magnon theory of a Schrödinger type, produces
various consequences in behavior of ferromagnets in comparison with
antiferromagnets.

While magnon-magnon interactions are described in detail in both cases, as
they follow from the effective Lagrangian, a minor puzzle seems to remain to
this day. Why do systems with the same SSB pattern
$\mathrm{SO}(3)\to\mathrm{SO}(2)$, and thus the same target space
$\mathrm{SO}(3)/\mathrm{SO}(2)=S^{2}$, behave differently? According to
analysis presented here, we see that the answer lies in the existence of
Lorentz-symmetry breaking order parameter – spontaneous magnetization
$\bm{\Sigma}\in\mathfrak{so}(3)^{*}\simeq\mathfrak{so}(3)\simeq\mathbb{R}^{3}$.
The unbroken subgroup, $H=\mathrm{SO}(2)$, is in fact the stabilizer
$\mathrm{Stab}(\bm{\Sigma})=\mathrm{SO}(2)$ and
$\mathrm{SO}(3)/\mathrm{Stab}(\bm{\Sigma})=S^{2}$ is a two-sphere of radius
$|\bm{\Sigma}|$ [55]. Thus, in the case of ferromagnets, the sphere $S^{2}$ is
generated as a coadjoint orbit which comes equipped with canonical sypmlectic
form, so the total number of physical degrees of freedom is one. Upon
quantization, it becomes the ferromagnetic magnon which is classified as a
type B Nambu-Goldstone boson [19].

##  V Coordinate expressions

Coordinate-free approach presented in previous sections is useful for general
discussions, but explicit form of effective Lagrangian is needed for practical
calculations. Since detailed accounts on coordinate expressions for various
$G$-invariant term exist [56, 57, 16, 48, 20, 58, 46], we will focus here only
on the one which is directly related to $\bm{\Sigma}$. Also, to compare
results presented here with existing literature, we shall assume
$\mathrm{M}=\mathbb{R}^{m+1}$ so that $\pi$ is a trivial bundle and that $G$
describes internal symmetry. Coordinate expressions also provide a direct
illustration of results obtained in previous section.

###  V.1 Coordinates on homogeneous space $G/H$

Suppose that the symmetry breaking occurs as described in Section III. To
discuss possible terms in effective Lagrangian, we first need to specify the
induced vector fields which describe symmetry transformations on $G/H$ and
corresponding jet bundle. If we choose $\\{u^{a}\\}$ as standard coordinates
on $G/H$, each point $\mathrm{a}\in G/H$ can be written as $\exp
u\cdot\bm{T}$, where we use abbreviation $u\cdot\bm{T}=u^{a}\bm{T}_{a}$ and
$\\{\bm{T}_{a}\\}$ constitute a basis of $\mathfrak{m}$. For given
$\bm{A}\in\mathfrak{g}$, the induced field $X_{\bm{A}}$ at a point
$\mathrm{a}\in G/H$ is given by given by [59]

$X_{\bm{A}}|_{\mathrm{a}}=\frac{\mathrm{d}}{\mathrm{d}\epsilon}\bigg{|}_{0}\Big{[}\exp\epsilon\bm{A}\exp
u\cdot\bm{T}\Big{]}_{\mathfrak{m}}\equiv(X_{\bm{A}})^{b}\frac{\partial}{\partial
u^{b}}\bigg{|}_{\mathrm{a}}$ (39)

where subscript $\mathfrak{m}$ denotes the restriction to
$T_{\mathrm{a}}(G/H)\simeq\mathfrak{m}$ and we have used abbreviation
$l(g,\mathrm{a})=g\mathrm{a}$. Now we specify a basis in $\mathfrak{g}$ so
that

$\Big{[}\bm{T}_{i},\bm{T}_{j}\Big{]}=f_{ij}^{\;\;k}\bm{T}_{k}$ (40)

and use BCH formula to evaluate exponetials. As a result, we obtain components
of induced vector fields as a power series in coordinates on homogeneous
space. The first few terms are

$(X_{\bm{A}})^{b}=A^{i}\left[\delta^{b}_{i}+u^{a}f_{ia}^{\;\;k}\delta^{b}_{k}+\frac{1}{2}u^{a}u^{c}f_{ia}^{\;\;k}f_{ck}^{\;\;j}\delta^{b}_{j}+\dots\right].$
(41)

As efficient algorithms for evaluating BCS terms exist [60], one can calculate
additional terms in a straightforward fashion. The 1-jet prolongation of
$X_{\bm{A}}$ is now given by [34]

