Gauge symmetry from decoupling

Gauge symmetries emerge from a redundant description of the effective action for light degrees of freedom after the decoupling of heavy modes. This redundant description avoids the use of explicit constraints in configuration space. For non-linear constraints the gauge symmetries are non-linear. In a quantum field theory setting the gauge symmetries are local and can describe Yang-Mills theories or quantum gravity. We formulate gauge invariant fields that correspond to the non-linear light degrees of freedom. In the context of functional renormalization gauge symmetries can emerge if the flow generates or preserves large mass-like terms for the heavy degrees of freedom. They correspond to a particular form of gauge fixing terms in quantum field theories.


I. Introduction
Gauge symmetries characterize the fundamental interactions -strong, electroweak and gravitational. Where do they come from? Their most important property is to protect gauge bosons and the graviton from being massive particles, at least perturbatively and in the absence of spontaneous symmetry breaking. From the viewpoint of a microscopic theory that features a large mass scale as the Planck mass, this property guarantees non-trivial long distance physics at length scales much larger than the Planck length. From the perspective of functional flow of the effective action gauge symmetries permit "life after the Planck mass" -not all particles get "heavy masses" ∼ M such that the flow would effectively stop.
For global symmetries such a behavior is well known in a generalized Wilsonian setting of functional flow equations. An enhanced symmetry always constitutes a partial fixed point if the flow equation is compatible with this symmetry. Once the effective action exhibits a symmetry exactly, the flow will not move away from the symmetry. This defines the partial fixed point property. Small deviations from the fixed point may either grow (IR-unstable) with the flow towards the infrared, or they may decrease (IR-stable). A macroscopic global symmetry can be generated by the flow in a more general setting if the partial fixed point is IRstable.
The situation for local gauge symmetries differs from the case of global symmetries. An exact local symmetry eliminates degrees of freedom which no longer belong to the spectrum of physical excitations. A continuous approach to a local symmetry needs therefore to eliminate continuously these additional degrees of freedom. The most straightforward way how this can be achieved is the generation (or preservation) of a large mass-like term for the additional degrees of freedom. The approach to local symmetry is then the process of decoupling of the "heavy degrees of freedom". This should realize local gauge symmetry for the description of the remaining light degrees of freedom.
A different point of view may state that gauge symmetries as diffeomorphism symmetry are fundamental, with no need or use of having them emerge from a more general setting. It is, however, difficult to realize diffeomorphism invariance in a regularized quantum field theory, in particular if the setting is discrete. In a discrete formulation it is possible to impose lattice diffeomorphism symmetry [12]. This symmetry is, however, not as strong as the continuous diffeomorphism invariance. The latter should be realized in the continuum limit. One is back to the question of emergence of local gauge symmetries from the flow towards the infrared.
The present paper explores the possibility that local gauge symmetries emerge from the decoupling of heavy degrees of freedom. We first investigate in sect. II the most simple case of only one light and one heavy variable with a linear gauge symmetry. Heavy degrees of freedom are defined by the presence of a large quadratic term in the action ("heavy mass"), while for the light degrees of freedom no such term occurs. Correspondingly, we divide the sources in the functional integral into "physical sources" J for the light degrees of freedom, and complementary sources H for the heavy degrees of freedom.
More generally, the physical sources obey a constraint formulated with an appropriate projector P , P T J = J. Correspondingly, the light degrees of freedomĝ are constrained variables. One defines the effective actionΓ(ĝ) for the light degrees of freedom by simply setting the heavy degrees of freedom to zero. Omitting the constraint for g and extendingΓ(g) to unconstrained variables g realizes the gauge symmetry. It corresponds to a redundant description, sinceΓ only depends onĝ despite its formal dependence on general variables g. More precisely, this procedure involves the map g →ĝ(g) which associates to each general variable g a constrained variableĝ(g), with Γ(g) =Γ ĝ(g) . The gauge transformations acting on g are the transformations that leaveĝ(g) invariant.
The constraint on the physical sources should correspond to the covariant conservation of currents for Yang Mills theories or to the covariantly conserved energy momentum tensor for quantum gravity. These constraints involve the macroscopic gauge fields or metric. We therefore consider in sect. III the case of a "field dependent projector" P (g). Such a field dependence of the projector renders the gauge transformations non-linear, as characteristic for Yang-Mills theories or gravity. The explicit construction of the map g →ĝ(g) leads to the concept of gauge invariant variableŝ g(g) that we discuss in sect. IV. These physical variableŝ g(g) obey differential constraints P (ĝ) ∂ĝ ∂g = ∂ĝ ∂g P (g) = ∂ĝ ∂g . (1) For Yang-Mills theories or gravity they will be generalized to gauge invariant fieldsÂ µ (x) or metricsĝ µν (x). The gauge invariant variablesĝ(g) correspond to trajectories in field space rather than being defined globally. Their specification involves "initial values". We discuss the macrophysical gauge invariant effective action and its properties in sect. V. In sect. VI we generalize our setting from the simple two variable case to N variables. Quantum field theories obtain in the limit N → ∞, as briefly outlined in sect. VII. The particular gauge symmetries of Yang-Mills theories or gravity specify the projector P and therefore the form of the effective gauge fixing term for the heavy modes. Within functional renormalization for the effective average action [13] this form appears in the formalism of gauge invariant flow equations involving a single gauge field [14]. Conclusions are presented in sect. VIII.

II. Gauge symmetry for light mode from decoupling of heavy mode
We will interpret the emergence of a gauge invariant effective action in terms of the decoupling of "heavy modes" or "heavy degrees of freedom". Gauge symmetry arises as a redundant description for the "light modes". Here the notion of "heavy" and "light" is associated to the presence or absence of a large quadratic term in the action, similar to the mass term for particles in quantum field theory. (Actually, only the relative size of the quadratic terms for light and heavy degrees of freedom matters.) In this section we describe the setting in its simplest version, with one heavy mode c and one light mode b, and a linear gauge symmetry.
The emergence of gauge symmetry can be sketched as follows: The effective action Γ[g] = Γ(b, c) has a large quadratic term in c and no such term for b. We can construct an effective theory for the light mode b by simply setting c = 0,Γ ForΓ(b) any dependence on the heavy field c is eliminated. In particular, the large quadratic term for c (which corresponds to the gauge fixing term) is no longer present. We may now formally reintroduce c by using g = (b, c), while maintaining the effective actionΓ. The formal appearance of g inΓ(g) is redundant, sinceΓ actually only depends on the light mode b. This redundancy is reflected by the invariance The shift symmetry under g → g +(0, c) corresponds to the local gauge symmetry once b and c are promoted to fields. Thus gauge symmetry arises as a formal invariance in a redundant description, which tells that the only physical degree of freedom in the effective theory is b, despite the formal appearance of g = (b, c). We demonstrate the basic idea here in its simplest form, with only one light variable b and one heavy variable c. The coefficient of the quadratic term for c is taken to be constant (field independent). In this form the usefulness of a gauge invariant formulation for the light mode is not yet clear -this will become more apparent in the next section where the projections on the light and heavy modes in field space are more complex. Nevertheless, many key features of our setting are already visible in this simplest example.

