Irreversible vierbein postulate: Emergence of spacetime from quantum phase transition

We formulate a model for quantum gravity based on the local Lorentz symmetry and general coordinate invariance. A key idea is the irreversible vierbein postulate that a tree-level action for the model at a certain energy scale does not contain an inverse vierbein. Under this postulate, only the spinor becomes a dynamical field, and no gravitational background field is introduced in the tree-level action. In this paper, after explaining the transformation rules of the local Lorentz and general-coordinate transformations in detail, a tree-level action is defined. We show that fermionic fluctuations can induce a nonvanishing gravitational background field.


Introduction
Quantum gravity-how to consistently quantize interacting spacetime fluctuations-is one of the most profound mysteries that has defied human challenges over the past century.On the other hand, it has been established experimentally that there indeed exist spacetime fluctuations that propagate over cosmic distances with the speed of light, namely the gravitational waves, consistently described by Einstein's general relativity; see e.g.Refs.[1,2] for classic examples.Given the tremendous success of the Standard Model (SM) of particle physics based on quantum field theory (see e.g.Ref. [3] for a review), it is natural to expect that the gravitational field, governing the observed spacetime fluctuation, must be quantized too.Whether it is really quantized or not will be experimentally explored within the forthcoming decades as a form of quantized free spacetime fluctuations, gravitons, on a curved classical background during inflation in observations of the B-mode polarization of cosmic microwave background [4,5,6] and further in direct observation of the cosmic gravitational-wave background [7]; see also Ref. [8].
It is well-known however that quantization of metric based on Einstein's general relativity is perturbatively non-renormalizable, requiring an infinite number of counterterms, spoiling its predictability at the quantum level due to the infinite number of free parameters; see e.g.Ref. [9] and references citing it.Furthermore, the truncation of the gravitational action up to the dimension-two Einstein-Hilbert term with the Ricci curvature scalar R yields a conformal mode that has a wrong-sign kinetic term, which makes Euclidean quantum gravity ill-defined for both the directions of Wick rotations such that either the wrong-sign mode or the other fields become exponentially growing along the imaginary time direction; see e.g.Refs.[10,11]; see also Appendix A in Ref. [12] for a simple review.
It is known that a higher-derivative gravity that involves the R 2 and R 2 µν terms in addition to the Einstein-Hilbert term is perturbatively renormalizable; however, in compensation, the unitarity of the theory is lost [13].If we allow the theory to discard the Lorentz symmetry, it could be perturbatively renormalizable [14], though the speed of light depends on particle species and hence we need additional fine-tunings; see also Ref. [15] and references citing it.In any case, it appears that the realization of a renormalizable theory of gravity in perturbation theory is difficult if we retain all the essential symmetries and properties, in particular, both the Lorentz symmetry and unitarity.
The above perturbative non-renormalizability argument is based on quantum theory around a free theory.In the Wilsonian viewpoint, the perturbative gravity is constructed around the Gaussian (trivial) fixed point.On the other hand, the notion of renormalizability in quantum field theory is generalized to the non-perturbative realm.This scenario is known as asymptotically safe gravity [16,17,18]; see also Refs.[19,20,21,22,23,24,25,26,27,28,29,30,31,32] for reviews.There is accumulating evidence that there exists a non-trivial (interacting) ultraviolet (UV) fixed point in gravitational systems by means of the functional renormalization group method.Quantum gravity is perturbatively non-renormalizable but might be non-perturbatively renormalizable.This situation is similar to O(N ) non-linear sigma model.(This will be discussed in Section 2.) The statements mentioned above are based on an assumption that the metric field (spin-2 symmetric tensor field) is the fundamental degree of freedom.The view that Einstein's general relativity is a local-Lorentz (LL) gauge theory [33] is found almost at the same time as the (non-abelian) gauge theory itself [34] and has been developed in Refs.[35,36].To write down the LL symmetry, it is essential to rewrite the metric degrees of freedom by the vierbein (tetrad) ones.The vierbein is also indispensable to writing down a spinor field on a curved space, namely, the matter field in our universe.In this sense, the vierbein degrees of freedom are more fundamental than the metric ones.In this paper, we postulate that the dynamical degrees of freedom that describe spacetime fluctuation are the vierbein and the LL gauge field. 5The simplest gravitational model with the vierbein and the LL gauge fields is the Einstein-Cartan gravity; see e.g.Ref. [38] for a review on classical Einstein-Cartan gravity and Refs.[39,40] for its quantization.
In this paper, we consider a model for gravity and matter based on the LL gauge symmetry as well as the invariance under the general coordinate (GC) transformation [sometimes interchangeably called diffeomorphism (diff)] at a certain energy scale Λ G [41].In particular, we postulate that its tree-level action admits the degenerate limit of vierbein [42,43].This forbids inverse vierbeins in the action and therefore we call it the "irreversible vierbein postulate".
Under the irreversible vierbein postulate, only spinor fields can have kinetic terms, while the other fields become dynamical due to the quantum effects of spinor fields below Λ G .The main purpose of this work is to demonstrate the generation of a spacetime background, i.e., the emergence of a non-vanishing background vierbein field, due to quantum fluctuations of the spinor field.This idea might also be viewed along the direction of pregeometry; see e.g.Refs.[41,44,45,46,47,48,49,50,51,52,53,54,55].
This paper is organized as follows: We start with a brief overview of degrees of freedom and symmetries in gravitational theories in Section 2. In Section 3, we introduce our notation and explain transformation laws under the LL and GC transformations in detail.In particular, together with Appendices A and B, we intend to highlight differences between earlier works and ours.Then, we implement the degenerate limit on the action in Section 4, where we refer to Appendices C and D for detailed calculations.After briefly explaining transformation laws for the background fields in Section 5, we demonstrate a generation of a non-vanishing flat background field of vierbein due to quantum effects of fermionic degrees of freedom in Section 6.In Section 7, we summarize this work and discuss future prospects.

Degrees of freedom of gravitational fields
In this section, we first review the ordinary minimal Einstein gravity in the metric formalism and then in the vielbein formalism.

Minimal Einstein gravity in metric formalism
It is known that the Einstein-Hilbert action as the metric formalism well-describes the classical gravitational interactions in D = 4. Here, M 2 P = 1/(8πG N ) is the Planck mass squared or inverse Newtonian coupling constant and Λ cc is the cosmological constant.The Ricci scalar curvature R(g) is given by the metric field g µν and its inverse g µν .
The metric field is a symmetric tensor, so it classically has D(D + 1)/2 degrees of freedom in D dimensional spacetimes.
It is known that the quantum theory based on the action (1) is non-renormalizable in terms of the perturbative expansion of G N [56].The simplest explanation for the perturbative non-renormalizability is the negative mass-dimensionality of G N .This may however be somewhat naive.Indeed, the Einstein-Hilbert action in three-dimensional spacetime is renormalizable even though the Newton coupling has the negative mass-dimensionality [57].This is because the Einstein-Hilbert action in three-dimensional spacetime becomes a topological theory and thus can be formulated as a Chern-Simons theory due to the peculiarity of the three-dimensional spacetime.In other words, this is because there are no propagating degrees of freedom of graviton.The simple dimensional counting of the coupling constant cannot fully capture the property of renormalizability.
Another viewpoint why the perturbation theory for the Einstein-Hilbert action becomes non-renormalizable is the existence of the inverse metric which is defined by This reversibility condition (2) for the metric field induces an infinite number of interactions: When one considers the metric fluctuation field h µν around a background field ḡµν , namely g µν = ḡµν + h µν , the inverse metric is expanded so as to satisfy Eq. ( 2) and is given by This series continues infinitely around a certain background field.That is, once the inverse metric is defined by Eq. ( 2), the Ricci scalar curvature in the action (1) generally contains an infinite number of vertices of metric fluctuations, whereas all vertices have a common coupling constant G N .In general, one cannot remove all the UV divergences arising from quantum loops including an arbitrary number of vertices by only a single coupling constant.