$X_{\bm{A}}^{(1)}=(X_{\bm{A}})^{b}\;\frac{\partial}{\partial
u^{b}}+(X_{\bm{A}})^{b}_{\mu}\;\frac{\partial}{\partial u^{b}_{\mu}}$ (42)

where $u^{b}_{\mu}$ are derivative coordinates,

$(X_{\bm{A}})^{b}_{\mu}=\frac{\mathrm{d}(X_{\bm{A}})^{b}}{\mathrm{d}x^{\mu}}$
(43)

and

$\frac{\mathrm{d}}{\mathrm{d}x^{\mu}}=\frac{\partial}{\partial
x^{\mu}}+u^{b}_{\mu}\frac{\partial}{\partial u^{b}}$ (44)

is the total derivative on $J^{1}\pi$.

The Lorentz-symmetry breaking order parameter represents the ground state
expectation value of the Noether charge and thus it may be viewed as a
$\mathfrak{g}^{*}$ valued timelike vector field
$\bm{\Sigma}\otimes\partial_{t}$. A term in effective Lagrangian, invariant up
to total derivative [see (8)] may be obtained by contracting it with a
pullback of $\mathfrak{g}$ valued one-form on $G/H$. In terms of local
coordinates on $G/H$, we have

$\displaystyle\left\langle\bm{\Sigma}\otimes\frac{\partial}{\partial
t},\phi^{*}c^{i}_{a}(u)\mathrm{d}u^{a}\otimes\bm{T}_{i}\right\rangle=\Sigma_{i}c_{a}^{i}(s)\frac{\partial\phi^{a}}{\partial
t}=(j^{1}s)^{*}c_{a}(u)\dot{u}^{a}\equiv(j^{1}s)^{*}L^{(0,1)}$ (45)

where $c_{a}(u)=\Sigma_{i}c_{a}^{i}(u)$, $\dot{u}^{a}=u^{a}_{0}$ and
$\langle,\rangle$ denotes natural paring between elements of $\mathfrak{g}$
and its dual, as well as paring between tensor fields on $\mathrm{M}$.

According to (8), the invariance of classical field theory corresponding to
$L^{(0,1)}$ under the action of $G$ is expressed as

$X_{\bm{A}}^{(1)}\Big{(}L^{(0,1)}\Big{)}\mathrm{VolM}=\mathrm{d}\big{(}\Delta^{\mu}_{\bm{A}}\mathrm{Vol_{\mu}M}\big{)}$
(46)

Since

$X_{\bm{A}}^{(1)}\Big{(}L^{(0,1)}\Big{)}=\dot{u}^{a}X_{\bm{A}}^{b}\partial_{b}c_{a}+\dot{X}_{\bm{A}}^{b}c_{b}=X_{\bm{A}}^{b}\big{(}\partial_{b}c_{a}-\partial_{a}c_{b}\big{)}\dot{u}^{a}+\frac{\mathrm{d}}{\mathrm{d}t}\Big{(}X_{\bm{A}}^{b}c_{b}\Big{)},$
(47)

by assuming only $\Delta^{0}_{\bm{A}}\neq 0$, and defining
$\Delta^{0}_{\bm{A}}=E_{\bm{A}}+X^{a}_{\bm{A}}c_{a}$, where $E_{\bm{A}}$ is a
function on $\pi$ linear in $\bm{A}$, we obtain equations which determine
functions $c_{a}(u)$

$X_{\bm{A}}^{b}\big{(}\partial_{b}c_{a}-\partial_{a}c_{b}\big{)}=\partial_{a}E_{\bm{A}}.$
(48)

The components $X^{b}_{\bm{A}}$ are given in (41), while the meaning of
functions $E_{\bm{A}}$ follows from (21). By letting $\bm{A}$ to be one of the
generators of $\mathfrak{g}$, we see that

$\Sigma_{i}=\left\langle\frac{1}{V}\int_{\bar{\mathrm{U}}}s^{*}E_{i}\;\mathrm{Vol_{0}M}\right\rangle_{\rm{g.s.}}$
(49)

and equation (48) can be solved as described in [16, 20, 19].

