2.1. Review of coset construction

In the bottom-up approach to effective field theory, one typically aims to construct an effective action by writing down all possible terms that are “manifestly invariant” under the salient symmetry group—here, physicists have in mind the case where the degrees of freedom transform as linear representations of the symmetry group, and the action invariants are correspondingly easy to construct. However, it is seldom obvious how to write down such terms when those symmetries are non-linearly realized by the degrees of freedom in which one is interested. A paradigm case of such non-linear realizations is given by Goldstone modes generated from spontaneous symmetry breaking (SSB; another paradigm case of non-linear realizations is in how spacetime translations act on fields).Footnote ³ Goldstone’s theorem guarantees the existence of gapless degrees of freedom, referred to as Goldstone modes (or simply “Goldstones”), in cases of SSB. The coset construction is a formal recipe for writing down an effective action in terms of such modes, which tend to dominate the dynamics at large length (low-energy) scales. Importantly for our purposes, the coset construction can be extended to deal with spacetime symmetries (like those involved in Corollaries V and VI), which are both more ubiquitous and more subtle.

The coset construction recipe is roughly as follows.Footnote ⁴ First, we identify a symmetry-breaking pattern, in the form of a “full-theory” symmetry group G (capturing the relevant physics at some higher-energy scale) which is broken to a subgroup $\tilde{G}$ at some lower-energy scale—with the goal of writing an effective action at the latter scale. We split the generators of G into three categories: the broken generators $X_a$ , the unbroken translation generators $\bar{P}_{\alpha}$ , and the other unbroken generators $T_b$ . Using this categorization, we then write down the coset parameterization U, which is the most general group element of the non-linearly realized symmetry group (the broken group plus unbroken translations):Footnote ⁵

(1) ${e^{{{\bar P}_\alpha }{x^\alpha }}}{e^{{X_a}{\pi ^a}(x)}},$

where $\pi^{a}(x)$ are the Goldstone modes. Using (1), we can evaluate the Maurer–Cartan 1-form $\Omega_{\rm{MC}} = U^{-1}dU$ , which is valued in the algebra of G and is invariant under G-transformations. It can thus be decomposed into terms linear in the generators of G:

(2) ${\Omega _{{\rm{MC}}}} = \Omega _{\bar P}^\alpha {{\bar P}_\alpha } + \Omega _X^a{X_a} + \Omega _T^b{T_b}.$

It can be shown that the 1-forms $\Omega_{\bar{P}}$ and $\Omega_X$ transform linearly under G, while $\Omega_T$ transforms like a connection form (or gauge potential). Furthermore, $\Omega_{\bar{P}}$ plays the role of a vielbein (which is trivial for our non-relativistic examples). In addition to the above 1-forms, which can be used to construct terms in the effective action that are strictly invariant under G, one also must include terms that are invariant up to a total derivative, known as Wess–Zumino–Witten (WZW) terms.Footnote ⁶

Before constructing the effective action, we should note a complication in the case of spacetime symmetry breaking. While Goldstone’s theorem guarantees the existence of one Goldstone for every broken internal symmetry, descriptive redundancies in spacetime symmetries can lead to fewer Goldstones resulting from broken spacetime symmetries.Footnote ⁷ More specifically, if the commutator of an unbroken translation generator with a broken generator contains another broken generator (i.e., $[P,X_1]=X_2$ ), then one can impose a constraint to eliminate one of the Goldstones.Footnote ⁸ However, what constraint to impose, or even whether to impose any such constraint, is not always obvious. Often there will be independent reasons to motivate the choice to eliminate certain Goldstones in favor of others (as is the case in the examples we consider).

The above machinery gives us a recipe for using the symmetry-breaking pattern to construct an effective action using terms composed of invariant combinations of the components of $\Omega_{\rm{MC}}$ and WZW terms: roughly speaking, we simply contract indices invariantly using combinations of the coefficients of the broken generators while using the coefficients of the unbroken generators to form covariant derivatives (see the references in footnote Footnote 4 for details). The resulting effective action will contain all possible combinations of Goldstones that are consistent with the symmetry-breaking pattern.