Non-linear and linear sigma models
The above situation is quite similar to the O(N ) non-linear sigma model which is a low energy effective model of pions, π i (i = 1, . . ., N − 1).Its action is given by where f π is the pion decay constant.This theory can be obtained from the spontaneous symmetry breaking in the O(N ) linear sigma model whose action reads Here ϕ i = π j , σ with i = 1, . . ., N and j = 1, . . ., N − 1.For m 2 < 0, a non-trivial vacuum is realized and O(N ) symmetry is broken into O(N − 1).As a consequence, the σ mode becomes massive and decouples from the low energy dynamics, while π i are massless and remain as effective degrees of freedom in the low energy regime.In this case, one has the constraint on fields (6) with which integrating out the σ mode in Eq. ( 5) (with the constraint σ = f 2 π − (π i ) 2 ) results in the action of the non-linear sigma model (4).The decay constant just corresponds to the vacuum expectation value f π = 2 |m 2 | /λ.To summarize, the nonlinear sigma model ( 4) is obtained from the expansion of the linear sigma model (5) around the vacuum (6).
An important fact is that for D > 2, the non-linear sigma model is perturbatively nonrenormalizable, while the linear sigma model is perturbatively renormalizable.The parameter f π , which arises from the consequence of the O(N ) symmetry breaking in the linear sigma model, is a free parameter in the non-linear sigma model.In particular, the massless pions are realized only at the vacuum at ⟨σ⟩ = f π in the linear sigma model as a consequence of the Nambu-Goldstone theorem.In this viewpoint, one has an inconsistency in the non-linear sigma model between the massless pion condition and a free choice of f π , and there is a range of validity for pion field fluctuations: π i ≲ f π .This makes the system non-renormalizable. 6

Minimal Einstein gravity in vielbein formalism
In the metric theory describing gravity, Eq. ( 2) may be regarded as the constraint analogous to Eq. ( 6).Following the argument above, the inconsistency at high energies in the metric formalism may be between an expansion of the metric field around a background field, e.g. a flat background metric ⟨g µν ⟩ = ḡµν = η µν , and the existence of massless metric fields.We expect that there exists an appropriate high-energy theory of the metric theory and the generation of a vacuum ⟨g µν ⟩ = ḡµν ̸ = 0 may imply the appearance of the massless metric field.The reversibility condition (2) for the metric field enforces a finite domain where quantum fields can fluctuate analogously to π i ≲ f π in the non-linear sigma model.Beyond such a domain, we expect that new degrees of freedom appear and participate in the dynamics.
We notice at this point that the gravitational theory with the action (1) is similar to the non-linear sigma model (5).Hence, we intend to construct a gravitational theory with new additional degrees of freedom analogous to the meson σ in the linear sigma model.
Having this viewpoint in mind, we are motivated to consider a theory for gravity with more degrees of freedom.Let us here deal with a formulation for the gravitational theory with vielbein e a µ (x) and LL gauge field ω a bµ (x), known as the Einstein-Cartan gravity based on SO(1, d) LL symmetry.Its minimal form of action is given by where |e(x)| = det a,µ e a µ (x) is the determinant of vielbein, and In this formulation, it seems that there are apparently D(3D − 1)/2 independent classical degrees of freedom because e a µ (x) and ω a bµ (x) have D 2 and D(D − 1)/2 degrees of freedom, respectively.In the action (7), however, there are no apparent kinetic terms for neither vielbein nor the LL gauge field, so these fields are auxiliary fields at this stage, i.e., they are not dynamical degrees of freedom yet.Imposing the equation of motion on ω a bµ (x), i.e. δS EC /δω µ = 0, one obtains its solution to ω a bµ (x) as a function of vielbein, namely the Levi-Civita spin connection: ω a bµ (x) = e Ω a bµ (x); see Eq. ( 73) below for its explicit definition.By substituting this into ω a bµ (x) in the field strength, the LL gauge field disappears from the action, and the action is written in terms of only vielbein with the kinetic term.At this point, the second term on the right-hand side of Eq. ( 7) just turns to the Einstein-Hilbert term written in terms of vielbein and the number of dynamical degrees of freedom is D(D + 1)/2 which is the same as that of the symmetric metric field, being the composite of vielbein fields g µν (x) = η ab e a µ (x) e b ν (x).From the fact above, it turns out that a certain condition between e a µ (x) and ω a bµ (x), such as the equation of motion for ω a bµ (x), reduces the original (tree-level) auxiliary degrees of freedom to the "dynamical" ones that have the kinetic term.A question now is whether such a condition can be generalized or not.
In this paper, we advocate the irreversible vierbein postulate [41]: At a certain energy scale Λ G the action for gravity must admit the degenerate limit of vielbein [42,43,61] in which arbitrary set of eigenvalues of vielbein goes to zero and hence the inverse vielbein cannot be defined.In a sense, the action at Λ G corresponds to the linear sigma model (5).The irreversible vierbein postulate shares the same assumption that, in the language of the linear sigma model, we do not take into account inverse powers of O(N ) invariant ϕ i ϕ i such as (ϕ i ϕ i ) −1 and (ϕ i ϕ i ) −2 .Even though they do not spoil renormalizability in terms of power counting, they do prevent defining the symmetric phase ⟨ϕ i ϕ i ⟩ = 0.In this sense, the irreversible vierbein postulate ensures a well-defined symmetric phase ⟨g µν ⟩ = 0.
Indeed, solving the equation of motion for the auxiliary field ω a bµ (x) requires the inverse vielbein.Thus, at Λ G , we cannot impose the equation of motion.If one introduced the inverse vielbein a priori, it could kinematically reduce the degrees of freedom.By contrast, in our postulate, we claim that the reduction of degrees of freedom takes place dynamically below the scale Λ G .Hence, the inverse vielbein is defined thanks to quantum dynamics.
We consider a gravitational theory which is based on SO(1, d)×GC in the degenerate limit which entails ⟨g µν ⟩ = 0 at the tree level.Its dynamics realizes ⟨g µν ⟩ ̸ = 0 and the massless metric field as a consequence of spontaneous symmetry breaking: SO(1, d) × GC → GC.Following sections, we explain the transformation laws under the LL and GC transformations and introduce corresponding gauge fields in detail.

Local-Lorentz and general coordinate transformations
We introduce mainly the fields and symmetries to clarify our notations and to construct an action.Our starting assumption is that at a certain scale Λ G , the action enjoys the LL and GC symmetries.In particular, the gravitational sector consists of the vielbein (vierbein in 4 dimensional spacetime) and the LL gauge field.
In this section, we clarify especially how the fields transform under the LL and GC symmetries in details.In Sec.3.1, we spell out the field content.In Sec.3.2 and 3.3, we present their transformation laws under the LL and GC symmetries, respectively.In Sec.3.4, we show the field strength for the LL gauge field and argue that we do not need an extra GC gauge field nor its field strength.Through this section, we work in d+1 spacetime dimensions.Later we will specify d = 3 when constructing a concrete action.