Higher order terms in effective Lagrangian can be found in standard ways. For
example, a term quadratic in derivatives of Nambu-Goldstone fields can be
constructed from unique $G$ invariant metric on $G/H$ built from the $H$
invariant inner product on $\mathfrak{m}$ [39]. By our assumption, the Lorentz
invariance is broken by $\bm{\Sigma}$, but if rotational symmetry remains, the
inverse Lorentz metric $\eta^{\mu\nu}\partial_{\mu}\otimes\partial_{\nu}$
splits into $\eta^{\mathrm{t}}\partial_{t}\otimes\partial_{t}$ and
$\eta^{\mathrm{s}}\delta^{rs}\partial_{r}\otimes\partial_{s}$ where
$\eta^{\mathrm{t}}$ and $\eta^{\mathrm{s}}$ are constants,
$\eta^{\mathrm{s}}<0$ and $r,s=1,2,\dots m$. If
$g_{ab}\mathrm{d}u^{a}\otimes\mathrm{d}u^{b}$ is $G$ invariant metric on
$G/H$, two additional terms in effective Lagrangian are

$\displaystyle\left\langle\frac{1}{2}\eta^{\mathrm{t}}\partial_{t}\otimes\partial_{t},s^{*}g_{ab}\mathrm{d}u^{a}\otimes\mathrm{d}u^{b}\right\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{2}\eta^{\mathrm{t}}(j^{1}s)^{*}g_{ab}(u)\dot{u}^{a}\dot{u}^{b}\equiv(j^{1}s)^{*}L^{(0,2)},$
$\displaystyle\left\langle\frac{1}{2}\eta^{\mathrm{s}}\delta^{rs}\partial_{r}\otimes\partial_{s},s^{*}g_{ab}\mathrm{d}u^{a}\otimes\mathrm{d}u^{b}\right\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{2}\eta^{\mathrm{s}}(j^{1}s)^{*}g_{ab}(u){u}^{a}_{r}{u}^{b}_{r}\equiv(j^{1}s)^{*}L^{(2,0)}.$
(50)

The function $(j^{1}s)^{*}\left[L^{(0,1)}+L^{(0,2)}+L^{(2,0)}\right]$
describes the dynamics of $\mathrm{dim}(G)-\mathrm{dim(H)}$ Nambu-Goldstone
fields. To identify type B NG fields, one proceeds as described in [19, 20].
First, a quadratic approximation to Lagrangian is made after which one
diagonalizes the term with a single time derivative which leads to the
separation of type A and type B NG fields.

###  V.2 Coordinates on bundle $G/H$

While, in general, local coordinates on $G/H$ mix two types of NG fields, a
different view on space $G/H$ enables us to separate type A and type B fields
from the start. That is, with a suitable choice of coordinates on $G/H$, we
can make distinction between two types of NF fields in the effective
Lagrangian more apparent.

As we noted in Section III, homogeneous space $G/H$ is in fact a bundle with
base space $G/\mathrm{Stab}(\bm{\Sigma})$ which is a symplectic manifold. The
symplectic structure is provided by $G$ invariant Kirilov-Kostant (KK)
symplectic form which is defined as follows. Let
$\bm{\Sigma}\in\mathfrak{g}^{*}$ denote the Lorentz-symmetry breaking
parameter, $\bm{\Sigma}^{\prime}=\mathrm{Ad}^{*}(g)\bm{\Sigma}$ be another
point on $G/\mathrm{Stab}(\bm{\Sigma})$ obtained from $\bm{\Sigma}$ by
coadjoint action of $g\in G$ and let $X_{\bm{A}}$ and $X_{\bm{B}}$ be two
induced vector fields on $G/\mathrm{Stab}(\bm{\Sigma})$ corresponding to
$\bm{A},\bm{B}\in\mathfrak{g}$. The KK form $\omega_{\bm{\Sigma}}$ at
$\bm{\Sigma}^{\prime}$ is defined as [42]

$\omega_{\bm{\Sigma}}(\bm{\Sigma}^{\prime})\Big{(}X_{\bm{A}}\big{|}_{\bm{\Sigma}^{\prime}},X_{\bm{B}}\big{|}_{\bm{\Sigma}^{\prime}}\Big{)}:=\bigl{\langle}\bm{\Sigma}^{\prime},[\bm{A},\bm{B}]\bigr{\rangle}.$
(51)

We can now use (23) to to decompose $\bm{A}$ and $\bm{B}$ as
$\bm{A}=\bm{A}_{\mathfrak{stab}(\bm{\Sigma})}+\bm{A}_{\mathfrak{n}}$ and
$\bm{B}=\bm{B}_{\mathfrak{stab}(\bm{\Sigma})}+\bm{B}_{\mathfrak{n}}$. Since
$\bm{\Sigma}$ is $\mathfrak{stab}(\bm{\Sigma})$ invariant, we have