Field independent source constraint
We first want to understand the circumstances under which our setting for a gauge invariant effective actionΓ arises from the generating functional W for connected correlation functions. Consider a function W that depends on two sources J and H, where ρ may depend on J. The sources J and H couple to the light and heavy degrees of freedom, respectively. The source constraint projecting on the "physical source" for the light degree of freedom simply reads H = 0. The "macroscopic fields" associated to J and H are Instead of fields we deal here, however, with simple variables b, c, J, H and simple functions W and Γ. The effective action is defined by the Legendre transform where the last identity uses Similarly, J is considered as a function of b and c, given by the solution of eq. (5). We will not need its explicit form. One has the usual identities In the limit γ/α → 0 the effective action decomposes into two separate pieces withΓ(b) the Legendre transform ofW (J) = (ρ/2)J 2 . For γ = 0 we may defineΓ(b) = Γ(b, c = 0) and observe for Γ (b) an additional term ∼ γ 2 . Consider now the two-component vectors g = (g 1 , g 2 ) = (b, c) and L = (L 1 , L 2 ) = (J, H), such that With b(g) = g 1 the functionΓ(b) can be written as Γ(g) =Γ b(g) , which actually only depends on the first component This realizes the property that the field equations only involve the physical source. The second equation (11) implies a "gauge symmetry" under the infinitesimal transformation For W we have for H = 0 the identity where b = b(J, H = 0). The second component, vanishes only for γ → 0. The projection on the light field or physical variable takes a simple form Similarly, the projection on the "physical source" J obeys This realizes a linear gauge symmetry where the projector is filed independent, similar to abelian local gauge theories as pure QED.

Physical effective action
Our example shows that gauge symmetry can be realized rather trivially by definingΓ(g) = Γ(g, c = 0). We will require further thatΓ is the "physical effective action", realized for physical sources, e.g. H = 0. Beyond the formal gauge invariance we will impose conditions such thatΓ(g) indeed describes the effective action for the light degrees of freedom,corresponding to a restriction to physical sources. The first condition simply states that for a restriction to physical sources the macroscopic variable for the heavy degree of freedom should vanish c(J, H = 0) = 0. (17) Otherwise the effective actionΓ(b) = Γ(b, c = 0) describes the system only for H = 0. For the second condition we require that the second derivative ofΓ yields the two point function for the light modes by inversion In terms of the projector P eq. (18) can be written as whereΓ andΓ (2) , W (2) denote the matrices of second derivatives, e.g.Γ Again, we require that eq. (18) or (19) holds for physical sources, i.e. H = 0. From eq. (14) and c = ∂W/∂L 2 one immediately concludes that the first condition (17) only holds for γ = 0. The projected second derivatives obey One infers the matrix identities and We want to evaluate these relations for H = 0, corresponding to c = γJ. In general, P does not vanish and Γ (2) P W (2) P does not necessarily equal P . For γ = 0, however, the source J becomes only a function of b for arbitrary H, such that Realizing thatΓ P , eq. (19) is obeyed for γ = 0. In this case also the r.h.s of eq. (24) vanishes due to ∂J/∂c = 0, implying We conclude that the two conditions (17), (19) amount to the condition γ = 0.
3. Generating function for connected n-point functions So far we have considered a given W and discussed its relation to the effective action Γ. We next want to realize W = ln Z as the generating function for connected correlation functions in a microscopic formulation. The usual functional integral for the partition function Z is here represented as a simple integral. We will see that our scenario can be realized in the usual setting with gauge fixing, but only provided that a particular form of the gauge fixing is chosen.
Let us define W (J, H) by the integral (27) Here S stands for the microscopic or classical action and S gf is "a gauge fixing term" that we first take as The gauge fixing term plays the role of the "heavy mass" for the heavy degree of freedom c ′ , and the decoupling limit will correspond to α → 0. Derivatives of W with respect to J and H yield the connected correlation functions for b ′ and c ′ . In particular, one has and If S is gauge invariant under the infinitesimal transformation δ ξ c ′ = −ξ, δ ξ b ′ = 0, it is independent of c ′ . In this case the integral (27) yields and therefore indeed γ = 0. This demonstrates how our setting can be realized in the most simple form. Already at this stage we arrive at one of the important conclusions of this paper. It is crucial that the gauge fixing is quadratic in the field c ′ and does not involve any linear term. Consider a different gauge fixing term Performing the Gaussian integration over c ′ one arrives now at where we have taken a gauge invariant microscopic action S (b ′ ). ForW (J) = (ρ/2)J 2 this produces a term linear in where The conditions (17), (19) for a physical effective action are therefore not realized for an arbitrary form of the gauge fixing! In the presence of a non-zero γ one may still define a gauge invariant effective actionΓ usinḡ Making the trivial extensionΓ(g) =Γ(g 1 , g 2 ) =Γ(g 1 ), one arrives at a gauge invariant effective actionΓ(g). The definition (36) corresponds to a "wrong expansion point" since H differs from zero for c = 0, but one may not care and be satisfied with ∂Γ/∂b = J. What goes wrong, however, is the connection between the second derivative ofΓ and the connected correlation function W P . For the projected second derivative,Γ the relationΓ P |H=0 = P no longer holds, such that the correlation function for the light modes for H = 0 cannot be extracted fromΓ.
The origin of this problem is apparent on the level of Γ where we can employ Γ (2) W (2) = 1. For one has and therefore The r.h.s. equals the projector only for γ = 0. Similarly, forΓ(b) defined by eq. (36) one has reproducing eq. (40). We conclude that W (2) P is no longer the inverse ofΓ (2) P in the projected space. This relation is crucial, however, in order to compute correlation functions fromΓ or to formulate an exact flow equation.