Field content
The gravity sector consists of the vielbein field e a µ (x) and the LL gauge field ω a bµ (x), where µ, ν, . . .(a, b, . . . ) run for the spacetime (tangent-space) indices 0, . . ., d (0, . . ., d).Here and hereafter, we make the dependence on a specific coordinate system x µ explicit on each chart, unless otherwise stated, since it is anyway necessary for any realistic calculation of a dynamical quantity; this will make a distinction between variable and constant more apparent.From the vielbein, we construct the metric field where the tangent-space metric and its inverse are η ab a,b=0,...,d = η ab a,b=0,...,d = diag(−1, 1, . . ., 1) , in which "diag" denotes the corresponding diagonal matrix.We note that the metric field g µν (x) can be constructed without using the inverse vielbein field e a µ (x), whereas construction of an inverse metric field g µν (x) does require the inverse vielbein.
The matter sector of an effective field theory consists of scalar, spinor, and 1-form fields ϕ(x), ψ(x) and A µ (x) with spin 0, 1/2, and 1, respectively. 7Precisely speaking, A µ (x) are the components of the 1-form field A(x) := A µ (x) dx µ , but we sloppily call these components a 1-form field too.Below, Ψ(x) will denote either ϕ(x) and ψ(x) fields collectively.Also, Φ(x) will denote any one of the fields, including both the gravity and matter sectors.
Here, we take the 1-form field A µ (x) rather than the corresponding vector field A µ (x) := g µν (x) A ν (x) as a fundamental degree of freedom because the former rather than the latter primarily appears in a gauge covariant derivative8 More explicitly, on a field Ψ(x) in the fundamental representation of a gauge group, the covariant derivative due to the gauge transformation of the gauge field A µ (x): or for an infinitesimal transformation U (x) = I + ϑ(x), where I is the identity matrix and the commutator is [A, B] := AB − BA.
In the following sections, for totally symmetric and anti-symmetric tensors, denoted here by S and A, respectively, we use the notation In particular, second-rank tensors are given by9

Local-Lorentz transformation
Here, we review the LL transformations on various fields to spell out our notation.The LL transformation is a local rotation of the tangent-space basis, therefore in particular it does not act on a spacetime-scalar field.

LL transformation on gravity sector
Under an LL transformation that satisfies the defining relation of the SO(1, d) symmetry the gravitational fields transform as where Λ −1 a b = Λ b a , with their indices being lowered and raised by the tangent-space metric and its inverse (9).Here and hereafter, a derivative such as ∂ µ := ∂ ∂x µ acts only on its neighbor: etc.
We will also use a matrix notation such as leading to and where the superscript "t" denotes the transpose.For an infinitesimal transformation Λ(x) = I + θ(x) in the matrix notation, or more explicitly Λ a b (x) = δ a b + θ a b (x), we have or more explicitly Note that the defining relation for SO (1, d) in Eq. ( 17), or (22), implies the antisymmetry θ ba (x) = −θ ab (x).
To summarize, the vielbein transforms as a fundamental representation of the LL symmetry, while being a spacetime 1-form.Recall that the Higgs field transforms as a fundamental representation of gauge symmetry while being a spacetime scalar.On the other hand, the LL gauge field transforms just as an SO(1, d) gauge field under the LL symmetry.
It is the transformation (19), or (24), that makes the LL-covariant derivative on an LLvector (spacetime-scalar) field V a (x), to be covariant: In the matrix notation, the above equations read and

LL transformation on matter sector
Now we turn to the matter fields.The bosonic matter fields transform as a scalar (namely, do not transform) under the LL symmetry: We here comment on the relation to the irreversible vierbein postulate which we will impose on the action.The 1-form field A µ (x) can be regarded as a composite field A µ (x) = A a (x) e a µ (x) made of the vielbein e a µ (x) and an LL-vector10 spacetime-scalar A a (x) that transforms as Even when we regard A a (x) as a fundamental degree of freedom, we can always construct A µ (x) without contradicting to the irreversible vierbein postulate.In contrast, if starting from the 1-form field A µ (x), we need the inverse vielbein field e a µ (x) to construct the LLvector spacetime-scalar field A a (x) = e a µ (x) A µ (x).That is, we cannot reconstruct A a (x) from A µ (x) under the irreversible vierbein postulate at the scale Λ G .
The fermionic matter field, spinor, transforms non-trivially under the LL symmetry.In the matrix notation, we may parametrize an LL transformation as that is, Now the spinor field transforms as where we define in which the LL generators on the spinor representation are with γ a being the gamma matrices that obey the Clifford algebra: Here the anti-commutator is defined by {A, B} := AB + BA.Among the matter fields, the spinor field is the only non-trivial representation under the LL symmetry.As stressed in Introduction, the LL symmetry is necessary to define a spinor field at all on a curved spacetime.Note also that in our treatment, the LL symmetry has no difference from the ordinary gauge symmetry other than that it is under a non-compact group SO(1, d).

General coordinate transformation
Next, we discuss the GC transformation x µ → x ′µ (x), where, throughout this paper, the prime symbol ′ exclusively denotes a quantity after the GC transformation and not a derivative.Accordingly, the bases for 1-form and spacetime vector transform as The GC transformation is generally identified with diffeomorphism (diff), while it is often said that the transformation under diff is given by the Lie-derivative (LD) transformation.One might regard that these three transformations were equivalent.Strictly speaking, however, GC/diff and the LD transformation should be distinguished.Indeed, the GC transformation introduced in this section is not given by the LD.A detailed comparison between the GC transformation and diff is given in Appendix A. In the main body of this paper, we use the terminology "GC" rather than diff.
In a matrix notation and writing similarly to Eq. ( 21) such as [M (x)] µ ν := M µ ν (x), the above GC transformations on the bases read It is important that the "matrix" M satisfies the extra for the GC transformation (42).Conversely, it is also true that any function M µ ν (x) that satisfies the condition (45) can always be written (locally) in terms of (d + 1) functions x ′µ (x) (µ = 0, . . ., d) as in Eq. (42).The transformation by M µ ν without the condition (45) corresponds to the general linear (GL) transformation, i.e.GL(d + 1). From By anti-symmetrizing λ and ν, we get Similarly, the derivative of the inverse function gives where vertical lines (in between the anti-symmetrization symbols in indices) denote that the indices between the vertical lines are not anti-symmetrized.For instance, in Eq. ( 47), the index α between the vertical lines is not anti-symmetrized.In actual computations such as will be done in Eq. ( 72), it is more convenient to use the coordinate notation on the righthand sides in Eq. ( 42) rather than to use these relations ( 42), ( 46), (47), etc. in the matrix notation on the left-hand sides in Eq. (42).The matrix notation is of use for more conceptual understanding.

GC transformation on fields
Under the GC transformation, the gravitational fields transform as and the matter fields as The transformed scalar field ϕ ′ (x ′ ) is defined to satisfy ϕ ′ x ′ (x) = ϕ(x) such that the pullback of the function ϕ ′ (x ′ ) by the function x ′ (x) becomes ϕ(x).Equivalently, the pullback of the function ϕ(x) by the inverse function Here, the spinor field also transforms the same as the scalar field (namely as the pullback of the function) under the GC transformation; see Appendix B for another point of view and our opinion on it.Note that the spinor field transforms as scalar under the GC transformation because it does not have any spacetime index, while the "LD transformation" gives different transformation laws from Eq. ( 51).They are discussed in Appendix B.
In the matrix notation, with Eq. ( 42), the GC transformation is, on the gravitational fields, and, on the matter fields, In the matrix notation, a spacetime vector V µ transforms like a fundamental representation under the GC transformation:

GC gauge field
Now we want to define a GC-covariant derivative.To this end, let us first suppose that there exists a GC gauge field that transforms as where we employ the matrix notation [Υ µ (x)] α β := Υ α βµ (x) similarly to Eq. ( 21).More explicitly, the transformation (58) means Here, we stress that a difference from the gauge transformations of the ordinary and LL gauge fields in Eqs. ( 13) and ( 19) (or ( 24)), respectively, is the last M −1 factor that rotates the spacetime index too.
Then one can construct a GC-covariant derivative on a spacetime-vector field V µ (x) and a 1-form field W µ (x): In the matrix notation, we write where the left derivative reads , etc. similar to Eq. ( 20).More explicitly, they are expressed as It is straightforward to check their covariance under the GC transformation: In the matrix notation, where we have suppressed the dependence on x in the right-hand side and have used the identity recall that we are employing the neighboring notation for derivatives as given in Eq. ( 20).More explicitly, the above transformations read We may separate the GC-gauge field Υ µ into symmetric and anti-symmetric parts: where parentheses and square brackets for indices are defined in Eq. ( 16).Note that we have mixed the indices β and µ that correspond to an internal gauge index and a spacetime index, respectively, for the case of the ordinary/LL gauge field.The symmetric and antisymmetric parts in the first and second terms of Eq. ( 68) have is the same as that of the GC conditions (45).This fact suggests that it is redundant for the GC symmetry.
Let us see that this is indeed the case.Under the GC transformation ( 59), the antisymmetric part of the GC-gauge field transforms homogeneously: where we have omitted the dependence on x in the right-hand side for simplicity and have used the GC-condition (45) in the last step.That is, the GC-covariance of the GC-covariant derivative is maintained even if we do not include the anti-symmetric part Υ α [βµ] (x).(Though it means that we do not need the anti-symmetric part at all in order to realize the GCcovariance of the GC-covariant derivative, this argument itself does not prohibit having the anti-symmetric part.)

Levi-Civita (spin) connection
Conventionally, the Levi-Civita connection g Γ has been used as the GC gauge field: which is the solution to the metricity condition on Υ: By construction, it has only the symmetric part ; recall the discussion in the paragraphs containing Eqs. (68) and (69).The transformation of the Levi-Civita connection can be found, as in any textbook of general relativity, e.g.Ref. [62], to be the same as Eq. ( 58), or (59): We note that the Levi-Civita connection requires an inverse metric g µν and hence an inverse vielbein e a µ .Therefore, it cannot be used under the irreversible vierbein postulate imposed on our action at the scale Λ G .That is, the GC gauge field is absent at Λ G since we do not further introduce it as extra degrees of freedom; see also footnote 11.We will come back to this point below.
Once the Levi-Civita connection g Γ µ is introduced (in our scenario, it is induced by quantum fluctuations below the scale Λ G ), then another LL-gauge field can also be induced, namely the Levi-Civita spin connection It is straightforward to check that the Levi-Civita spin connection e Ω µ (x) transforms in the same way as the LL gauge field ω µ (x) under the LL and GC transformations.

Field strengths, Riemann tensor, and GL(d + 1)
Now we come back to the consideration for the general gravitational gauge fields ω and Υ.
The field strengths for the ordinary and LL gauge fields are given in the matrix notation, respectively as 12 More explicitly, the LL field strength reads We can rewrite the field strength as a commutator of the covariant derivatives ( 10) and (27), or (29), on the fundamental representation: ω It is important that the field strengths reduce to functions ( 74) and ( 75) when deriving their covariance ω 12 For a compact gauge group such as introduced in footnote 8, it is more common to use from the covariance of the covariant derivatives ( 12) and (30), respectively, as follows: Now let us take the commutator of the GC-covariant derivative (60), or (62), on a spacetime-vector field V (x) = V µ (x) ∂ µ that transforms as a fundamental representation under GC transformation: In the matrix notation, or more explicitly, The last term is peculiar to the GC-field strength: The anti-symmetric part Υ ρ [νµ] (x) is not only in vain in covariantizing the GC-covariant derivative but also is an obstacle to making the GC-field strength a function rather than a differential operator.This fact disfavors an introduction of the extra GC gauge field Υ µ with the anti-symmetric part.See also the discussion around Eq. ( 69) for the redundancy of the anti-symmetric part.
Let us comment on the relation to the Levi-Civita (spin) connection.It is noteworthy that if the GC gauge field is identified with the Levi-Civita connection (70), the GC-field strength becomes the Riemann tensor itself: where . This can be shown as follows: or more explicitly, the right-hand side is nothing but the Riemann tensor.Under the assumption (84), the anti-symmetric part of the GC gauge field Υ α βµ does not take part and play any role.In the same manner, we can define the Riemann tensor from the LL-field strength with the Levi-Civita spin connection, i.e.Υ α βµ ≡ e Ω α βµ together with a vielbein and its inverse: where e in which Under the irreversible-vielbein postulate, we expect that physically the following scenario takes place: In the action at Λ G , there is no specific background of vielbein.The quantum dynamics induces a non-trivial background vielbein ēa µ (x) that has its inverse ēa µ (x) everywhere, namely, a non-degenerate ēa µ (x), on which the expectation value of the LL gauge field should become the Levi-Civita spin connection: where This way, our formulation would reproduce the conventional covariant derivative acting on the spinor field in the metric formulation.It suffices therefore that only the LL-field strength exists to construct the Riemann tensor at lower energies below Λ G if the physical expectation (91) is met.We do not need to prepare the GC-field strength as a source for the Riemann tensor from the beginning at Λ G .We comment on the application order of transformations, namely the GC transformation after a gauge transformation, or the other way around.The GC transformation acts on any spacetime indices, so that gauge transformations are affected by it.In other words, one may thinks that elements of their transformations do not commute.Fortunately, our definition of the GC transformations ( 48)-( 52) commute with any gauge transformations, while diff defined by the LD transformation does not.This means that the former is given by a direct product "GC × gauge", while the latter is by the semi-direct product "GC ⋉ gauge".These facts are discussed in Appendix A. 3.
Finally, let us comment on the GL(d + 1) theory.If we do not impose the GC conditions ( 45), ( 46), (47), etc. and we regard M µ ν as a general (d + 1) × (d + 1) matrix, then the theory becomes a GL(d + 1) gauge theory; see e.g.Ref. [63,64].This theory might be of interest in itself, but we do not go in this direction and do not introduce the extra GC gauge field Υ µ at Λ G because of the above-mentioned points: (i) unnecessity of its anti-symmetric part for the covariance of GC-covariant derivative, (ii) the GC-field strength becoming a differential operator rather than a function due to the anti-symmetric part, and (iii) unnecessity as a source for constructing the Riemann tensor.

Summary on covariant derivatives
For general LL and GC gauge fields ω and Υ, respectively, we summarize our notation for the covariant derivatives: LL only: GC only: LL and GC: 4 Action under irreversible vierbein postulate In this section, we construct a "tree" action given at a certain scale Λ G based on the LL and GC symmetries, i.e.GC × SO(1, 3).As discussed in Introduction, a central assumption at Λ G is the irreversible vierbein postulate that forbids the action at Λ G to contain an inverse of the vielbein.In Sec.4.1, we start with the introduction of the Levi-Civita tensor which is independent from vielbein and inverse-vielbein.Then, in Sec.4.2, the definition of the irreversible vierbein postulate is explained and the action respecting this postulate is shown.
It follows that where |g(x)| is always negative due to the Lorentzian signature.
From the Levi-Civita symbol, we define the Levi-Civita tensor for the LL transformation, and for the GC transformation, Note that the Lorentzian signature leads to which follows from the p = d case of more general identities: for 0 ≤ p ≤ d, etc. 13 Using the Levi-Civita tensor, we can write down the volume element in terms of the local coordinate system of each chart: where ⋆ is the Hodge dual, which is defined for a p-form α(x) Note that from the vielbein, the LL gauge field, and its field strength, we can construct GC-scalar 1 and 2-form fields such that

