Quantum Space-Time Symmetries: A Principle of Minimum Group Representation

We show that, as in the case of the principle of minimum action in classical and quantum mechanics, there exists an even more general principle in the very fundamental structure of {\it quantum space-time}: This is the principle of {\it minimal group representation} that allows to consistently and simultaneously obtain a natural description of the spacetime dynamics and the physical states admissible in it. The theoretical construction is based on the physical states, average values of the Metaplectic group $Mp(n)$ generators: the double covering of $SL(2C)$ in a vector representation, with respect to the {\it coherent states} carrying the spin weight. Our main results here are: (i) A connection between the Metaplectic symmetry generators and the physical state dynamics. (ii) The ground states are coherent states, of Perelomov-Klauder type of the Metaplectic group dividing the Hilbert space into {\it even} and {\it odd} states. (iii) The physical states have spin contents $s = 0,\; 1/2, \;1,\; 3/2$ and $2$. (iv) The generators introduce a natural supersymmetry and a superspace whose line element is the geometrical Lagrangian of our model. (v) A coherent physical state of spin 2 is obtained naturally related to the metric tensor. (vi) This is {\it naturally discretized} by the discrete series in the $n$ number representation, reaching the classical (continuous) space-time for $n$ $\rightarrow\infty$. (vii) A relation emerges between the coherent state metric eigenvalue $\alpha$ and the black hole entropy through the Planck length. The lowest level of the quantum space-time spectrum, $n = 0$ and its characteristic length, yields a minimum entropy for the black hole history.


I. INTRODUCTION
A key concept for a full quantum theory of gravity is quantum space-time as well as for quantum theory in its own.The basic motivation of this paper is to demonstrate that, as in the case of classical and quantum mechanics in which the minimum action is the regulating and determining principle, there is an even more general principle that intervenes in the very and fundamental structure of quantum space-time: this is the interplay between dynamics and symmetry or alternatively matter/energy and space-time.The maximum simplicity to achieve this goal is based on the Metaplectic group M p (n) which is the double covering of the S p (2C) group, and which for the illustrative case that we intend to establish here we fix to M p (2).We characterize quantum space-time as originating from a mapping P (G, M ) between the real space-time manifold M and the quantum phase space manifold of a group G. Once one component of the momentum P operator is identified with the time T , the space-time metric of M is found using the metric g ab on the phase space group manifold.The group's compactness or noncompactness determines the metric's signature; nonetheless, noncompact groups are necessary for the majority of physical situations of interest because of the real space-time signature and its hyperbolic structure..The quantum space-time established from the harmonic oscillator's phase space represents the more obviously fundamental examples of this development.In the case of the normal (real frequency) oscillator, refer to Refs [1], [2], [3], and the mapping (X, P ) → (X, iT ), or alternatively (X, P ) → (X, T ) in the situation of the imaginary frequency ( the inverted oscillator) which appear in many physical examples, in particular in cosmology (e.g; in the propagation eqs of classical and quantum perturbations).The quantum space-time algebra of non commutative operator coordinates is the quantum oscillator algebra.The line element arises from the Hamiltonian (Casimir operator) and its discretization yields the quantum space levels.The zero point energy yields the new quantum region splitting the light cone origin because the classical generating lines X = ±T are replaced by the curves X 2 − T 2 = [X, T ] which are the quantum hyperbolae due to the non-zero space and time commutators, and generate in particular the quantum light cone.[4], [5] The inverted oscillator is associated to the hyperbolic space-time structure, while the normal oscillator yields euclidean (imaginary time) signature (quantum gravitational instantons).
An important point in this principle of minimum group representation is the description of quantum spacetime symmetries: The "algebro-pseudo-differential correspondence" plays a key role.This correspondence establishes that a radical operator (e.g. a Hamiltonian) is equivalent in the context of the metaplectic description, to a Majorana-Dirac type operator with internal variables in the oscillator representation .This correspondence is exemplified in the expression Eq.( 13), Section IV in this paper.This algebraic interpretation is significant because it allows for a connection with pseudodifferential operators and semigroup (Fourier-Integral) representations, as shown below.[6]: In this theoretical and physical context, the resulting solution consists of two types: the basic state and the observable physical state,which is bilinear with respect to the basic state (e.g.,the mean value).The basic state is a coherent state corresponding to the Metaplectic group, which is the double covering of the SL(2C) group, [6]- [10].We use as our example, Ref [6], a N = 1 superspace with an invertible and nondegenerate supermetric, where the unconstrained quantization is precisely carried out using novel techniques based on coherent states and keeping the Hamiltonian's form.Thus, from the discrete spectrum of the states themselves, a discrete structure of the spacetime automatically arises without any prescription of discretization.
Due to the Metaplectic representation (double covering of the SL(2C)) of the coherent state solution representing the emergent spacetime, the crossover from the quantum micro-scopic regime to the macroscopical regime (classical or not) is natural and consistent.This important fact allows us to conciliate apparently different pictures as that of a macroscopical quantum gravity regime and that of a dynamical quantum microscopic picture (the complete process of black hole emission in all its stages being a clear example).
Despite its simplicity, the framework introduced here have provided physically and geometrically significant responses concerning an accurate description of quantum gravity.
It is convenient to think of this kind of coherent states as arising from a Lie group G operating on a Hilbert space H through a unitary, irreducible representation T .The set of vectors ψ ∈ H such that ψ = T (g) ψ 0 for some g ∈ G. is what we describe as the coherent state system{T, ψ 0 } for a fixed vector ψ 0 .We define the states |ψ⟩ corresponding to these vectors in H as generalized coherent states.We briefly describe now the relevant symmetry group to achieve the realization of the Hamiltonian operator of the problem.Specifically, this group is the metaplectic M p (2) group as the groups SU (1, 1) and Sp(2) that are topologically covered by it.In function of the (q, p) operators, or equivalently the operators (a, a + ) of the standard harmonic oscillator, the generators of M p (2) are: With the commutation relations, The commutation relations can be written as: . Therefore, the oscillator states | n ⟩ of the number operator are eigenstates of the T 3 generator