The coset construction has been successfully applied in multiple areas of physics, especially in high-energy particle physics and condensed matter physics. The technique was originally established to construct effective actions for light mesons (e.g., pions), which non-linearly realize chiral symmetries. Since then, it has been used to study Fermi liquids (Penco Reference Penco2020), relativistic membranes and superfluids (Delacretaz et al. Reference Delacretaz, Endlich, Monin, Penco and Riva2014), gauge theories (Goon et al. Reference Goon, Joyce and Trodden2014), and generic condensed matter systems (Nicolis et al. Reference Nicolis, Penco, Piazza and Rattazzi2015). Its power lies in its generality and in the ubiquity of the symmetry-breaking analysis—lots of systems in which we are interested break “full-theory” symmetries, and all we need to know to write down an effective action is the symmetry-breaking pattern (though more information may be needed to interpret the effective action in terms of appropriate degrees of freedom).

2.2. Interpreting the coset construction

In the previous section, we summarized the method for constructing effective actions from our knowledge of a generic symmetry-breaking pattern. However, how should one interpret this “symmetry-breaking pattern?” The answer is less than clear in many reviews of the coset construction, and typically proceeds by means of an analogy with more detailed cases of SSB. In this section, we propose a novel and minimal interpretation of “symmetry-breaking pattern” that does not draw on an explicit analogy with SSB (though the latter is an interesting topic that we leave to a future investigation), and furthermore reveals a fruitful connection to the kinds of symmetry principles discussed in the philosophy literature (e.g., Newton’s Corollaries V and VI).

In our view, what is typically meant (in the context of the coset construction) by “the symmetry G gets broken to $\tilde{G}$ ” is that (in some relevant low-energy context) we have the following schematic empirical subsystem–environment decomposition:

1. (i) The subsystem is such that the statesFootnote ⁹ related by “broken” subsystem symmetries—call these ground states Footnote ¹⁰—are indistinguishable relative to some set of scales (which we might conceptualize as relative to a physical “subsystem observer”).

2. (ii) Correlatively, we have an environment with respect to which the ground states are physically distinct, and thus relative to which the notion of “broken” symmetry makes sense.

We stress that the above ground states are not the kinematic states of the desired EFT. Rather, they form a starting point for the goal of the coset construction, viz. to describe the dynamics of small perturbations (“Goldstone modes”) around these ground states (the necessary kinematic information is thus introduced at the level of the Goldstones).

This interpretation makes good sense of the physics literature. In a representative piece of reasoning, Delacretaz et al. (Reference Delacretaz, Endlich, Monin, Penco and Riva2014) write that “… any state of matter which is not the vacuum must break at least some space-time symmetries. For instance, even a state with a single point particle at rest breaks boosts by selecting a preferred reference frame” (2). The physical image here is clearly not one of a lonely particle in an empty universe, but rather a particle subsystem whose being “at rest” makes sense relative to some reference frame provided by the environment (e.g., the fixed stars). Furthermore, our interpretation dissolves the (merely) apparent tension between the claim that the free particle breaks boost symmetries, and the uncontroversial statement that boosts are symmetries of the free particle. The former sense in which “boosts are broken” refers to a schematic notion of an environment with respect to which the “breaking” can be said to take place, whereas the latter sense (in which boosts are symmetries) refers to our (i) above, viz. the indistinguishability of ground states with respect to a suitably defined subsystem observer. Not only are these statements compatible, but they are mutually constitutive of the relevant notion of “symmetry breaking pattern.”

Our interpretation, and the above example, bring us tantalizingly close to a symmetry that has been much discussed by philosophers, viz. that of Newton’s Corollary V: “[w]hen bodies are enclosed in a given space, their motions in relation to one another are the same whether the space is at rest or whether it is moving uniformly straight forward without circular motion” (Newton et al. Reference Newton, Cohen, Whitman and Budenz1999). Evidently, the symmetry in question is the uniform boosts already considered in Galileo’s “ship” thought experiment. This case has been analyzed in terms of subsystem–environment decomposition in the “empirical significance of symmetry” literature (Greaves and Wallace Reference Greaves and Wallace2014; Teh Reference Teh2016; Murgueitio-Ramírez and Teh Reference Murgueitio-Ramírez and Teh2021), which argues that boost symmetry is empirically significant because it is a subsystem symmetry that effects a relational difference between the subsystem and environment. It is thus tempting to forge an explicit connection between statements like Corollary V above and our interpretation of the coset construction’s symmetry-breaking pattern. Indeed, it has recently emerged that Newton himself originally thought of Corollary V as a fundamental “law” rather than a corollary (of the laws of motion) in an early draft of the Principia (DiSalle Reference DiSalle2020). This raises the question of whether (inverting the procedure of the published Principia) one could in a suitable sense start with Corollary V and derive the free particle equations of motion.