$\left\langle\bm{\Sigma},[\bm{A}_{\mathfrak{stab}(\bm{\Sigma})},\bm{B}]\right\rangle=\left\langle\bm{\Sigma},\mathrm{ad}(\bm{A}_{\mathfrak{stab}(\bm{\Sigma})})\bm{B}\right\rangle=-\left\langle\mathrm{ad}^{*}(\bm{A}_{\mathfrak{stab}(\bm{\Sigma})})\bm{\Sigma},\bm{B}\right\rangle=0,$
(52)

so that in the definition of $\omega_{\bm{\Sigma}}(\bm{\Sigma})$ we may
restrict $\bm{A}$ and $\bm{B}$ to their components within
$\mathfrak{n}=\mathfrak{g}/\mathfrak{stab}(\bm{\Sigma})$. If we pick a chart
with coordinates
$z^{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\alpha}}$,
where
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\alpha}},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\beta}}=1,2\dots\mathrm{dim}(G/\mathrm{Stab}(\bm{\Sigma}))$,
we can expand $\omega_{\bm{\Sigma}}$ as

$\omega_{\bm{\Sigma}}=\frac{1}{2}(\omega_{\bm{\Sigma}})_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\alpha}}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\beta}}}\;\mathrm{d}z^{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\alpha}}}\wedge\mathrm{d}z^{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\beta}}}.$
(53)

and, to the lowest order, it is given by

$\omega_{\bm{\Sigma}}=\frac{1}{2}\Sigma_{i}f_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\alpha\beta}}}^{\;\;\;\;i}\mathrm{d}z^{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\alpha}}}\wedge\mathrm{d}z^{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\beta}}}.$
(54)

Since $\mathrm{d}\omega_{\bm{\Sigma}}=0$, we can write
$\omega_{\bm{\Sigma}}=\mathrm{d}c_{\bm{\Sigma}}$, where

$c_{\bm{\Sigma}}=\frac{1}{2}\Sigma_{i}f_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\alpha\beta}}}^{\;\;\;\;i}z^{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\alpha}}}\mathrm{d}z^{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\beta}}}$
(55)

is an one-form on $G/\mathrm{Stab}(\bm{\Sigma})$ which can be used to
construct $L^{(0,1)}$

$\left\langle\frac{\partial}{\partial
t},\phi^{*}(c_{\bm{\Sigma}})_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\beta}}}(z)\mathrm{d}z^{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\beta}}}\right\rangle=\frac{1}{2}\Sigma_{i}f_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\alpha}}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\beta}}}^{\;\;\;\;i}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\phi_{\mathrm{B}}^{\alpha}}}\partial_{t}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\phi_{\mathrm{B}}^{\beta}}}=L^{(0,1)}\bigl{(}{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\phi_{\mathrm{B}}}}},\partial_{t}{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\phi_{\mathrm{B}}}}}\bigr{)}.$
(56)

It can be shown, using arguments similar to those which lead to (48), that
$L^{(0,1)}\bigl{(}{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\phi_{\mathrm{B}}}}},\partial_{t}{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\phi_{\mathrm{B}}}}}\bigr{)}$
changes by a total derivative under the infinitesimal action of $G$. We can
now also choose a local chart on fibers over $G/\mathrm{Stab}(\bm{\Sigma})$.
By denoting these coordinates as
$v^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\bar{\alpha}}}}$,
where
${{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\bar{\alpha}}}},{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\bar{\beta}}}}=1,2\dots\mathrm{dim}(\mathrm{Stab}(\bm{\Sigma})/H)$,
we arrive at the lowest order effective Lagrangian

$L(\phi,\partial_{t}\phi,\partial_{s}\phi)=\frac{1}{2}\Sigma_{i}f_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\alpha}}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\beta}}}^{i}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\phi_{\mathrm{B}}^{\alpha}}}\partial_{t}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\phi_{\mathrm{B}}^{\beta}}}-\frac{1}{2}(\eta^{\mathrm{s}}_{\mathrm{B}})_{{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\alpha}}}{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\beta}}}}\nabla{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\phi_{\mathrm{B}}^{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\alpha}}}}}\cdot\nabla{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\phi_{\mathrm{B}}^{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\beta}}}}}+\frac{1}{2}(\eta^{\mathrm{t}}_{\mathrm{A}})_{{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\bar{\alpha}}}}{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\bar{\beta}}}}}\partial_{t}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\phi_{\mathrm{A}}^{\bar{\alpha}}}}\partial_{t}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\phi_{\mathrm{A}}^{\bar{\beta}}}}-\frac{1}{2}(\eta^{\mathrm{s}}_{\mathrm{A}})_{{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\bar{\alpha}}}}{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\bar{\beta}}}}}\nabla{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\phi_{\mathrm{A}}^{\bar{\alpha}}}}\cdot\nabla{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\phi_{\mathrm{A}}^{\bar{\beta}}}}+O(\phi^{4})$
(57)