III. THE MP(2) VECTOR REPRESENTATION AND ITS COVERINGS
The commutation relation that specifies the generators L i is the main feature of the specific representation that was introduced in [2] : The representation above is a non-compact Lie algebra with the following matrix form: that obey, in a geometrical way: We want to remark is the following equivalence : The generators in the representation of Eq.( 2) fulfil the relation: where T i are the Metaplectic generators namely [6], [7]: Proof : We can write the generators L i in matrix form as The representation Eq. ( 2) is faithful, we take into account that σ k enter as a "metric" in the sense given in Ref [15], that is, it introduces the signature in the quadratic terms in a and a + Eq.( 9) explicitly giving rise to the expression Eq.( 5).Therefore, we have: Consequently, and by inspection, Eq.( 2) coincides with Eq.( 9): Thus, the equivalence Eq.( 5) is proved.

IV. SYMMETRY AND DYNAMICS PRINCIPLE: STEPS TO FOLLOW
A fundamental component of the dynamic description is the square root type Hamiltonian or Lagrangian, which is, in theory, a non-local and non-linear operator.This is because the right physical spectrum is generated by the invariance under reparametrizations both as a Lagrangian and as a corresponding Hamiltonian.The fundamental principles of our strategy here are based on certain elements that are explicitly mentioned in the sequel: 1.The invariant action (i) Considering the space-time-matter structure, the geometric Lagrangian (functional action) of the theory is the elementary distance function, which is defined as the positive square root of the line element.
At least, the line element's symmetry matches that provided by the super-Poincaré or Cartan-Killing form of Osp(1,2), enabling a bosonic realization based on the a and a + operators of the conventional harmonic oscillator.This leads to the metric being nondegenerate and having extra odd (fermionic) coordinates.