Gauge invariant effective action without microscopic gauge invariance
Consider next the case where the action S(b ′ , c ′ ) in eq. (27) depends on c ′ , We will show that our scenario can be realized even for a non-gauge invariant microscopic action (42), provided we take α → 0 in the gauge fixing term (28). We first keep only the term linear in c ′ . (Higher order terms can be combined with the gauge fixing term. We will be interested in the limit α → 0 for which they can be neglected.) Performing the Gaussian integral over c ′ yields (43) This produces a linear term in H corresponding to γ = 0, where the expectation value r 1 is evaluated with the action and in presence of the source J.
In the limit α → 0 the influence of the term linear in H becomes negligible. In this limit one has S ′ =S. In particular, for r 1 = rb ′ with constant r one has Corrections ∼ γ 2 /ρα in eq.(40) vanish ∼ αρr 2 for α → 0. Furthermore, the term ∼ r 2 c ′2 in eq.(42) can be neglected as compared to c ′2 /2α. For α → 0 the term exp(−c ′2 /2α) becomes δ(c ′ ), up to an irrelevant constant factor. Therefore also all higher order terms in the expansion (42) become negligible. We conclude that our setting with a gauge invariant effective actionΓ[g] and the properties (17), (19) is realized for α → 0 even if the microscopic action is not gauge invariant, e.g. r 1 = 0. The gauge symmetry violating terms are "projected out" by the gauge fixing term for α → 0. Let us describe this issue in more detail. Up to first order in α one finds withW (J) = lnZ(J), and (49) Thus W has the same form as eq.(31) up to the "linear term" αT H. The linear term does not affect W (2) P .
In the presence of this linear term the expectation value of c ′ no longer vanishes for H = 0 and nonzero small α, Inverting this equation, together with one obtains the effective action The second derivative of Γ reads such that for α → 0 the relation is obeyed. ExtendingΓ (b) toΓ (g) we realize eq.(11) and gauge invariance ofΓ according to eq.(12). The projected second functional derivative Γ P can be computed from Γ (g).
Since for α → 0 one has c ∼ α, the influence of the higher order terms in the expression (42) is suppressed by higher powers of α. For example, one has This type of expression appears (multiplied with r 2 (b ′ )) if we insert the expression (42) into the integral over c ′ . For α → 0 such terms can be neglected. We conclude that for α → 0 the functionsW (J) andΓ(b) can be computed by replacing S →S(b ′ ) in the integral (27). They remain related by a Legendre transform.

Gauge symmetry from decoupling
This simple finding has an important consequence: the limit α → 0 "projects out" any gauge symmetry violating term in the "microscopic action" S. We do not need to start with a gauge invariant microscopic action S in order to obtain a gauge invariant effective actionΓ. In particular, we may add an infrared cutoff violating gauge invariance. It may introduce a term ∼ r 1 c ′ , but the influence of this gauge symmetry breaking vanishes if we choose a gauge fixing with α → 0.
For the present example this has a very simple interpretation. For α → 0 the variable c ′ corresponds to a "heavy degree of freedom", while b ′ can be viewed as a "light degree of freedom". In the limit α → 0 the heavy degree of freedom "decouples" and the effective action for the light degree of freedom no longer feels the influence of the heavy degree of freedom. Gauge invariance expresses the fact that after omission of the heavy field the effective action for the light fieldΓ only depends on b. The generating functions W (J) andΓ(b) are related by a Legendre transform. What is important in this setting is not the gauge invariance of the microscopic action S, but rather the correct form of the quadratic term in the heavy fields S gf . In the decoupling limit α → 0 it is the form of S gf that determines the precise gauge symmetry of the effective action for the light modes.
We also note that for α → 0 an inappropriate gauge fixing term (32) cannot be "cured" by any finite non-zero source H. A finite linear term cannot modify the relation c = ǫb, such that the conditions for a physical effective action do not only follow from H = 0, but from any finite H.
Our simplest example gives already a glance how gauge symmetry can be realized by the flow to the infrared. It is sufficient that a term Γ gf of the form (9) is generated, and that α reaches the decoupling limit α → 0. Nothing else is required for the microscopic setting, provided Γ gf is the only term in the effective action that diverges for α → 0.
We will see next how these properties generalize to settings more closely related to Yang Mills theories and gravity. The main conceptual difference will be that light and heavy field cannot be defined globally, but only locally in field space.

III. Non-linear gauge symmetry
The situation for quantum gravity or non-abelian gauge theories differs in one important aspect from the setting of the previous section. In gravity the covariant conservation of the energy momentum tensor amounts to a constraint on the source that involves the metric. Similarly, the physical sources for non-abelian gauge theories are covariantly conserved currents, such that the constraint involves the macroscopic gauge field. We therefore have to generalize our setting to the case of a constraint on the source that depends on a macroscopic "field" g = (g 1 , g 2 ). In this section we will again consider only two variables g 1 and g 2 .
The projector depends now on the macroscopic field g and therefore indirectly on the physical source. Therefore the precise meaning of light and heavy degrees of freedom will be modified. In particular, the "physical variable"ĝ for the light degree of freedom can no longer be defined globally but rather obeys a differential constraint.

Field dependent projector
The main new ingredient of this section is the dependence of the projector P on the macroscopic field g and thereby indirectly on the source L. The projection property or constraint for the physical source, will define the precise gauge symmetry. We first consider a symmetric projector that obeys Here η is a function of g 1 and g 2 such that the projector P = P (g) depends on g.
The physical source J obeys the field dependent constraint IfΓ(g) couples to the physical source, this implies gauge invariance under the infinitesimal transformation as follows directly from The gauge invariance ofΓ follows from the redundant description, using unconstrained variables g while the source is constrained. We observe that the gauge variation (63) of the macroscopic variable involves g in a non-linear way, as characteristic for the field dependence of gauge transformations in gravity or non-abelian gauge theories. We may expand g around a given "expansion point" or "background"ḡ Infinitesimal fluctuations h can be split into "physical fluctuations" f , and "gauge fluctuations" a, Consider now the gauge transformation of the background fieldḡ This transformation can equivalently be accounted for by an infinitesimal change of a, leaving f andḡ invariant, An infinitesimal gauge fluctuation a can therefore be viewed as the change ofḡ under gauge transformations, hence the name "gauge fluctuation". For finite f and a the gauge transformation of g induces additional terms in δ ξ f and δ ξ a. Due to the dependence of η on g a simple global gauge degree of freedom no longer exists. The choice of the expansion pointḡ is completely arbitrary -any value of g can be used as expansion point. It makes no sense to split g globally into heavy and light degrees of freedom. However, for each g the physical and gauge "directions" are well defined by the projections of an infinitesimal variation h. Also the meaning of the projected second derivativeΓ (2) P = P TΓ(2) P is well defined. In sect. IV we will use the projections (66), (67) of the fluctuations in order to define the physical variableĝ by a differential constraint.