where
$(\eta^{\mathrm{s}}_{\mathrm{B}})_{{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\alpha}}}{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\beta}}}}$
are coefficients defining a $\mathrm{Stab}(\bm{\Sigma})$-invariant inner
product on $\mathfrak{g}/\mathfrak{stab}(\bm{\Sigma})$, while
$(\eta^{\mathrm{t}}_{\mathrm{A}})_{{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\bar{\alpha}}}}{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\bar{\beta}}}}}$
and
$(\eta^{\mathrm{s}}_{\mathrm{A}})_{{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\bar{\alpha}}}}{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\bar{\beta}}}}}$
are proportional to each other and define an $H$ invariant inner product on
$\mathfrak{stab}(\bm{\Sigma})/\mathfrak{h}$ [61, 43], and we have neglected a
term containing
${\partial_{t}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\phi_{\mathrm{B}}^{\alpha}}}\partial_{t}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\phi_{\mathrm{B}}^{\beta}}}$
[20]. This Lagrangian describes the dynamics of noninteracting NG fields with
type A and type B fields decoupled from each other. Further simplification can
be achieved with the use of Darboux coordinates around the origin in
$G/\mathrm{Stab}(\bm{\Sigma})$ in which $\omega_{\bm{\Sigma}}$ takes a
diagonal form. This choice of coordinates may not be optimal for other terms
in the effective Lagrangian, however. Note also that when viewing $G/H$ as a
bundle, $\bm{\Sigma}$ need not be explicitly included in (56) since the
information about the Lorentz-symmetry breaking order parameter is already
encoded into space $G/\mathrm{Stab}(\bm{\Sigma})$ via $\omega_{\bm{\Sigma}}$
and contraction of $\phi^{*}c_{\bm{\Sigma}}$ with $\partial_{t}$ simply, from
the point of view of effective Lagrangian, acknowledges that Lorentz symmetry
breaking did occur. Thus, even in the case when $G/H$ may be viewed as an
internal space (i.e. when the group $G$ describes an internal symmetry), the
symplectic manifold $G/\mathrm{Stab}(\bm{\Sigma})$ also reflects the
properties of spacetime $\mathrm{M}$, namely the existence of a timelike
vector field $\bm{\Sigma}\otimes\partial_{t}$ whose component is the vacuum
expectation value of Neother charge.

##  VI Explicit symmetry breaking

To study the effects of explicit symmetry breaking on the spectrum of Nambu-
Goldstone fields, imagine that we describe a system with a $G$-invariant
Hamiltonian $\mathsf{H}$. If the symmetry of the ground state is $H\subset G$
invariant, and if it develops a $\mathrm{Stab}(\bm{\Sigma})$-invariant
Lorentz-symmetry breaking order parameter $\bm{\Sigma}\in\mathfrak{g}^{*}$ as
a vacuum expectation value of Noether charge, the spectrum of Nambu-Goldstone
fields contains both type A and type B excitations [see(36) and (32)]. Suppose
that the symmetry of the model is explicitly broken by finite charge density
in the ground state. The system is now described by a modified Hamiltonian
$\widetilde{\mathsf{H}}=\mathsf{H}-\mu^{i}Q_{i}$, where $\mu^{i}$ is the
chemical potential corresponding to the Noether charge $Q_{i}$ [27, 62, 28,
29, 63]. As we saw in Section II, the Noether charge is an element of
$\mathfrak{g}^{*}$, $\bm{Q}=Q_{i}\bm{T}^{*i}$, so that $\mathsf{H}$ and
$\widetilde{\mathsf{H}}$ differ by $\left\langle\bm{Q},\bm{\mu}\right\rangle$
where $\bm{\mu}=\mu^{i}\bm{T}_{i}$ is a constant element of $\mathfrak{g}$
corresponding to the chemical potential. This observation allows us to
identify $\widetilde{G}$, the symmetry group of $\widetilde{\mathsf{H}}$, as
the stabilizer of $\bm{Q}$: $\widetilde{G}=\mathrm{Stab}(\bm{Q})$ which we
assume to be connected, simply connected Lie group. To account for subsequent
spontaneous symmetry breakdown, we must take $H\subset\widetilde{G}$. Also, we
shall assume that explicit symmetry breaking does not change the symmetry
group of the ground state and, with respect to the behavior of stabilizer
$\mathrm{Stab}(\bm{\Sigma})$, we distinguish between two interesting cases
below. As we illustrate in examples following general discussion, our
arguments also hold for theories defined using Lagrangian formalism where
chemical potential $\bm{\mu}$ appears as the temporal component of the gauge
field.