Irreversible vierbein and action
Let us now construct an action invariant under GC × SO(1, 3) symmetry.Hereafter, we work in the d = 3 spatial dimensions, assuming that it is already settled down to be so at Λ G .We call the vielbein for the d + 1 = 4 spacetime dimensions the vierbein, accordingly.
The formulation of the irreversible vierbein postulate starts by imposing regularity under the limit of any zero eigenvalues λ a → 0 of vierbein among four eigenvalues in four-dimensional spacetime.Obviously, in such a case, the inverse vierbein cannot be defined.In other words, the inverse vierbein contains divergences.Then, the irreversible vierbein postulate at Λ G states that the action at Λ G does not diverge even for the (not necessarily simultaneous) zero eigenvalue limit of vierbein.We call this kind of limit the degenerate limit [42,43,61].
There are some cases where, even when a term is apparently written down using the inverse vierbein, the irreversible vierbein postulate does not forbid such a term in the degenerate limit.An important observation for this is that some inverse vierbeins can be absorbed into the determinant |e(x)| from the volume element.More specifically, from the identities (107) and 13 See footnote 9 for the normalization. 14Or else, one may first define we obtain Though the left-hand sides of these equations appear to have the inverse vierbeins, the righthand sides do not.Only these combinations of the inverse vierbeins can be used to write down the action at Λ G without contradicting the irreversible vierbein postulate.
Let us now write down the action explicitly.First, it turns out that the kinetic terms for GC scalar and vector fields contain the inverse vierbeins all of which cannot be simultaneously absorbed into the volume element via Eqs.( 115)-(118): They contain the inverse metric g µν that is symmetric for its indices (See Appendix C for more explicit discussions):15 Therefore, these terms are forbidden and thus GC scalar and vector fields are not dynamical at Λ G .The kinetic terms for vierbein e a µ and LL gauge fields ω a bµ cannot be introduced due to the same reason.
On the other hand, the spinor kinetic term is consistent with the irreversible vierbein postulate because it contains only single inverse vierbein and thus we can use Eq.(115): Unlike the ordinary gauge theory, the LL symmetry has a fundamental representation that is a spacetime vector, the vierbein.It allows one to write down an LL-invariant term constructed from a single field strength: This term has become compatible with the irreversible vierbein postulate thanks to Eq. ( 116).Note that as mentioned in the paragraph containing Eq. (92), it is not necessary to introduce the term with the field strength corresponding to the GC transformation Υ F.
Finally, one can also write down the cosmological constant term: Barring the topological terms (see discussion below), the terms (120), (121), and (122) are the only combinations that are consistent with the irreversible vierbein postulate: where Z ψ , X ω , and V are arbitrary functions of spacetime scalars at x constructed by matter fields, say, ϕ(x), ψ(x) ψ(x), etc.For example, V includes the mass term for spinor M ψ ψ(x) ψ(x) and the cosmological constant Λ cc as well as the ordinary scalar potential.Note that the kinetic term for the Rarita-Schwinger field which is a spin 3/2 field is also compatible with the degenerate limit (see Appendix C.3).In this work, we do not take this into account.
Various combinations of fields can be contracted with the LL metric η ab and the totally anti-symmetric LL-tensor ϵ [abcd] to yield topological terms of the action, which are summarized in Appendix D. In general, one may multiply, on these "topological" terms, arbitrary functions of GC scalars such as Z ψ , X ω , and V in Eq. (123) so that they become dynamical (non-topological).Since all such interactions are higher dimensional, we neglect them in this paper.The inclusion of these terms might be of interest in itself, which we leave for future study.
Finally, we stress again the reason why we impose the irreversible vierbein postulate at the tree level.A typical criticism may be as follows: The existence of inverse vierbein in the tree action is harmless since terms with inverse vierbeins behave as e −O(e −1 ) → 0 for e → 0 within the path integral.On the other hand, the standard model of particle physics assumes that the action does not have inverse power of the fields such as 1/H † H, where H is the Higgs field, even though such terms can be perfectly consistent with all the gauge and spacetime symmetries.The absence of inverse power is particularly noteworthy in the effective field theory picture because such negative-power terms are more relevant than the normal ones towards the IR direction.We can interpret this as the requirement of the existence of the weak-field limit H → 0 for the action such that the symmetric phase ⟨H⟩ = 0 is well-defined. 16 Local-Lorentz and general coordinate transformations for Background fields One of main purposes in this paper is to demonstrate the generation of non-trivial background fields due to quantum effects in four dimensional spacetimes.This will be done in Section 6.
In this section, assuming that background vielbein and LL gauge fields are induced in arbitrary spacetime dimensions, we discuss their transformation laws and covariance.Those are important for understanding of the low energy effective theory from the action (123).

Invertible background vielbein and Levi-Civita (spin) connection
To begin with, we introduce a certain gravitational background Φ = (ē, ω), while do not assume a classical background for the matter fields for simplicity: Ψ = ( φ, ψ, Āµ ) = 0 Here, an important assumption is that the background vielbein field ēa µ (x) is invertible that allows the inverse background vielbein ēa µ (x) and the inverse background metric where the background vielbein is defined to satisfy In general, the full vielbein and metric fields will also become invertible, where e and g represent their quantum fluctuations, respectively.However, we hereafter raise and lower the spacetime indices by the background vielbein and/or metric.In particular, we can now give recall the argument below Eq. ( 33).The invertible background vielbein also allows us to write down the background Levi-Civita (spin) connection: where etc. Note that Γ and Ω are solely made of the vielbein and its inverse.In the previous language, The background spin connection Ω transforms the same as the background LL gauge field ω under the background LL and GC transformations.

Background covariance
For the given background field of the gravitational fields Φ, we define the following LLbackground-covariant (only) derivatives denoted by Dµ : for matter fields and for gravitational fields We also define the LL-and-GC-background-covariant derivative Dµ .It acts the same as Dµ on the matter fields without the spacetime indices, Ψ, whereas on the gravitational fields, 17 So far, ω is pretty much unconstrained. 18In this paper, we assume that the background vielbein should obey the metricity Dµ ēa ν (x) = 0, (140) 17 For a given background ē and ω, we may always construct a background GC gauge field that is defined to satisfy the metricity: We can explicitly check that γλ νµ is invariant under the background LL transformation. 18We might end up with the following expression: In the language of differential forms, this can be written as the background LL gauge field ωa bµ (x) be Eq.( 91), and the background GC connection be the Levi-Civita one recall Eq. ( 70).

Global background Lorentz invariance after spontaneous symmetry breaking
For a general background ēa µ , there remains a local SO(1, 3) × GC background symmetry: Āµ (x) and The Lorentz transformation under the global SO(1, 3) is nothing but an accidental symmetry arising only when we take the flat background ēa µ = δ a µ for which ωa bµ = 0.That is, the gobal SO(1, 3) is the "diagonal subgroup": From Eq. ( 145), one has19 x µ → Λ µ ν x ν , (this is mere a reparametrization) (148) so to say Hence, under this transformation in the diagonal subgroup, the kinetic term of spinors transforms as usual, even though we assign them only a trivial representation under GC.

Dynamical generation of flat spacetime from spinor loop
In this section, we derive the effective potential for a vierbein background field, assuming it to be a flat spacetime background, and then demonstrate that indeed a non-vanishing flat vierbein background is induced by quantum effects of the fermion field.