Generating functions from fluctuation integral
We next investigate how a field dependent constraint on the source is realized within the formulation of a "functional" integral. The partition function Z(L) is defined as The fluctuating variables c ′ and b ′ are eigenvectors of the projector P (g), and obey The projector P (g) depends on the macroscopic field g = (g 1 , g 2 ), which is a function of the sources, g = g(L). The precise choice of g(L) will be discussed later. The general source L in eq. (71) can be decomposed into J and H, with J the "physical source", The definition of the generating function for connected correlations W involves both g andḡ in addition to L, This occurs since depends on g via the projector and on the "expansion point"ḡ. We will proceed later to a particular choice of g(g), auch that W (L, g) = W L, g,ḡ(g) depends on L and g. Such a choice is understood implicitly in the following and we do not write the dependence onḡ explicitly. We also will take later the limit α → 0. The appearance of the field dependent projector introduces an unfamiliar feature in the definition of the functional integral. The formulation of the partition function Z involves the macroscopic field g. More precisely, the argument of the exponential in eq. (71) has no longer a purely linear dependence on the source L. The "gauge fixing term" ∼ 1/α depends on c ′ . In turn c ′ depends on g via eq. (77) and therefore on the sources, g = g(L). As a consequence, the integral (71) is only defined implicitly. Formally, eq.(71) becomes an integro-differential equation for Z since the r.h.s. depends on g(L). In general, the macroscopic variable g no longer equals the expectation value of the microscopic variable g ′ . The relation g = g ′ is not needed, however.
In the limit α → 0 the "non-linear formulation" of the partition function resembles in many aspects a "linear background formulation" where the projector is formulated with a fixed background variableḡ instead of the macroscopic variable g. The reason why we formulate the gauge fixing term in terms of the projector P (g) and use a dynamicalḡ(g), rather than employing a fixedḡ and P (ḡ), arises from the coupling of the "light variable" b ′ to the physical source J, which involves P (g) according to eq. (61). Indeed, with our construction using P (g) the source term obeys the simple relation This ensures that the light vector b ′ couples to the physical source, while the heavy vector c ′ couples to H. We do not impose at this point that S is a gauge invariant function of g ′ . Our implicit construction seems perhaps cumbersome. We will never need, however, to solve the integrodifferential equation explicitly.
We observe that the quadratic term ∼ c ′2 /α is close to a particular background gauge fixing. The latter would be realized if we use a fixed background fieldḡ in the projector, i.e. replacing P (g) by P (ḡ) in eqs. (73), (75).
We write the generating function with and h ′ = g ′ − g. One observes the standard relations Here partial derivatives are taken at fixed g andḡ =ḡ(g). Both g ′ and h depend on g.
For fixed g we defineΓ as the Legendre transform ofW , where L is the source associated to h by inverting the second equation (81). One has the usual relations for the first and second derivatives taken at fixed g, and observing that the source associated to h is the same as the one associated to g ′ = g + h, one sees thatΓ(h, g) can be identified with the Legendre transform of W (L, g) in eq. (79), taken at fixed g, It will be our aim to extract fromΓ(h, g) a gauge invariant effective actionΓ(g).
In order to proceed we need to specify the relation between the sources L and macroscopic variable g, L = L(g), as well as the choice ofḡ(g). Then h is, in principle, computable as a function of g. Inserting h(g) in eq. (85) the effective actionΓ will only depend on a single variable g. We will begin in the next section by identifyingḡ(g) with the "physical variable"ĝ(g).

IV. Physical variable
The physical variableĝ plays an important role for the construction of the gauge invariant effective actionΓ. Indeed, the construction of a gauge invariant actionΓ(g) from the more general effective action Γ(g) without gauge invariance relies in a first step on the restriction of the variable g to the "physical variable"ĝ. The latter obtains by eliminating the heavy degree of freedom, imposing the constraint c = 0.

Physical variable from physical fluctuation
An arbitrary variable g can be decomposed into a physical variableĝ and a "gauge variable"ĉ, g =ĝ +ĉ. (87) We therefore can also writê The split (87) of the macroscopic variable g into physical and gauge variables is a key for our construction. It will find its equivalent on the level of quantum field theory, corresponding to physical and gauge degrees of freedom. We therefore discuss this concept in some detail in our simple two-variable model. Consider two neighboring variables g =ḡ and g =ḡ + h.
This decomposition of fluctuations will be the basis for the definition of physical and gauge variables and the decomposition (87). Physical variables are defined such that the difference between two neighboring physical variables is a physical fluctuation. Let two neighboring physical variables,ĝ andĝ ′ =ĝ+ĥ, differ by an infinitesimalĥ. Physical variablesĝ are defined such thatĥ is a physical fluctuation, h = f . Formally, physical variables obey the constraint Here V (g) obeys the differential relation Applying the constraint on two neighboring physical variables yields for the difference For infinitesimalĥ = h this implies Comparing with eq.(67), h = f + a, and using, eq. (93) implies a k = 0 and we identify indeed The family of physical variablesĝ corresponds to a "trajectory" in the space of variables g where two neighboring points are connected by a physical fluctuation f . Since the trajectory only has to obey a differential equation the family of physical variables is specified uniquely if a suitable initial valueḡ 0 is chosen. From there it can be spanned by subsequent additions of physical fluctuations f .