###  VI.1
$\widetilde{\mathrm{Stab}(\bm{\Sigma})}={\mathrm{Stab}(\bm{\Sigma})}$

Consider first the situation in which, with respect to the action of both $G$
and $\widetilde{G}$, $\mathrm{Stab}(\bm{\Sigma})$ remains the same. Since the
spontaneous symmetry breaking pattern is $\widetilde{G}\to H$, the number of
physical degrees of freedom carried by type A and type B NG fields is given by

$\widetilde{N}_{\mathrm{A}}=\mathrm{dim}\Bigl{(}\mathrm{Stab}(\bm{\Sigma})/H\Bigr{)},\hskip
28.45274pt\widetilde{N}_{\mathrm{B}}=\frac{1}{2}\mathrm{dim}\biggl{(}\widetilde{G}/\mathrm{Stab}(\bm{\Sigma})\biggr{)}.$
(58)

We see that the reduction of symmetry $G\to\widetilde{G}$ does not change the
number type A NG fields. On the other hand, the number of type B NG fields is
reduced and

$N_{\mathrm{mNGF}}=N_{\mathrm{B}}-\widetilde{N}_{\mathrm{B}}=\frac{1}{2}\mathrm{dim}\biggl{(}G/\mathrm{Stab}(\bm{\Sigma})\biggr{)}-\frac{1}{2}\mathrm{dim}\biggl{(}\widetilde{G}/\mathrm{Stab}(\bm{\Sigma})\biggr{)}$
(59)

is the number of physical degrees of freedom carried by ”massive” NG fields.
One way to understand (59) is to recall that, for $\widetilde{G}\to H$
symmetry breaking pattern, NG fields may be viewed as excitations generated by
space-time depended $\widetilde{G}$ transformation of the ground state
configuration [1]. All classical field configurations generated by elements
$g\in G$ not contained in $\widetilde{G}$ do not satisfy assumptions of the
Goldstone theorem and thus need not be ”massless”. That this is indeed the
case may be showed, for example, by using effective Lagrangian and
substitution $\partial_{0}\phi^{a}\to\partial_{0}\phi^{a}-(X_{i})^{a}\mu^{i}$,
together with additional terms containing no derivatives of NG fields [27]. In
particular, $L^{(0,1)}$ receives a contribution
$s^{*}\left\langle\bm{\Sigma}(z),\bm{\mu}\right\rangle$ [27], where
$\bm{\Sigma}(z)=\mathrm{Ad}^{*}(\exp z\cdot\bm{T})\bm{\Sigma}$,
$\bm{T}\in\mathfrak{n}\simeq\mathfrak{g}/\mathfrak{stab}(\bm{\Sigma})$ and
$z^{\alpha}$ are local coordinates on $G/\widetilde{G}$. Since
$\mathrm{Ad}(g)\bm{\mu}=\bm{\mu}$, for $g\in\widetilde{G}$, this term is
$\widetilde{G}$ invariant and vanishes as $\mu_{i}\to 0$. The series expansion
for $\Sigma_{i}(z)$

$\Sigma_{i}(z)=\Sigma_{i}+z^{\alpha}f_{\alpha
i}^{\;\;k}\Sigma_{k}+\frac{1}{2}z^{\alpha}z^{\beta}f_{\alpha
i}^{\;\;l}f_{\beta l}^{\;\;k}\Sigma_{k}+\dots$ (60)

which is useful for explicit calculations, can be obtained using the
definition of coadjoint action.

The associated bundle which includes effects of explicit symmetry breaking may
be build starting from the diagram

${G}$${G/H}$${G/\mathrm{Stab}(\bm{\Sigma})}$${G/\widetilde{G}}$$\textstyle{\widetilde{\pi}_{\mathrm{P}}}$$\textstyle{\pi_{\tilde{\mathrm{P}}}}$$\textstyle{f}$$\textstyle{f^{\prime}}$$\textstyle{\widetilde{f}}$
(61)

since the fiber of
$\bigl{(}G/\mathrm{Stab}(\bm{\Sigma}),\widetilde{f},G/\widetilde{G}\bigr{)}$
is $\widetilde{G}/\mathrm{Stab}(\bm{\Sigma})$ and this is the target space for
type B excitations generated by SSB pattern $\widetilde{G}\to H$, while the
total space $G/\mathrm{Stab}(\bm{\Sigma})$ hosts both massless and massive
type B NG fields. In other words, massive NG fields may be viewed as
excitations which take values on $G/\widetilde{G}$ which, since
$G/\widetilde{G}$ is the base space of bundle $G/\mathrm{Stab}(\bm{\Sigma})$,
is a symplectic manifold carrying the KK form induced by $\bm{\Sigma}$. The
number of physical degrees of freedom carried by massive NG fields is thus
$(1/2)\mathrm{dim}\bigl{(}G/\widetilde{G}\bigr{)}$, in agreement with (59).
Also, the fiber of $\bigl{(}G/H,f^{\prime},G/\widetilde{G}\bigr{)}$ is
$\widetilde{G}/H$, which is the target space for all massless excitations in
this case. Further construction of associated bundle may be conducted as
outlined in Section IV.