Effective action for conformal mode
We study now the dynamical symmetry breaking of the LL symmetry in four-dimensional spacetime.A central object for the observation of such a symmetry breaking is the effective potential for the vierbein field.To obtain it, we assume a background field for vierbein ēa µ and investigate the effective potential.The degenerate limit would enforce such a minimum to be located at ēa µ = 0. What we want to see in this section is whether quantum effects generate a non-trivial expectation value of ēa µ or not.In this section, we consider the quantum effects of the spinor fields at the one-loop level on the effective potential for vierbein, while we deal with the vierbein and the LL gauge fields as classical fields.The one-loop approximation might be justified by a large number of spinor degrees of freedom in the SM that is 90 without including the right-handed neutrinos, whereas we do not exclude the possibility of large effects from other sectors; we will come back to this point later.
We start with separating the vierbein field into the background and fluctuation as in Eq. ( 126).For simplicity, we concentrate on a constant background field in this paper.At the tree level of our action (123), the potential of the (constant) vierbein field is simply given by Here, we suppose that the kinetic term of vierbein, which will be generated dynamically below Λ G , will take a "correct" (negative) sign so that the action is given by . This assumption will be discussed at the end of this section.For a negative cosmological constant Λ cc < 0, we have an unbounded potential and |ē| = 0 is an unstable extremum, while for Λ cc > 0, the potential is bounded and has a minimum at ē = 0.The former induces the spacetime background already at the tree level.How about the latter?
We next consider quantum corrections to the effective potential.At the level of the action (123) at Λ G , only the spinor fields are dynamical and give the leading effects on the effective potential.Here, we compute the effective potential for an assumed background vierbein field value ēa µ (x), taking into account the spinor one-loop correction.In order to evaluate the spinor action concretely, let us here assume a flat spacetime background, i.e., we parametrize where C is a dimensionless constant.This parametrization is special case that all eigenvalues λ a of vierbein take the same value, namely, ēa C, C, C).For such a flat background, the equation of motion may entail ωabµ = 0.
Hence, the point C = 0 corresponds to the degenerate limit for the eigenvalues, and thus is identified with a "symmetric phase" of GC × SO(1, 3) in analogy with the ordinary Higgs mechanism [67].Needless to say, the point C = 0 has no background spacetime at all and is intractable.Our strategy to handle this "symmetric" point to compare with the broken phase 20 C ̸ = 0 is first computing the effective potential for C > 0 and then examining the limit C ↘ 0.
Let us derive the effective potential for C arising from the spinor loop.The assumption (151) for the background field gives |ē| = C 4 and ēa µ = C −1 δ µ a .In order to obtain the one-loop effects from a spinor under the background (151), it suffices to take its quadratic terms: where we have included the spinor mass term coming from V and have written γ µ := δ µ a γ a .21Now we redefine the spinor field as ψ(x where Integrating out the spinor field yields the effective action where Tr acts on all internal degrees of freedom involved in the spinor field, e.g.eigenvalues of the covariant derivative, spinor space, and so on.The second term in Eq. ( 154) may give a non-trivial form of the effective potential.
One can perform the Fourier transformation to obtain where we have taken the spacetime volume from the momentum-space delta function δ 4 (0) in the trace as (2π) 4 δ 4 (0) = C 4 d 4 x, which can be naturally understood by first performing the heat-kernel expansion and then taking the limit (151).Thus, the effective potential for C under the flat background (151) is given by where we have performed the Wick rotation such that p 0 = ip 0E and p i = p iE .If we neglect the second term corresponding to the quantum correction from the spinor field, the effective potential would be simply V eff (C) = Λ cc C 4 and hence, for Λ cc > 0, the effective potential would have a vacuum C = 0, implying ēa µ = ⟨e a µ ⟩ = 0. Let us now perform the loop momentum integral in Eq. (156).To this end, we need regularizations.Here, we attempt to employ the momentum cutoff and dimensional regularization.The use of the momentum cutoff such that 0 while by dimensional regularization, we obtain where The momentum regularization case (157) is rather complicated than the dimensional regularization case (158).To understand Eq. ( 157), let us consider the chiral limit (m f → 0) for which The quartically divergent terms ∼ Λ 4 G can be subtracted by the additive renormalization for the cosmological constant Λ cc .More specifically, we prepare counter terms for Λ cc such that δΛ cc + Λ cc,R δ Λcc where δΛ cc additively subtracts terms proportional to Λ 4 G , while the counter term δ Λcc multiplicatively subtracts divergent terms.Therefore, by employing an appropriate renormalization condition, we would obtain This counter-term does not contribute to the running of the cosmological constant.Indeed, such a prescription is analogous to the mass-independent scheme in scalar theories: For scalar mass term, we give δm 2 + m 2 R δ m 2 and subtract quadratic divergences ∼ Λ 2 by δm 2 , while logarithmic divergences are removed by m 2 R δ m 2 and the running effects of scalar mass originate from m 2 R δ m 2 , but not from quadratic divergences.The cancellation between the Λ 4 G terms and δΛ cc is nothing but the cosmological constant problem [68].We do not intend to solve this problem in this work.
Next, we consider the last term on the right-hand side of Eq. (157).It seems that this term cannot be subtracted because no counter term proportional to C 2 exists in the original action (123).However, this is not the case.The momentum-cutoff regularization explicitly breaks the symmetries GC × SO (1,3).In such a case, one has to estimate symmetry-breaking effects from the corresponding Ward-Takahashi identity and add counter terms to the action.Therefore, the last term on the right-hand side of Eq. ( 157) should be also removed from the effective action.To summarize, the effective potential under the momentum-cutoff regularization reads This is compatible with the result from dimensional regularization (158) together with the identification log Λ G ↔ 2/ε.After removing the power divergences, which do not affect the running of theory parameters, the effective potential for C becomes in the sense of bare perturbation theory around the scale Λ G .In Fig. 1, we plot the effective potential (162) as a function of C. Here, we set Λ G = m f = 1 and Λ cc = 0.1 for displaying the effective potential.It is expected that the effective potential could yield a non-zero expectation value of C, i.e., ⟨e a µ ⟩ ̸ = 0 due to the quantum tunneling effects.However, at this level of the approximation, the effective potential becomes a "run-away" form.Thus, a non-trivial stable vacuum cannot be determined.So far, there are mainly two interpretations within the current model: (i) Inclusion of higher-order effects such as loop effects of the vierbein and the LL gauge field stabilize the effective potential.
(ii) The run-away behavior of C implies the cosmological evolution of the scale factor [69].
In the tree-level action (123), there are no kinetic terms for the vierbein and the LL gauge field, so that these fields do not contribute to the effective potential at leading order.In possibility (i), after the kinetic terms are induced by the fermionic quantum effects, the effective potential receives loop effects from the vierbein and the LL gauge field and could be stabilized.
The field C can be actually regarded as a conformal factor because the parametrization (151) with C = e σ gives the Weyl rescaling ḡµν = η ab ēa Here σ is called the dilaton, the conformal field, or the scalaron, depending on the situation, and is associated with the scale symmetry.Indeed, in the potential (162), powers of C reflect the canonical scaling of dimensionful parameters such as Λ cc and m f .Another possible interpretation is therefore that C is regarded as a renormalization scale.The change of C may give the running of renormalized couplings.The evolution of C may be reasonable for realizing the expanding universe in cosmology. 22In this sense, the run-away potential is not 22 Obtaining a run-away potential for vierbein has been discussed in Ref. [69] in a different mechanism: A background metric is assumed to be a cylinder of topology R × M 3 with an arbitrary three manifold M 3 , i.e., ds 2 = C 2 (t)(−dt 2 +dx 2 ) with time-dependent factor C(t) and the effective potential for C 2 (t) is derived.In this scenario, a "wrong-sign" kinetic term of C(t) is crucial to obtain an unbounded potential for C(t).excluded from possible scenarios.Moreover, the run-away potential may be also related to the wave function in the universe [70].In any case, an important fact is that the solution ⟨C⟩ = 0 is an unstable vacuum, and then a non-vanishing vacuum is realized.
Finally, we comment on the kinetic term of the conformal mode C in vierbein.In the argument above, we have assumed a "correct-sign" kinetic term of the conformal mode and then have obtained a run-away effective potential.Conversely, if the conformal mode has a "wrong-sign" kinetic term, the effective potential is unstabilized by the cosmological constant term and is bounded by the fermion loop effect.Consequently, we obtain a stable vacuum at The sign of the kinetic term for the conformal mode highly depends on interactions between gravitational fields and matter fields.Indeed, depending on the interaction between the scalar curvature R = R αβ αβ (see Eq. ( 85) for the definition of the Riemann tensor) and scalars, the sign of the kinetic term for the conformal mode varies; see e.g.Appendix A in Ref. [12].In our model, it depends on whether or not the Planck mass in Eq. ( 121) is regarded as a function of scalars.Therefore, we do not specify the sign of the kinetic term for the conformal mode.Nonetheless, we stress that in any case, background vierbein has a non-trivial expectation value thanks to fermion loop effect.