Physical and gauge variables from trajectories in field space
For our simple example the physical variablesĝ are represented by a line in the two-dimensional space of macroscopic fields g. This line is specified by a choice of an initial conditionḡ 0 . Two neighboringḡ 0 are equivalent if they generate the same trajectoryĝ for physical fields. This is realized if the two neighboring initial valuesḡ 0 differ by a physical fluctuation. In contrast, different lines of physical metrics are induced if the two neighboring initial values differ by a gauge fluctuation.
Inversely, for a given g we can follow the differential equation (97) forĉ until the line g −ĉ(τ ) intersects the line of physical variablesĝ(σ) for some σ. For this purpose σ needs not to be known -the projector in eq. (97) only depends on g(σ, τ ), and for fixed σ andĝ(σ) we have the differential equation P g(σ, τ ) ∂ τ g(σ, τ ) = 0. This procedure can be visualized in fig. 1. For a fixed g, indicated by the filled square, one follows the trajectory for fixed but unknown σ until it intersects the line of physical metrics (open square), thus determiningĝ(σ), and correspondinglŷ c(τ, σ). For every g we can construct in this wayĝ(g) and c(g) and realize the decomposition (87). We indicate the variablesĝ(g) andĉ(g) by arrows in fig. 1. In the following we will assume the absence of bifurcations, at least in the region close toĝ(σ) which will be needed for our purposes. With this assumption the decomposition (87) exists and is unique.
In principle, the choice of the initial valueḡ 0 , that is needed for a unique specification ofĝ(g), is arbitrary. An obvious possible choice is to identifyḡ 0 with the expectation value < g ′ > in the absence of sources,ḡ The difference betweenĝ and < g ′ > arises then only for L = 0.

Differential constraints
We can use this construction in order to derive two differential constraints for the map g →ĝ(g). An infinitesimal change dĝ =ĥ obeys P (ĝ)dĝ = dĝ, and we infer or An infinitesimal gauge variation of g, dg = a, obeys P (g)dg = 0. It changesĉ, corresponding to a change of τ at fixed σ, but leavesĝ invariant, This entails the constraint or By construction, the physical variableĝ is gauge invariant. This obtains formally from eq. (63) by virtue of the constraint (102), Similarly, a physical variation of g, dg = f , changesĝ and leavesĉ invariant. This implies the relation We are interested in the region of very small |ĉ|, with typically values that vanish as α goes to zero. Infinitesimalĉ obey the simple condition P (ĝ)ĉ = 0.

Gauge invariant variable
The physical variableĝ is a two-component variable. Due to the differential constraint (103) it is gauge invariant. The valley corresponding toĝ(g) is a one-dimensional hypermanifold. We may therefore construct a single gauge invariant variable s corresponding to this hypermanifold.
For this purpose we employ normalized vectors w and v that are eigenstates of the projector P , with For the projector (59) they are given explicitly by The eigenvectors of P obey We further introduce vectors U and V that are related to w and v by a differential relation The difference between U and w or V and v reflects the dependence of w and v on g, which arises since η depends on g. The gauge invariant variable s is constructed as Indeed, U T (g)g is invariant, The gauge invariant variable s can be extended to gauge invariant field combinations in Yang-Mills theories or gravity. This is mainly an argument of existence, since an explicit construction of U may be difficult.
This can be interpreted as the effective action for the light variable, with heavy variableĉ eliminated. At this pointĝ is a constrained variable. The effective action for the light variable can be extended to the gauge invariant effective action Γ(g) =Γ(ĝ(g)). ( This description is redundant sinceΓ(g) depends formally on an arbitrary variable g despite the fact that it only involvesĝ. Gauge symmetry expresses this redundancy. By virtue of the differential constraint (103) one has The gauge variation ofΓ therefore vanishes Of course this is a direct consequence of the gauge invariance of the physical variableĝ. From eq. (117) one also concludes that the first variation ofΓ is some generalized physical source The definition (115) is motivated if Γ involves a large quadratic term For α → 0 the field equations derived by variation of Γ will be solved forĉ = 0. Settingĉ = 0 eliminates the heavy mode, such thatΓ is indeed the effective action for the light mode.
We may use fig. 1 in order to visualize our construction of the gauge invariant effective actionΓ(g). The effective action Γ(g) (without gauge symmetry) is a function over the (g 1 , g 2 ) plane. For small α it has a deep valley along the line of the physical variableĝ, with second derivative perpendicular to this line ∼ 1/α. The actionΓ(ĝ) has support only on the line for the variableĝ. The extension Γ(g) =Γ ĝ(g) has again support in the whole plane. It is constant along the lines g(σ, τ ) with fixed σ, taking the valueΓ ĝ(σ) .

Macroscopic emergence of gauge symmetry
At this point we may formulate the general condition how an effective gauge invariant theory arises from a more general effective action Γ(g) that is not gauge invariant. Two conditions are sufficient: (i) Γ(g) contains a term ∼ĉ 2 with a coefficient ∼ 1/α that exceeds all other relevant scales. Hereĉ is defined by the decomposition g =ĝ +ĉ, withĝ obeying a differential constraint. We will take later α → 0 such that smallĉ obeys in linear order P (ĝ)ĉ = 0. (ii) A possible term linear inĉ in Γ(g) should have a coefficient that remains finite for α → 0. These two conditions are realized if Γ(g) contains a gauge fixing term (120) and no other terms diverge ∼ α −1 .
As a consequence of these two conditions the field equations can be projected into a "heavy" and a"light" sector, and where g =ĝ +ĉ. Here we use the relation (105), and eq. (121) reads more precisely The coefficients A 0 , A 1 for the expansion in powers ofĉ in eq. (122) remain finite for α → 0. The dots on the r.h.s. of eqs. (121), (122) denote terms that vanish forĉ → 0 if c/α is finite. For α → 0, and finite sources L, the solution of eq. (121) impliesĉ = 0. This can be inserted into eq.
(122) such that only A 0 (ĝ) matters. For the light degrees of freedomĝ we can defineΓ(ĝ) = Γ(g,ĉ = 0). The gauge invariant effective actionΓ(g) obtains then by dropping the constraint onĝ and extendingΓ(g) =Γ ĝ(g) . We next show that forĉ = 0 the l.h.s of eq. (122) can be written as ∂Γ/∂g, For this purpose we write such that Eq. (124) follows from eqs. (117) and (105). We recover eq. (119). We emphasize that the conditions leading to a gauge invariant effective actionΓ(g) for the light degrees of freedom are purely formulated on the "macroscopic level", e.g. in terms of the effective action Γ(g). No particular assumption on the microscopic physics that leads to these conditions is required. This opens the possibility that gauge invariance emerges as a result of the "renormalization flow" from microphysics to macrophysics. It is sufficient that this flow produces or keeps a term ∼ĉ 2 /α that is huge on the scales of the effective theory for the light degrees of freedom, and that no huge term linear inĉ is generated. The precise relation between Γ(g) andΓ(g), as defined by eq. (86) is not important. Also the sourcesL,J,H may differ from L, J, H.