Extended Hamiltonian of the system
The geometric Hamiltonian, which is the fundamental classical-quantum operator, is obtained from (i) in the conventional manner.
From the perspective of the physical states, this universal Hamiltonian (square root Hamiltonian) has an enlarged phase space because it includes a zero moment P 0 characteristic of the entire phase space at its highest level.
Time "disappears" from the dynamic equations in a proper time system when the evolution coincides with the time coordinate.This is prevented by including a zero momentum P 0 , which would otherwise lead to the arbitrary nullification of the Hamiltonian.

Relativistic wave equation and the algebraic interpretation
(iv) The Hamiltonian H s , rewritten in differential form, defines a new relativistic wave equation of second order and degree 1/2 (square root form).This fact can be reinterpreted as a Dirac-Sudarshan type equation of positive energies and internal variables (e.g.oscillator type variables) contained as components of the auxiliary or internal vector L α : having the basic solution-states of the system, a para-Bose or para-Fermi interpretation of | Ψ ⟩.This gives rise to the main justification for an algebraic interpretation of the radical operator: we have a clean action at operator level and a consistent number of states of the system (the Lagrange multiplier method eliminates the square root in a non physical way doubling the spectrum of physical states).With the Mp(2) interpretation we can also describe a complete multiplet spanning spins from ( 0, 1/2, 1, 3/2, 2 ).This is a consequence of the fact that with the fundamental states and the allowed vectorial generators, the tower of states is finite and all the states involved are physical, as it must be in the physical context.

V. STATEMENT OF THE PROBLEM
Geometrically, we take as the starting point the functional action that will describe the world-line (measure on a superspace) of the superparticle as follows: where θ ), and the dot indicates derivative with respect to the parameter τ , as usual.The above Lagrangian was constructed considering the line element (e.g. the measure, positive square root of the interval) of the non-degenerated supermetric where a superspace ( 1, 3 |1 ) is composed by the bosonic term and the Majorana bispinor , with coordinates (t, x i , θ α , θ α), being the Maurer-Cartan forms of the supersymmetry group are: As our manifold have extended to include fermionic coordinates, it is natural to extend also the concept of trajectory for a point particle to the superspace.Consequently, we take the coordinates x (τ ), θ α (τ ) and θ .
The Hamiltonian in square root form, namely m 2 − P 0 P 0 − P α |Ψ⟩ = 0, is constructed defining the supermomenta as usual and the Lanczos method for constrained Hamiltonian systems was used, due the nullification of this Hamiltonian Therefore, an algebraic realization of the pseudo-differential operator (square root) does exist in the case of an underlying Mp(n) group structure: Therefore, both structures can be identified: e.g.
, being the state Ψ the square root of a spinor Φ (on which the "square root" Hamiltonian operates) in such a manner that it can have the bilinear expression Φ = ΨL α Ψ. Equation and Eq. ( 15) in the context of our work has its equivalent second order Dirac-Like operator in the expression given by Equation and Eq.( 16).This type of operator has been developed by Majorana, Dirac (e.g.[18], [19], and others [20] ) containing internal variables of the harmonic oscillator type, and in our original and particular case, it gives an algebraic interpretation to the radical operator, with two fundamental objectives fulfilled: interpreting the action of the square root operator, and describing the relationship between the physical (bilinear) states and the fundamental (basic) states, as described in detail in Section VI here.Equation ( 16) is nothing more and nothing less than the algebraic interpretation of the radical operator: a Majorana Dirac type operator, that is to say, a equation with internal variables in the sense of Dirac, Majorana, and others refs, e.g.[18], [19], [20] with different spinorial decomposition structure.The curly brackets in Equation ( 16) define the limit of the equivalence with the radical operator expression given by Eq (15).
The key observation here is that the operability of the pseudo-differential "square root" Hamiltonian can be clearly interpreted if it acts on the square root of the physical states.
The square root of a spinor certainly exist in the case of the Metaplectic group, [15], [17], [18], [19] making our interpretation Eq. ( 15) and Eq. ( 16) fully consistent from both the relativistic and group theoretical viewpoint.
Regarding Equation ( 16) we want to emphasize that our paper refers to the role of the generator of the metaplectic group both in the dynamics and in the physically admissible states of the model.We stress that the variables of the harmonic oscillator are internal from the point of view of the equations, and the origin of these variables is the faithful and fundamental representation of the symmetry of the generators of the dynamics of the spacetime through the physical states, such as the mappings of the generators in that particular representation, as explained in the paper.
The concept and underlying logic of Equations ( 15) and ( 16) are clear: Quantum symmetries contain -give rise to-the classical structure.The physical states, as well as the metric (spin 2) are emergent under the action of the symmetry operator via Equation (15).
(Moreover, concrete examples of this concept can be found in Ref. [21] by these authors).
Spin and supersymmetry do not need the Minkowskian structure.The clear example can be seen for the case of spin 2, in Sections VIII, IX and X of this paper.
In the next paragraph, we will describe these states (truly spinorial and relativistic ones) coming from the algebraic correspondence.