Summary and discussion
In this paper, we have proposed a model for quantum gravity based on the LL gauge and GC symmetries.In Section 2, we have explained our viewpoint towards constructing a quantumgravity model.We have summarized transformation laws under these symmetries and the corresponding covariance carefully in Section 3. Our main claim is to impose the irreversible vierbein postulate on the tree-level action at the scale Λ G such that the action does not contain inverse vierbein in invariant operators under the LL gauge and GC symmetries.This postulate also prohibits, at Λ G , the background field of the gravitational fields, especially vierbein.It has been shown in Section 4 that with matter fields, only three types of terms are admitted in the tree-level action (123): the cosmological constant, the linear term in the field strength of the LL gauge field, and the kinetic term for spinor, with their couplings being GC-and LL-invariant functions of matter fields.This means that only the spinor can behave as the dynamical quantum field at the lowest level.In Section 6, we have argued the generation of a non-vanishing background vierbein field.There, supposing that a flat background vierbein field is induced, ēa µ = Cδ a µ , we have discussed the effective action for C. Depending on the signs of the kinetic term of C and of the cosmological constant, the symmetric vacuum ⟨C⟩ = 0, which is consistent with the irreversible vierbein postulate, can be a stable minimum of the potential at the tree level, and then the fermionic fluctuations can make it unstable so that a non-vanishing value of ⟨C⟩ is realized.This implies the generation of spacetime background.Transformation laws for background fields have been summarized in Section 5.
With the non-trivial background vierbein generated by the quantum dynamics, we can discuss an effective theory.In the low energy regime, gravitational interactions are welldescribed by metric theories in which only two degrees of freedom within 10 degrees of freedom of symmetric tensor are physical, while our model (123) has 4 2 +6×4 = 40 degrees of freedom at the tree level since vierbein and the LL gauge fields have 16 and 6 × 4 = 24 degrees of freedom, respectively.This discrepancy can be understood by the Higgs mechanism: The vierbein field plays the role of the Higgs field in generating the background spacetime as its vacuum expectation value.The LL gauge field eats 6 degrees of freedom in the vierbein field and becomes massive [41].The remaining 10 degrees of freedom of the vierbein field are the same as those of the symmetric tensor field.At the quantum level, only 2 degrees of freedom remain due to the subtraction of 8 degrees of freedom by the gauge fixing and the ghost field.This picture provides a similar analogy with the non-linear sigma model from the linear sigma model as discussed in Section 2. Hence, in high energy, there are heavy modes for describing gravitational interactions which cannot be observed in low energy experiments.See Refs.[71,72,73,61,74,75,67,76] for a similar mechanism in the GL(4) case.
In this paper, we have not explored renormalizability of the theory because the existence of its further UV theories is unclear.String theory might be a high-energy theory for our model.Another possible candidate within the realm of quantum field theory is spinor gravity [77,78,79,80], in which vierbein and LL gauge fields are composites of spinor fields.This can be studied by renormalization group with the compositeness condition [81]. where This implies Conversely, a function Θ µ ν (x) that satisfies the condition (169) can always be written (locally) as Eq. ( 168).The condition (169) is the infinitesimal version of the GC condition (45).
For the infinitesimal transformation (165), the variation of the bases becomes of the gravitational fields and of the matter fields The 1-forms e a (x) = e a µ (x) dx µ , ω a b (x) = ω a bµ (x) dx µ , and A(x) = A µ (x) dx µ are trivially invariant under the GC transformation: For the infinitesimal GC-gauge transformation M α β (x) = δ α β + Θ α β (x), the GC-gauge field (as well as the Levi-Civita spin connection, both if exist) transforms as The next-to-last term comes from the rotation of the spacetime index and is peculiar to the GC-gauge field, compared to the ordinary and LL gauge transformations in Eqs. ( 14) and (25) (or ( 26)), respectively.The last term is the inhomogeneous transformation that characterizes the gauge-field transformation.
It is important that the anti-symmetric part has a vanishing inhomogeneous term under the GC transformation: due to the condition (169) for the GC transformation.That is, the anti-symmetric part is in vain for covariantizing the GC-covariant derivative.This is the infinitesimal version of the discussion in the last paragraph in Sec.3.3.2.

A.2 Lie-derivative transformation
Choosing the coordinate system (chart) x for each open subset U of a manifold M can be regarded as a diffeomorphism from U to R d+1 .The GC transformation between two different coordinate systems is a map between two different diffeomorphisms U → R d+1 from the same U ; see the upper panel in Fig. 2. Instead, one may consider two different maps from U to the same coordinate values in R d+1 .The transformation between these two maps can also be regarded as a diffeomorphism.We can define a derivative of such a diffeomorphism, namely the Lie derivative, which we will call the Lie-derivaitive (LD) transformation below; see the lower panel in Fig. 2. In the literature [82,83,39,40], the self diffeomorphism group on M is called the diffeomorphism, or diff in short, and this LD transformation can be regarded as its infinitesimal. 23 Lie derivative along an infinitesimal vector field ξ µ (x) is defined as the difference between fields on two distinct spacetime points that happen to have the same coordinate value x ′ before and after the infinitesimal GC transformation (165): where ξ(x) in an argument (of Φ in the first line, in this case) denotes the (d + 1) variables ξ 0 (x) , . . ., ξ d (x) , whereas those in other places denote the differential operator ξ(x) := ξ µ (x) ∂ µ .In other words, the Lie derivative (182) is the difference between the original field at GC-transformed point and the GC-transformed field.In terms of the Lie derivative, the GC transformation can be written as By construction, the Lie derivative does not change the basis, unlike the GC transformation (170) and (171).We spell out the concrete forms using Eqs.( 172)-(176): and (In the language of differential geometry, where the exterior derivative on the vector field ξ(x) = ξ µ (x) ∂ µ is given by dξ(x) := ∂ µ ξ ν (x) dx µ ∂ ν and the inner product, defined through that of the bases dx λ , ∂ ν = δ λ ν , reads ) One might find it uneasy to transform the coordinate bases as in Eqs. ( 170) and (171).Then it might be tempting to define another transformation, which we call the Lie derivative (LD) transformation: An advantage of the GC transformation over the LD one is that the former commutes with gauge symmetries whereas the latter does not [82,83,39,40] because the latter is a difference between two distinct spacetime points, which are gauge-transformed differently.