Gauge symmetry from microscopic formulation
The crucial point for the emergence of macroscopic gauge symmetry from the decoupling of the heavy degree of freedom is the presence of the gauge fixing term (120) in Γ, with α → 0. We will show that this obtains in the microscopic formulation (71). The functions Γ(g) andΓ(g) may differ by a term that does not diverge for α → 0. This difference will not affect the divergent gauge fixing term (120) in Γ(g).
We identify the dynamical backgroundḡ(g) withĝ(g). For the leading divergent term for α → 0 the saddle point approximation becomes valid, such that andΓ f in does not diverge ∼ α −1 for α → 0. Indeed,Γ gf corresponds to the "classical approximation", while "loopcorrections" do not diverge ∼ α −1 . The expectation value c ′ (g) is evaluated in the presence of sources and obeys, by virtue of eq. (77), The precise relation between g and L does not matter in this context. For smallĉ we can expand the propagator P (g) = P (ĝ + c) inĉ. The leading term in the constraint (129) takes the form which coincides with the projector equation (106) for infinitesimalĉ. This suggests that for a suitable choice of the macroscopic variable g the quantities c ′ (g) andĉ(g) can be identified in the limit α → 0. The relation g(L) can be defined implicitly by the relation between g and g ′ . We choose the macroscopic variable g such that With this choice andḡ =ĝ the relations indeed implyĉ = c ′ . This closes the argument that Γ(g) contains a gauge fixing term (120) in the limit α → 0, and therefore establishes the gauge invariant effective action Γ(g) for the light degree of freedom. We observe that the condition (131) does not fix the choice of the macroscopic variable g uniquely. One possible choice could simply be g = g ′ , which is equivalent to b ′ = 0. We will admit, however, the more general choice (131), for which we only require the condition One may use the remaining freedom in the choice of g(L) (compatible with eq. (133)) and the precise relation between Γ andΓ in order to "optimize" the properties of Γ(g). For example, this freedom is used in ref. [14] in order to obtain a simple form of a gauge invariant flow equation for a scale-dependentΓ[g]. We conclude that macroscopic gauge symmetry can emerge from microphysics in a rather general setting. The microscopic formulation (71) is only an example for a much wider class of microscopic settings that can lead to macroscopic gauge invariance. For α → 0 the precise form of the microscopic action S is arbitrary. The only thing that fixes the gauge symmetry is the form of the diverging effective action Γ gf for the heavy degree of freedom. In the context of flow equations it is sufficient that α flows towards zero, even if it does not vanish on the microscopic level. The generic emergence of gauge symmetry for the effective actionΓ(g) does not yet guarantee that the properties of Γ(g) are simple. For quantum field theories this concerns, in particular, locality properties ofΓ(g).

VI. Multi-component variables
In this section we proceed towards the construction of the gauge invariant effective action for quantum field theories. The main conceptual issues can already be understood by the two-component examples of the preceding two sections. The way to quantum field theory proceeds by a rather straightforward generalization to N -components, and finally to the limit N → ∞. The indices of the Ncomponent vectors will then contain spacetime coordinates or momenta.

Multi-component vectors
It is straightforward to generalize the microscopic and macroscopic variables g ′ and g, as well as the sources L, to N -component vectors. The projectors P and (1 − P ) depend again on the macroscopic variable g. The number of eigenvalues 1 and 0 of P needs not to be equal. If P has M eigenvalues 0 we have M "gauge degrees of freedom", c ′ = (1 − P ) h ′ , and N − M "physical degrees of freedom", b ′ = P h ′ . The projector is not necessarily symmetric. We use covariant vectors g i , g ′ i and contravariant vectors L i for the sources, such that The physical sources J i obey As before, the relation implies gauge invariance under infinitesimal transformations according to The projector 1 − P has M eigenvectors v s for the eigenvalue one, and N −M eigenvectors w u , u = 1 · · · N −M , for the eigenvalues zero (or eigenvalues one of P ), In terms of these eigenvectors the gauge transformation takes the form Contravariant vectors are related to covariant vectors by (For symmetric projectors P T = P one can use D ij = δ ij such that there is no difference between g ′i and g ′ i .) We also employ and similarly for w i u , obeying (For P T = P we can choose F ts = δ ts .) The normalization is chosen as and the orthogonality of eigenspaces implies v s i w i u = 0.
Using v i s v s i = M , w i u w u i = N − M , as well as the projector properties, one finds 2. Physical and gauge degrees of freedom We will employ again the physical and gauge degrees of freedomĝ i andĉ i and the decomposition Their construction is analogous to the two-variable model of sect. IV. For any given g i the physical and gauge fluctuations f i and a i obey Starting from a given initial valueḡ 0,i we subsequently add physical fluctuations f i in order to construct the N − Mdimensional hypermanifoldĝ i . This manifold may be parametrized by N − M variables σ u asĝ i (σ u ). The physical degrees of freedom obey the differential equation For everyĝ i (σ u ) we then constructĉ i (σ u , τ s ) by subsequently adding gauge fluctuations. They are solutions of the differential equations We again assume the absence of bifurcations in the relevant region of smallĉ such that the decomposition (148) exists andĝ(g) andĉ(g) are unique. The map g →ĝ obeys simple differential properties. First, an infinitesimal difference between two physical degrees of freedom is a physical fluctuation. It therefore obeys implying for the derivatives A simple constraint therefore applies to the partial derivatives ∂ĝ j /∂g k , and not toĝ j . Second, changing g j by a gauge fluctuation changesĉ i but does not affectĝ i , This implies the constraint The differential relations (154) and (156) constrain the variation of the hyperfaceĝ(g) with g. They correspond to eq. (1). We also generalize the relation (105), such that infinitesimalĉ obey The infinitesimal gauge transformation of g i , obeys This is a transformation in the gauge direction that can be realized by Indeed, the gauge invariance ofĝ follows from the differential constraint (156),(??), For infinitesimalĉ we can define These variables correspond to c in the two-variable model. With the transformation (160) one has In conclusion, we have decomposed the macroscopic variables g i into gauge invariant physical variablesĝ i and gauge degrees of freedomĉ i . Gauge transformations only act on c i . The decomposition is, in general, not global andĝ i only obeys differential constraints for its dependence on g. The precise definitionĝ therefore depends on the choice of an "initial value"ḡ 0 .
We can again construct N − M gauge invariant variables s u (g) which obey the defining relation and are formally given by with U i u obeying the differential equation Both s u and U i u are uniquely specified once the "initial values" for the solution of eq. (166) are given for someḡ 0 . The gauge invariance of s u follows from Generalizing g i to fields the variables s u become gauge invariant field combinations. Their number corresponds to the physical degrees of freedom, as obtained by the number of degrees of freedom in g minus the number of gauge degrees of freedom. While these gauge invariant field combinations exist, they are difficult to construct explicitly in practice.
The M -dimensional hypersurface spanned byĝ i (σ u ) + c i (σ u , τ s ) for fixed σ u andĝ(σ u ) constitutes a manifold of constant s u (g) = U i u (g)g i , since vanishes if h i is a gauge fluctuation.