###  VI.2
$\widetilde{\mathrm{Stab}(\bm{\Sigma})}\subset{\mathrm{Stab}(\bm{\Sigma})}$

Suppose that, under the action of $\widetilde{G}$, the order parameter
stabilizer is reduced to
$\widetilde{\mathrm{Stab}(\bm{\Sigma})}\subset\mathrm{Stab}(\bm{\Sigma})$. The
numbers of physical degrees of freedom corresponding to type A and type B NG
fields are now

$\widetilde{N}_{\mathrm{A}}=\mathrm{dim}\Bigl{(}\widetilde{\mathrm{Stab}(\bm{\Sigma})}/H\Bigr{)},\hskip
28.45274pt\widetilde{N}_{\mathrm{B}}=\frac{1}{2}\mathrm{dim}\biggl{(}\widetilde{G}/\widetilde{\mathrm{Stab}(\bm{\Sigma})}\biggr{)}.$
(62)

While the number of type A NG fields reduces, we see that none of
$N_{\mathrm{A}}=\mathrm{dim}\bigl{(}\mathrm{Stab}(\bm{\Sigma})/H\bigr{)}$ of
them becomes massive. Thus, the number of physical degrees of freedom carried
by massive NG comes from the difference in type B NG fields and is given by

$N_{\mathrm{mNGF}}=N_{\mathrm{B}}-\widetilde{N}_{\mathrm{B}}=\frac{1}{2}\mathrm{dim}\biggl{(}G/\mathrm{Stab}(\bm{\Sigma})\biggr{)}-\frac{1}{2}\mathrm{dim}\biggl{(}\widetilde{G}/\widetilde{\mathrm{Stab}(\bm{\Sigma})}\biggr{)}$
(63)

which obviously reduces to (59) if
$\widetilde{\mathrm{Stab}(\bm{\Sigma})}=\mathrm{Stab}(\bm{\Sigma})$ and agrees
with [27].

To gain some additional insight into the formula (63), consider the following
diagram

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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>(64)

In comparison with (61), we have three additional bundles: $f_{1},f_{2}$ and
$f_{3}$. The fiber of $f_{1}$ is $\widetilde{\mathrm{Stab}(\bm{\Sigma})}/H$,
which is the target space for type A NG fields corresponding to
$\widetilde{G}\to H$ spontaneous symmetry breakdown, and, as we have seen,
they don’t affect the number of massive NG fields. On the other hand,
$G/\widetilde{\mathrm{Stab}(\bm{\Sigma})}$ is the target space for type B NG
fields of $\widetilde{G}\to H$ as well as for massive NG fields. It may be
viewed as the total space of both $f_{2}$ and $f_{3}$. Since respective fibers
of these bundles are
$\mathrm{Stab}(\bm{\Sigma})/\widetilde{\mathrm{Stab}(\bm{\Sigma})}$ and
$\widetilde{G}/\widetilde{\mathrm{Stab}(\bm{\Sigma})}$, both of which are
symplectic manifolds and represent two subspaces of
$G/\widetilde{\mathrm{Stab}(\bm{\Sigma})}$ which support massless type B NG
fields, the number of physical degrees of freedom carried by massive fields is

$N_{\mathrm{mNGF}}=\frac{1}{2}\mathrm{dim}\biggl{(}G/\widetilde{\mathrm{Stab}(\bm{\Sigma})}\biggr{)}-\frac{1}{2}\mathrm{dim}\biggl{(}\widetilde{G}/\widetilde{\mathrm{Stab}(\bm{\Sigma})}\biggr{)}-\frac{1}{2}\mathrm{dim}\biggl{(}\mathrm{Stab}(\bm{\Sigma})/\widetilde{\mathrm{Stab}(\bm{\Sigma})}\biggr{)}$
(65)

which is the same as (63). We can also add that, according to

${\widetilde{G}/H}$${\widetilde{G}/\widetilde{\mathrm{Stab}(\bm{\Sigma})}}$${\widetilde{G}/\mathrm{Stab}(\bm{\Sigma})}$
(66)

the type B NG fields separate into two additional categories: the ones which
take values on $\widetilde{G}/\mathrm{Stab}(\bm{\Sigma})$ and the others with
target space
$\mathrm{Stab}(\bm{\Sigma})/\widetilde{\mathrm{Stab}(\bm{\Sigma})}$. We also
note that