VI. PHYSICAL STATES FROM SYMMETRIES
Generators (dynamical symmetries) being into a oscillator-like vector representation (spinorial) are mapped through their mean values with respect to the basic states (the Mp(n) coherent states) giving rise to the observable physical states.That is to say, there is an interrelation between symmetries and physical states.This gives rise to the first important consequence that, taking into account the unobservable basic states, the bilinear states that are observable can only contain spins (0, 1/2, 1, 3/2, 2).
Next, we will provide a brief theoretical justification to the above construction and then, in the following Section, we describe the emergent space-time discretization mechanism.
It can be noticed that the family of representations can be increased, e.g. as those of the Hilbert space operators in the Weyl representation for a great variety of groups, and asymmetric representations of various forms.In our case here, the big group involved is the Metaplectic group M p (2) (the covering group of SL(2C)).This important group M p (2) is also closely related with the para-Bose coherent states and squeezed states (CS and SS).
Let us consider the concept of generalized coherent states (CS) based in a Lie group G acting on a Hilbert space H through a unitary, irreducible representation T in the following.The coherent state system {T, ψ 0 } is defined as the set of vectors ψ ∈ H such that ψ = T (g) ψ 0 for some g ∈ G, given a fixed vector ψ 0 .These vectors' equivalent states in H are known as generalized coherent states (states |ψ⟩).
The following coherent state reproducing Kernel for any operator A (not necessarily bounded) serves as the foundation for our analysis: where α and α ′ are complex variables that characterize a respective coherent state, and g is an element of M p ( 2 ).The possible basic CS states are classified as: with the following independent, non-equivalent, symmetric and anti-symmetric combinations The important fact in order to evaluate the kernels Eq. ( 17) is the action of a and a 2 over the states previously defined and similarly for the states Ψ .
The dynamical structure of (quantum) spacetime clearly encodes the metric through the coherent basic states, solutions of the Equations ( 15) and ( 16).Therefore, the spacetime structure defined in this paper through the metrics in Equations ( 20)-( 25) fully and rigorously respect all the properties required in the fundamental quantum regime, as well as in the classical domain.
Precisely, the generalized coherent states here generate a map that relates the metric, solution of the wave equation g ab to the specific subspace of the full Hilbert space where these coherent states live.Moreover, there exists for operators ∈ M p (2)an asymmetric -kernel leading for our case the following λ = 1 state : This is so because the non-diagonal projector involved in the reconstruction formula of L ab is formed with the Ψ 1/4 and Ψ 3/4 states which span completely the full Hilbert space.
Observation 1 : Due to the non observability of isolated basic states, the spin zero physical states appear as bounded states ( g g ), where g ab ( t, s, w ) and g ab ( t, s, w ) are given by the bilinear expressions Eqs (25).
Observation 2 : Each kernel represents a global physical state composed by fundamental states that separately are basic and unobservable.
Notice that the spectrum of the physical states are labeled not only by their spin content λ, but also by the "eigenspinors" corresponding to the vector representations of L ab and L ab respectively, (maps over a region of H).