Gauge invariant effective action
Consider now an effective action Γ(g) that contains a term We will take α → 0 and assume that no other parts in Γ diverge in this limit. For T ij we assume that it has no zero eigenvalues on the projected space corresponding tô c i obeying eq. (157). In other words,ĉ i T ijĉ j = 0 implieŝ c i = 0. This is sufficient for the extraction of a gauge invariant effective actionΓ(g).
For finite sources the field equations require for the solutionĉ Inserting this partial solution into Γ(g) yields the effective action for the light degrees of freedomĝ i , The gauge invariant effective actionΓ(g) is defined as the extensionΓ We can use eq. (156), for establishing the gauge invariance ofΓ(g). The first derivative obeys implying a vanishing gauge variation We conclude that a general effective action of the form is projected onto a gauge invariant effective actionΓ(g) = Γ ĝ(g) for the light degrees of freedom provided the limit α → 0 is taken. Here ∆Γ is assumed to remain finite for α → 0 (or diverge less fast than α −1 ), and it is defined such that it vanishes forĉ = 0. This projection is realized by the solution (171) of the field equation for the heavy degree of freedom. It corresponds to the "decoupling of the heavy modes". The "functional" integral (178) leads for α → 0 precisely to an effective action of the form (177). Gauge invariance of S(g ′ ) is not needed. The gauge degrees of freedom are given bŷ and the physical variablesĝ are gauge invariant. They obey the differential constraints (153), (156). The argument proceeds in parallel to sect. V, relying on the validity of the saddle point approximation for the leading singular term in the limit α → 0. We conclude that the construction of a gauge invariant effective actionΓ(g) can be extended to an arbitrary number of fields. The field dependent projector P (g) needs not to be symmetric. On the level of the "functional" integral the crucial ingredient is the form of the "gauge fixing term" (169), being quadratic in the projected fluctuation fields c ′ i , and therefore dependent through the projector on the macroscopic field g. The limit α → 0 leads to an effective action for the light fields. Its arguments are the physical fieldsĝ i , andΓ[ĝ] turns to a gauge invariant action if the constraint onĝ is dropped. The limit N → ∞ does not pose any particular problem in this construction. We can therefore promote our construction to quantum field theories, and the integrals to functional integrals.

VII. Quantum field theories
The extension to quantum field theories is conceptually straightforward. It corresponds to the limit N → ∞, where the index i comprises now a space-time label x ν as well as Lorentz and internal indices.

Yang-Mills theories
For Yang-Mills theories the multi-component vector g i stands for the gauge field A z µ (x). The projector on physical modes P (g) is given by [15] It depends on the macroscopic field A µ through the covariant derivative Eqs. (180) and (181) involve matrices in the adjoint representation, e.g.
For the particular case of abelian gauge theories one has D µ = ∂ µ and P becomes independent of A µ . This generalizes the simple setting of sect. II. For the decomposition the physical gauge fieldsÂ µ (A) obey the differential constraint (For non-abelian gauge theories no global relation of the type P µ ν (Â)Â ν =Â µ is obeyed.) The gauge degrees of freedomĉ µ obey and for infinitesimalĉ µ one has Assume now that Γ[A] contains a gauge fixing term For α → 0 the solution of the field equation is found for finite sources L aŝ From the effective action for the light modes the gauge invariant action follows as By virtue of the differential constraints (184) one has This is obeyed for covariantly conserved sources For a gauge variation the conservation of the current (193) implies gauge invariance ofΓ Starting from a microscopic formulation the macroscopic gauge fixing term (187) is realized for a microscopic gauge fixing with This corresponds to Landau gauge fixing with a dynamical background field. With one has and therefore For α → 0 the leading order saddle point approximation generates in Γ the required term (187), witĥ We conclude that the macroscopic emergence of gauge symmetry is realized if the microscopic gauge fixing term takes the specific form of the Landau gauge fixing (196), (197).