A.3 (Semi-)direct product of general coordinate and gauge transformations
Since the GC transformation acts on spacetime indices, it acts on those of the gauge field.Therefore, when we apply both GC and a gauge transformations for a system, one may worry about the order of transformations, that is, a GC transformation after a gauge transformation or the other way around.To clarify this, we show explicit computations.
The local version of which is opposite to Eq. ( 201).In a gravitational theory based on the global Poincaré transformation, we infer to realize the symmetry breaking

B Comment on Lie derivative on spinor
In this appendix, several definitions of the Lie derivative acting on spinor are argued.
Once the background covariant derivative D is defined for the matter field Ψ, we may consider a parallel transport with respect to D: Here, we stress that this parallel transport differs from the GC transformation in Eqs. ( 50)- (52) in the sense that the transport (214) compares fields on physically distinct points that happen to have the same coordinate values before and after the GC transformation.The parallel transport (214) induces another Lie derivative of Ψ: On spinors, this is nothing but the Lie derivative introduced by Weyl [85], if ωa bµ is identified to the Levi-Civita spin connection (130). 25ne may further extend the above definition to the following form [86] (see also Ref. 24 In case of SU (2) → U (1) breaking, i.e.T ≃ SU (2)/U (1), we write This extension is motivated by the fact that on the flat Minkowski background ēa µ = δ a µ and ωa bµ = 0, there remains the global SO(1, d) invariance under namely ξ µ (x) = θ µ ν x ν + • • • .This is the same as Eq.(166) with Eq. (168).Hence, Eq. (217) becomes equivalent to the Lie derivative obtained when the background GC transformation reduces to the global SO(1, d) on the flat background, namely when M µ ν (x) → Λ µ ν : The transformation (219) corresponds just to the global SO(1, d) transformation for spinor.
The definition (217) may however be a the detour notation in our case.As discussed in Section 5.3, in our formulation, the global Lorentz SO(1, d) transformation is accidentally realized as a diagonal subgroup of SO(1, d)×GC, so that the detour notation is not necessary.

C Degenerate limit of vierbein
In this appendix, we explain the detailed definition of the degenerate limit and show its application for explicit several examples.

C.1 General definition of degenerate limit
As discussed in Introduction, we assume that the action at Λ G admits the weak field limit e a µ (x) → 0 just as the SM action does for the limit H(x) → 0, etc.In particular, we postulate that the action admits the degenerate limit for any combination of the (d + 1) 2 = 16 components of the vierbein [41].This requirement puts a severer constraint on the action than just requiring a simultaneous limit for all the components, as we will see.The degenerate configuration of the vierbein appears in the topology change of the background spacetime and is expected to play an important role in quantum gravity [42,43].
In general, a vierbein e a µ has four eigenvalues in four-dimensional spacetime.In the degenerate limit, at least one eigenvalue goes to zero, resulting in the determinant of vierbein to be zero: |e| → 0. Note that the degenerate limit does not necessarily mean the null limit e a µ → 0 for all the 16 components.Let us see this fact using a specific example.By a rescaling e a µ → Ce a µ , we obtain |e| → C 4 |e| and g µν → C −2 g µν , and hence the term does not diverge in the null limit C → 0. However, this term is in general divergent in a degenerate limit as we will see below.Now, we define the degenerate limit.The power for the degenerate limit can be counted by |e| rather than by the overall normalization factor C: The inverse of vielbein and metric, e a µ and g µν , contain one and two factors of |e| −1 , respectively: where the transpose of the cofactor matrix of e a µ is denoted by C a µ , which remains finite in the degenerate limit.Hereafter, we write the power of |e| as Each upper greek index of vielbein or metric serves extra −1 power of |e|, and its power −1 cancels the power +1 from a lower index of metric or vielbein.This can be explicitly seen as follows.The D 2 = (d + 1) 2 degrees of freedom of vielbein can be parametrized as where (λ 0 , . . ., λ d ) ∈ R D with λ 0 < 0 and λ i > 0 (i = 1, . . ., d) and each of Λ, 2 degrees of freedom. 27In the degenerate limit, an eigenvalue λ a goes to zero: λ a → 0. The determinant reads while the metric and its inverse are gives the power ⌊e a µ g µν ⌋ = −1.In one more example, for ∇ µ e a ν = ∂ µ e a ν − e a λ Γ λ νµ with the Levi-Civita connection (70), we have which gives e a λ Γ λ νµ = −1 because the first term does not contain a vierbein whose λ ′ leg is to be contracted.
Other examples are in order: The Levi-Civita connection Γ λ νµ contains two extra inverse power of |e| coming from g µν , ⌊Γ µ ρσ ⌋ = −2, while the Levi-Civita spin-connection has ⌊Ω a bµ ⌋ = −2 because of e a λ g λρ = −1 and hence e a λ Γ λ σµ = −1 (contraction of e b σ with Γ λ σµ does give the additional power −1 because the latter contains the index from ∂ σ , which does not come from vielbein nor from metric).Note that the contraction of Γ λ νµ with e a ν does raise the power by 1 because the former contains the second term in the parentheses in Eq. ( 70), ∂ ν g µλ , whose lower ρ index comes from the derivative.

C.2 Concrete examples
Here, we more explicitly show the degenerate limit on various terms and list the terms prohibited by having a negative power of eigenvalues of e a µ .• The kinetic term of a scalar is prohibited because from Eqs. ( 224) and ( 226) When some eigenvalues become zero λ a → 0, the matrix contains a divergent component ∝ λ −1 a .We note that Eq. ( 116) cannot be used in Eq. ( 230 and thus diverges for λ a → 0. Therefore, the gauge kinetic term is not compativble with the degenerate limit.
These terms are forbidden in the action consistent with the degenerate limit.
In contrast, the following operators can admit the degenerate limit: • The cosmological constant term is obviously not divergent for λ a → 0 thanks to Eq. ( 224).
• The linear term in F ab µν , namely This term does not contain any inverse of eigenvalues of the vierbein field, so that it is free from divergences for λ a → 0. Note that the mass term of the spinor field is also accepted.
Hence, imposing the degenerate limit on the action accepts only these three terms.In particular only spinor fields have their kinetic terms, i.e. become dynamical.

C.3 Rarita-Schwinger field
Here we examine if the spin-3/2 Rarita-Schwinger field is compatible with the irreversible vierbein postulate.
When we regard the lower-indexed ψ µ as the fundamental field, its free action is The former action is consistent with the irreversible vierbein postulate, whereas the latter is not.The compatibility of the Rarita-Schwinger field with the irreversible vierbein postulate is contingent upon whether we consider the field with upper or lower indices as the fundamental entity.

D Topological terms
There are four topological terms: (i) the Immirzi term, (ii) the Nieh-Yan invariant, (iii) the Pontryagin index, and (iv) the Euler number; see e.g.Ref. [88] for others. 29More specifically, they are given in the language of differential forms by These terms are compatible with the degenerate limit.However, barring the field-dependent couplings as discussed in the second-last paragraph in Section 4, they do not affect the quantum dynamics of the vierbein and LL gauge fields since these topological terms give the propagators of neither vierbein nor LL gauge fields.
ω F ab µν (x) is the field strength of LL gauge field ω a bµ (x).Note here that LL indices a, b, . . .are anti-symmetric in ω abµ (x) and ω F abµν (x), namely ω abµ (x) = −ω baµ (x) and ω F abµν (x) = − ω F baµν (x), due to the SO(1, d) algebra; see Section 3.4 below for details.It is worth stressing that the action (7) does not contain the LL gauge kinetic term ω F abµν (x) ω F abµν (x), whereas the existence of the term e a µ (x) e b ν (x) ω F ab µν (x) is peculiar in the Einstein-Cartan theory, as compared to an ordinary gauge theory that does not have such a term.

Figure 1 :
Figure 1: Plots of the potential as a function of C. The dashed line shows the tree-level potential (150), V tree (C) = Λ cc C 4 , while the effective potential (162) is depicted by the solid line.We set Λ G = m f = 1 and Λ cc = 0.1 and assume a correct-sign kinetic term for C.

Figure 2 :
Figure 2: Schematic figure for the GC (upper) and LD (lower) transformations.