VII. SUPERMETRIC AND EMERGENT SPACETIME
The Lagrangian density from the action Eq.( 14) represents a free particle in a superspace with coordinates z A ≡ x µ , θ α , θ • α .In these coordinates, the line element of the superspace reads, It is important to notice that following the steps detailed in Section IV, the quantization is exactly performed providing the correct physical and mathematical interpretation to the square root Hamiltonian, and the correct spectrum of physical states.
Without lose of generality, and for simplicity, we take the solution Eq. ( 20) to represent the metric and with three compactified dimensions ( s = 2 spin fixed), we have : where the initial values of the metric components are given by or, explicitly, The bosonic and spinorial parts of the exponent in the superfield solution Eq. ( 27) are, respectively, where β are constant spinors, ω = 1/| a | and the constant c 1 ∈ C, due to the obvious physical reasons and the chiral restoration limit of the superfield solution.We see in the next Section the associated emerging discrete space-time structure.

VIII. SUPERSPACE AND DISCRETE SPACETIME STRUCTURE
Let us see in this Section how the discrete spacetime structure emerges naturally from the model under consideration.Expanding on a basis of eigenstates of the number operator: we have Then, It follows According to the equation above, the splitting of ψ into the fundamental states of the metapletic representation is the only explanation that makes sense.
Consequently, at the macroscopic level, the arbitrary constants A and B govern the spectrum's classical behavior.Without losing generality, we assume for the purposes of this However, we will come back to this crucial point later.This is the outcome of the SO(2, 1) group's breakdown into two irreducible representations of the metaplectic group M p(2), spanning even and odd n, respectively.
Let us highlight the important property of the state | ψ(0 ) is invariant to the action of the operators a and a † .This is a consequence of the fact that in the metaplectic representation the general behaviour of these states are:

A. Statistical distributions and classical limit
From the Poissonian distribution for the coherent states we can see: It is different from the individual distributions defined from each one of the two irreducible representations of the metaplectic group Mp(2) (which span even and odd n respectively): Notice that in despite of the different form between the above equations, the limit n → ∞ is the same for both: the sum of the two distributions arising from the Mp(2) irreducible representations (IR), and for the SO(2, 1) representation as it must be.
Taking this into account, the explicit form of | α + ⟩, |α − ⟩ are given by : where all the possible odd n dependence is stored in the parameter ξ.
Consequently, |α + ⟩ connects only with even vectors of the basis number and |α − ⟩ with the odd vectors in the basis number.Therefore, using the decomposition Eq. ( 35) and decomposing the base number |n⟩ into even and odd, we obtain the following explicit result for the space-time metric : The expression above is an important pillar of our findings here: In this equation the discrete structure of the spacetime is shown explicitely as the fundamental basic feature of a consistent quantum field theory of gravity.
On the other hand, in the limiting case n → ∞ our solution for metric goes to the continuum one as, it must be: Similarly, for the lower part (spinor down) of the above equation, we obtain: Therefore, when the number of discrete levels increases, our metric solution goes to the general relativistic continuum "manifold" behaviour : as expected.
IX. THE LOWEST N = 0 LEVEL AND ITS LENGTH Is not difficult to see that for the number n = 0 the metric solution takes the value This evidently defines an associated characteristic length for the eigenvalues α, α * because the metric axioms in a Riemannian manifold.In principle, fundamental symmetries as the Lorentz symmetry can be preserved at this level of discretization due to the existence of discrete Poincare subgroups of this supermetric.