We now propose a general recipe of this kind for Corollary V-type statements. First, interpret such a statement from the perspective of the “empirical significance of symmetry” literature, i.e., the relevant subsystem symmetry—e.g., a uniform boost—effects a relational difference between the subsystem and its environment.Footnote ¹¹ Next, following the above discussion, take these “relational difference”-effecting subsystem symmetries to be the ones that count as “broken” in the coset construction (the ones that do not effect any physical difference will count as “unbroken”). We then have the input of a symmetry-breaking pattern (as we interpreted it above), and we can run the coset construction on this input to produce an effective action. In the next subsection, we demonstrate that when Corollary V is understood as a symmetry-breaking pattern along these lines, the coset construction gives us the (effective) Newtonian free particle action.

2.3. Example: Free, non-relativistic point particle

Given the previous section’s interpretation of the coset construction, we understand Corollary V as fixing a particular symmetry-breaking pattern, relative to the expected full symmetry group of a non-relativistic spacetime theory. We now wish to relate this understanding to a pre-existing derivation in the physics literature on coset construction—namely, the effective action for a non-relativistic, free point particle. Our approach here largely follows the calculation in Rothstein (Reference Rothstein2018);Footnote ¹² our novel contribution is to (i) connect this calculation to Corollary V, and (ii) to correct the symmetry group used in Rothstein’s calculation—rather than the Galilean group, we use the Bargmann group, which is the correct symmetry group for a potential-based formulation of Newtonian gravitation, cf. Teh (Reference Teh2018).

Following the approach outlined in the previous section, we schematically consider the motion of a (non-relativistic) free point particle (or a sufficiently point-particle-like collective coordinate), understood as taking place in a subsystem relative to some faraway reference body that operationalizes the environment. The important thing for our purposes is that such a schematic subsystem can be understood to “break boosts and spatial translations” in exactly the sense required by our interpretive recipe—both boost and translation subsystem symmetries will effect a relational difference between the subsystem and its environment (the reference body). Thus, we take the symmetry-breaking pattern to be $G \mapsto \tilde{G}$ , where G is the Bargmann group (the central extension of the Galilean group) and $\tilde{G}$ consists only of rotations, time translations, and the central mass symmetry.

Given the interest in describing point particle dynamics, we assume a “worldline action,” meaning that the Goldstones are only functions of time t. In this sense, we are making an assumption about the kinematics of our EFT degrees of freedom. The coset construction recipe then gives us the following coset parameterization and Maurer–Cartan form:Footnote ¹³

(3) $U = {e^{Ht}}{e^{\vec P\cdot\vec y(t)}}{e^{\vec K\cdot\vec \eta (t)}}\quad \to \quad {\Omega _{{\rm{MC}}}} = {\Omega _H}H + \Omega _P^a{P_a} + \Omega _K^a{K_a} + \Omega _L^{ij}{L_{ij}} + {\Omega _M}M,$

where $\vec{y}(t)$ and $\vec{\eta}(t)$ are Goldstones corresponding to broken translations and boosts respectively. Following the procedure outlined in section 2.1, we arrive at the following ingredients for constructing an effective action:

(4) $\Omega _K^a = - {{\ddot y}^a}\quad {\rm{and}}\quad {\Omega _M} = - {1 \over 2}{{\dot y}^2},$

where the former enters the effective action by straightforwardly contracting indices and the latter typically enters as a connection.

To proceed, we must consider two points. First, although the $\Omega_M$ term would typically enter the effective action as a connection (modifying the time derivative), it turns out that it is also a WZW term (because it is invariant up to a total derivative). Therefore, according to the coset construction recipe, $\Omega_M$ by itself should be included in the effective action—which makes sense, because it is clearly proportional to the expected kinetic term! Second, terms constructed via contractions of $\Omega_K^a$ will tend to lead to unstable theories according to Ostrogradsky’s theorem (Woodard Reference Woodard2015). Therefore, such terms should not contribute at leading order, but should be understood as subleading corrections; indeed, Rothstein (Reference Rothstein2018) argues that these are just scale-dependent finite-size effects, which will be small for a relatively particulate object.

Thus, taking into account the kinetic term as a WZW term and neglecting higher-order perturbations, the effective action is

(5) $S = \displaystyle\int d t \; \left( { - {{{c_1}} \over 2}{{\dot y}^2} + \cdots } \right).$

We identify the undetermined constant as $c_1 = -m$ , giving us an effective action corresponding to force-free motion.