Gravity
For gravity the vector g i corresponds to the metric g µν (x). The explicit construction of the projector P µν ρτ (x, y) on the physical fluctuations is more involved than for Yang-Mills theories. It has been discussed in ref. [16]. We present here only the structural aspects.
The physical sources are denoted here by K µν = K νµ and obey the constraint, where Γ ν µρ (g) is the connection formed with the metric g µν .
The effective actionΓ will be constructed such that it obeys With the constraint (202) this results in diffeomorphism invariance ofΓ. Multiplying eq.(203) with ξ µ and performing an integration over x yields after partial integration Employing eq.(203) and using the symmetry of g µν and K µν one obtains where With infinitesimal ξ µ = g µν ξ ν we recognize in eq.(206) the variation of the metric with respect to an infinitesimal diffeomorphism transformation. The interpretation of the constraint (202) is straightforward. The energy momentum tensor T µν is related to K µν by such that the constraint (202) expresses the covariant conservation of the energy momentum tensor A restriction to sources corresponding to conserved energy momentum tensors implies gauge invariance of the effective action. The latter can also be expressed by the local identity which follows from the combination of eqs.(203) and (202). Inversely, local gauge invariance ofΓ implies a conserved energy momentum tensor as reflected by the relation (202). Let us next specify the projector P on the physical fluctuations and sources. Consider for a given metricḡ µν a neighbouring metricḡ µν + h µν . We split the metric fluctuations h µν into "physical fluctuations" f µν and "gauge fluctuations" a µν , according to the decomposition into a vector and divergence free tensor part Here the covariant derivative D µ involves the connection formed withḡ µν . Also the lowering and raising of indices is performed withḡ µν andḡ µν , respectively. An infinitesimal gauge transformation (206) ofḡ µν can be realized by a µ → a µ − ξ µ , with invariant f µν . This motivates the naming of the fluctuations f µν and a µν . We can write the decomposition of h formally in terms of the projector P , P 2 = P , namely P a = 0, P f = f. (212) More explicitly, the projector is defined by two conditions: the first states that for arbitrary vectors a µ one has while the second expresses the projector property, (This also relates a to h by a = h − f = (1 − P ) h.) The explicit construction of the projector is not as simple as for Yang-Mills theories, for a discussion see ref. [16]. We decompose the macroscopic metric g µν into the "physical metric"ĝ µν and the gauge modeĉ µν g µν =ĝ µν +ĉ µν .
The family of physical metrics is characterized by the property that two neighbouring physical metrics differ by a physical fluctuation f µν . This is a differential relation in function space A unique manifold of solutions to the differential relations (218) needs the specification of some "initial value"ḡ 0 µν , from which the other physical metrics can be obtained by subsequently adding physical fluctuations f µν . Dealing with such a constraint in practice is rather cumbersome.
Dropping the formal constraint will lead gauge symmetry and result in important practical simplification.
It is important that the constraint on physical metrics is only formulated for the infinitesimal difference between two such metrics, f ν µν; = 0. There is no corresponding "global constraint" onĝ µν . For this reason the concept of a family of physical metrics is somewhat hidden A second reason is the differential character of the constraint which requires the choice of some initialḡ 0 µν . Differentḡ 0 µν lead to different families of physical metrics. The precise choice is arbitrary.
Using the decomposition (217) we write an arbitrary functional of the macroscopic metric in the form Our setting is realized if Γ [g] takes the form with a generalized "gauge fixing term" Γ gf at least quadratic inĉ µν . The first derivative of Γ yields sourcesL, The solutions withL µν = K µν correspond toĉ µν = 0, such that Γ gf vanishes if the solution is inserted. This extends to arbitraryL µν if the coefficient ∼ α −1 in front of the gauge fixing term diverges. As usual, the effective action for the "light modes"ĝ µν obtains from Γ by inserting the solution of the field equation for the "heavy modes"ĉ µν Γ[ĝ µν ] =Γ[ĝ µν ,ĉ µν = 0].
The gauge invariant effective actionΓ will then obtain by dropping the formal constraint onĝ µν , The gauge fixing term is no longer present. We finally connect our setting to a gauge fixed functional integral with a particular gauge fixing Here the covariant derivative D µ is formed with the macroscopic metric g µν . Together with its inverse g µν the macroscopic metric is also used to lower and raise indices. Decomposing one finds that S gf is indeed quadratic in c ′ µν and does not involve b ′ µν , Identifyingĉ µν = c ′ µν (227) induces in the effective action the gauge fixing term Γ gf , for which c ′ is replaced byĉ in eq. (226). The particular "physical gauge fixing" (224) has been advocated in ref. [16] and used for flow equations in ref. [17].

VIII. Conclusions
We have investigated the possibility that local gauge symmetries emerge macroscopically from microscopic laws that do not necessarily exhibit these symmetries. As a basic concept, the flow of a scale dependent effective action from short distances (microphysics) to large distances (macrophysics) may generate the gauge symmetries. This seems indeed possible.
The mechanism for the dynamical generation of a local symmetry differs, however, profoundly from the case of a global symmetry. A general effective action Γ may be written as a gauge invariant partΓ and a gauge violating part ∆Γ, Γ(g) =Γ(g) + ∆Γ(g). (228) For a dynamical emergence of a global symmetry ∆Γ should flow towards zero in the infrared. In contrast, a local symmetry can be realized dynamically if ∆Γ diverges in the infrared in a particular way. This happens if ∆Γ constitutes a gauge fixing term which is quadratic in the gauge fluctuations. The quadratic term separates heavy from light modes. Eliminating the heavy modes eliminates ∆Γ, leavingΓ as the effective action for the light modes. At this point the light modes correspond to constrained fieldsĝ(g). Dropping the constraint by the extensionΓ(g) =Γ ĝ(g) results in a redundant description that exhibits the gauge symmetry. This mechanism works for a particular form of a diverging gauge fixing term ∆Γ ∼ Γ gf . The gauge fixing term defines a hypermanifold of light fieldsĝ(g) for which it does not contribute. The gauge modes are perpendicular to this hypermanifold, and the gauge invariance ofΓ(g) expresses thatΓ does not depend on the gauge modes. The characterization of the gauge modes, and the corresponding projector P on physical fluctuations, is determined by the form of Γ gf . For a given gauge symmetry, as a local gauge group for Yang-Mills theories or diffeomorphisms for gravity, the properties of physical fieldsĝ(g) and gauge modeŝ c(g) are fixed (up to some initial values). This corresponds to a particular class of "physical" gauge fixing terms that produce for the light fields the wanted gauge symmetry. For Yang-Mills theories the Landau gauge with dynamical background field belongs to this class. For gravity the covariant conservation D ν h µν = 0 for metric fluctuations h µν is a physical gauge fixing. Other gauge fixing terms may lead to a projection on heavy and light modes that do not correspond to the wanted gauge symmetry.
It is possible that the required gauge fixing term is generated during the flow, even if not present at the microscopic level. This applies, in particular to the diverging coefficient ∼ α −1 in front of the gauge fixing term. The value α = 0 is a (partial) fixed point to which the flow may be attracted in the infrared.
The generation of a suitable gauge fixing term is sufficient for a realization of a local gauge symmetry for the effective action for the light modes. Local gauge symmetries can therefore arise in a rather general context. This does not imply, however, that all such gauge symmetric effective actions belong to the same universality class as standard Yang-Mills theories or quantum gravity. While local gauge symmetry is a crucial ingredient, it is presumably not sufficient for the determination of the universality class. In addition,Γ(g) should have suitable locality properties, for example admitting a derivative expansion on scales where perturbation theory applies (sufficiently above the confinement scale). Also the generalized measure contributions (Faddeev-Popov determinant or associated ghost sector) should be present.
Having established the way how local gauge symmetries can arise dynamically, the focus will have to concentrate on the properties of universality classes in order to find out under which circumstances the known fundamental interactions could emerge as a "long-distance-property".