LUTION
The black hole entropy, S = k B A bh /4 l 2 P where A is the horizon area and l P ≡ ℏG/c 3 is the Planck length as is well known, was first found by Bekenstein and Hawking [25] using thermodynamic arguments of preservation of the first and second laws of thermodynamics.Also found by Bekenstein was an information theory proof in which black hole entropy is treated as the measure of the "inaccessible" information for an external observer on an actual internal configuration of the black hole in a given state.Such state is described by the values of mass, charge, and angular momentum.
• Therefore, (and because radiation is emitted at quantized frequencies corresponding to the differences between energy levels), quantum gravity implies a discretized emission spectrum for the black hole radiation.
• The spectral lines can be very dense in macroscopic regimes leading physically no contradiction with Hawking's prediction of a continuous thermal spectrum in the semiclassical regime.
• From the point of view of our approach here: • If we now suppose simply that the constants A, B in the state solution eg.Eq. ( 27), Eq. ( 35), are different, A ̸ = B we have : • Therefore, the thermal (Hawking) spectrum at the macroscopic or semiclassical level does not appear.
• This fact is clearly explained because un exact balance between the superposition of the two irreducible representations of the Metaplectic group is needed.This will lead as a result, non classical states of radiation in the sense of [28] as can be easily seen putting, for example, one of the constants, B (or A) equal to zero: • Notice that only the up spinor part survives in this case, and the classical thermal limit is not attained.This is so, even in the continuous limit for this case, in which the number of levels increases accordingly to • Consequently, in such a case where A = 0, (or B = 0), the spectrum will takes only even (or odd) levels becoming evidently non thermal. Therefore, • In the case A = B the thermal Hawking spectrum is attained at the continuum classical gravity level eg, the Poissonian behaviour of the distribution is complete.
• In the cases, with A ̸ = B, the spectrum is non classical, and quantum properties of gravity are manifest at the macroscopic level.

XII. CONCLUDING REMARKS
Here we have demonstrated that there is a principle of minimal group representation that allows us to consistently and simultaneously obtain a natural description of the dynamics of spacetime and the physical states admissible in it.
The theoretical construction is based on the fact that the physical states are, roughly speaking, average values of the generators of the metaplectic group Mp(n) in a vector representation, with respect to the coherent states that are not observable (carrying the weight of spin).Schematically, we have the following picture where M ab = L ab (L ab ): In summary: (1) We demonstrate that there is a connection between dynamics, given by the generators of the symmetries, and the physically admissible states.
(2) The physically admissible states are mappings of the generators of the relevant symmetry groups covered by the metaplectic group, in the simplest case according to the chain: M p(2) ⊃ SL(2R) ⊃ SO(1, 2)) through a bilinear combination of basic states.
(3) The ground states are coherent states defined by the action of metaplectic group (Perelomov-Klauder type), these states divide the Hilbert space into even and odd states, and are mutually orthogonal.They carry a weight of spin 1/4 and 3/4 respectively.
(4) From the basic states combined symmetrically and antisymmetrically, two other basic states can be formed.These new states manifest a change of sign under the action of the creation operator a + .
(6) A symmetry of the superspace is formed by a realization of the generators with bosonic variables of the harmonic oscillator as Lagrangian.Taking a line element corresponding to such superspace a physical state of spin 2 can be obtained and related to the metric tensor.
(7) The metric tensor is discretized simply by taking the discrete series given by the basic states (coherent states) in the number n representation, consequently the metric tends to the classical (continuous) value when n → ∞.
(8) The results of this paper have implications for the lowest level of the discrete spectrum of space-time, the ground state associated to n = 0 and its characteristic length, in the black hole history of black hole evaporation.
(9) Moreover, recently we have successfully applied this general approach in physical scenarios of current interest obtaining coherent states of quantum space-times for black holes and de Sitter space-time, in our Ref [21].

4 .
Basic states of representation and the spectrum of physical states The basic states | Ψ s ⟩ belong to the group Mp(n) and have a spin weight s = 1/4, 3/4 in the simplest case Mp(2): They contain even and odd sectors (s = 1/4, 3/4) in the number of levels of the Hilbert space respectively and therefore, they span non-dense irreducible spaces.In this way, states that are bilinear in fundamental functions (corresponding to Ψ s=1/4, 3/4 , form the full physical spectrum.In the case of the Metaplectic group M p (2), these fundamental functions are f 1/4 and f 3/4 , having a spin weight s = 1/4 and 3/4 respectively.A physical state characteristic of M p (2) is given by Φ µ = ⟨ s | L µ | s ′ ⟩ with (s, s ′ = 1/4, 3/4 ), and L µ being the vector representation of one of the generators of M p (2).