$\widetilde{N}_{\rm A}+2\widetilde{N}_{\rm
B}+2\widetilde{N}_{\rm{mNGF}}={\rm{dim}}\bigl{(}G/H\bigr{)}-{\rm
dim}\biggl{(}\mathrm{Stab}(\bm{\Sigma})/\widetilde{\mathrm{Stab}(\bm{\Sigma})}\biggr{)}={\rm{dim}}\bigl{(}G/H\bigr{)}-\bigl{(}N_{{\rm
A}}-\widetilde{N}_{{\rm A}}\bigr{)}$ (67)

is the difference between total number of fields in $G\to H$ and
$\widetilde{G}\to H$ SSB patterns when
$\widetilde{\mathrm{Stab}(\bm{\Sigma})}\subset{\mathrm{Stab}(\bm{\Sigma})}$,
since those NG fields of type A, which would be generated by elements of $G$
in $\mathrm{Stab}(\bm{\Sigma})$, but not in
$\widetilde{\mathrm{Stab}(\bm{\Sigma})}$, disappear from the spectrum. Thus,

$N_{\rm{mNGF}}=\frac{1}{2}\biggl{(}G/\tilde{G}\biggr{)}-\frac{1}{2}\biggl{(}\mathrm{Stab}(\bm{\Sigma})/\widetilde{\mathrm{Stab}(\bm{\Sigma})}\biggr{)}$
(68)

and we see that the target for space massive NG fields is again a symplectic
manifold, $G/\widetilde{G}$, but their number is reduced when compared to the
case $\mathrm{Stab}(\bm{\Sigma})=\widetilde{\mathrm{Stab}(\bm{\Sigma})}$.
Additional constrains can further decrease the number of massive NG fields
(See Example VI.3.3).

Finally, it may happen that the explicit symmetry breaking term changes not
only $\mathrm{Stab}(\bm{\Sigma})$, but $H$ also. Depending on the relationship
between
$G,\widetilde{G},\mathrm{Stab}(\bm{\Sigma}),\widetilde{\mathrm{Stab}(\bm{\Sigma})},H$
and $\widetilde{H}$, where $\widetilde{H}$ denotes the symmetry group of the
new ground state, one can identify massless and massive NG fields in each
particular case using already established methods.

###  VI.3 Examples

####  VI.3.1 Heisenberg antiferromagnet

Heisenberg antiferromagnet is a famous example of
$\mathrm{SO}(3)\to\mathrm{SO}(2)$ SSB. Due to the nature of exchange
interaction, none of the generators (components of total spin operator)
develop vacuum expectation values and the spectrum contains two NG fields of
type A. The situation changes if the antiferromagnet is put into magnetic
field $\bm{B}$. Suppose the field is oriented along the easy axis of
antiferromagnetic system. This field explicitly breaks original
$G=\mathrm{SO}(3)$ of Heisenberg Hamiltonian by reducing it to
$\widetilde{G}=\mathrm{SO}(2)$. If the field is strong enough, the spins will
rearrange themselves in such a way that, at first, easy axis is perpendicular
to magnetic field and, eventually, the canted phase will emerge [64]. The
canted phase is characterized by complete breakdown of internal
$\mathrm{SO}(3)$ symmetry and a non-zero value of net magnetization (See
Figure 1). Thus,
$\mathrm{Stab}(\bm{\Sigma})=\widetilde{\mathrm{Stab}(\bm{\Sigma})}=\mathrm{SO}(2)$
and $H=\\{e\\}$. Still, after SSB, there is only one type A NF field and this
is related to the fact that $\widetilde{G}=\mathrm{SO}(2)$ is Abelian. The
spectrum also contains massive NG fields whose number is determined by (59).
This gives $(3-1)/2=1$ degree of freedom carried by massive NG field, which is
a well known result [64, 65, 66, 67, 27].

The lowest-order Lagrangian which describes the spin waves in this case may be
constructed as follows. Let
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}v}$ denote
the coordinate on $S^{1}$ so that
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi_{\mathrm{A}}}\equiv{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}v}\circ{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi_{\mathrm{A}}}$
be a local representative of type A NG field, and let us choose coordinates on
$S^{2}$ in such a way that two massive NG fields are collected into a complex
field
${\color[rgb]{0,0.9,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.9,0.8}\psi_{\rm
M}}=\sqrt{|\bm{\Sigma}|/2}({\color[rgb]{0,0.9,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.9,0.8}\phi_{\rm