3.1. Gravitational dynamics

In this section, we first isolate the local (in time) central mass charge (Abelian) symmetry and pose the question of what effective Goldstone dynamics emerge when we break this local symmetry to a global subgroup (i.e., the symmetry-breaking pattern of Corollary VI); the answer will turn out to have the form of a gravitational potential action. In the next subsection, we follow a standard prescription for coupling this action to the effective point particle theory of Corollary V, viz. by gauging the coset construction of section 2.3.

To proceed, we leverage Goon et al.’s (Reference Goon, Joyce and Trodden2014) insight that gauge symmetries can be understood as non-linearly realized by the gauge fields.Footnote ¹⁶ Thus, the coset construction can be used to write down an effective action for gauge fields (which turn out to be the Goldstones) by taking the full local gauge symmetry and “breaking” it down to a global subgroup where the symmetry transformation is linear. In the case at hand, the local symmetry group is generated by the mass charge M and has the parametrization $g = e^{\alpha(x)M}$ , where $\alpha(x)$ is some arbitrary function (in a moment, we will set $\alpha$ to be just a function of time, since this is the case relevant to Corollary VI).

Suppressing calculational details, we can use the coset construction to find the following building blocks for the effective action:Footnote ¹⁷

(6) $\Omega _P^\alpha = d{x^\alpha },\qquad {\Omega _M} = d{x^\nu }{A_\nu },\qquad {\Omega _\nu } = - {1 \over 2}{F_{\mu \nu }}d{x^\mu },$

with other components of $\Omega_{\rm{MC}}$ being composed of covariant derivatives of the field strength tensor. While $\Omega_M$ transforms like a connection (as one might expect of a gauge field), the other components can be used to build invariant terms in the effective action. The lowest-order such term is simply the usual quadratic kinetic term $F\wedge\star F \propto F_{\mu\nu}\,F^{\mu\nu}d^4x$ . As we mentioned earlier, in our inertial coordinates the gauge field takes the form $A=\phi dt$ , and so the kinetic term reduces to the usual kinetic term for the gravitational potential, yielding

(7) $S = \displaystyle\int {{d^4}} x[{c_1}{(\nabla \phi )^2} + \cdots ],$

with $c_1=1/8\pi G$ .

3.2. Coupling to the point particle

Finally, we consider how to couple the gravitational field introduced above to the previously discussed point particle action via the coset construction. Since Corollary VI already suggests a candidate gauge field theory of force (whose action we derived in the last section), we follow the prescription from Delacretaz et al. (Reference Delacretaz, Endlich, Monin, Penco and Riva2014) for how to couple a point particle EFT to a gauge field. More precisely, we return to the symmetry-breaking pattern of Corollary V (discussed in section 2.3), gauge the global central mass symmetry so that it becomes local, and run the coset construction on this data—the resulting action then includes a coupling between the effective particle and the gauge field.

We begin by revisiting the Maurer–Cartan form for the free particle, equation (3), in section 2.3 and “gauge” the construction by replacing the exterior derivative d with the covariant derivative $d + AM$ (where A is the gauge field correponding to the local mass gauge symmetry and M is the mass central charge generator). In our choice of coordinates, the “gauged” MC form is

(8) ${\Omega _{{\rm{MC}}}} = {U^{ - 1}}({\partial _t} + \phi M)Udt = dt({\Omega _H}H + \Omega _P^a{P_a} + \Omega _K^a{K_a} + \Omega _L^{ij}{L_{ij}} + {\Omega _M}M).$

The calculation remains mostly the same, since M commutes with all other generators, with just one extra term in the expression for $\Omega_M$ . Eliminating $\vec{\eta}$ as before, we have

(9) ${\Omega _M} = - {1 \over 2}{{\dot y}^2} + \phi .$

This leads to the following effective action (ignoring terms that lead to instabilities):

(10) $S = \displaystyle\int d t \; \left( {{1 \over 2}m{{\dot y}^2} - m\phi + \cdots } \right).$

Finally, we can combine (10) with (7) to yield

(11) $S = \displaystyle\int d t \;\left[ {{1 \over 2}m{{\dot y}^2} - m\phi - \displaystyle\int {{d^3}} x{1 \over {8\pi G}}{{(\nabla \phi )}^2}} \right],$

or, with the understanding that the point particle mass density is $\rho(\vec{x},\vec{y})=m\delta^3(\vec{y}-\vec{x})$ ,

(12) $S = \displaystyle\int d t \;\left[ {{1 \over 2}m{{\dot y}^2} - \displaystyle\int {{d^3}} x\left\{ {\rho \; \phi + {1 \over {8\pi G}}{{(\nabla \phi )}^2}} \right\}} \right].$