Geometric Event-Based Quantum Mechanics

We propose a special relativistic framework for quantum mechanics. It is based on introducing a Hilbert space for events. Events are taken as primitive notions (as customary in relativity), whereas quantum systems (e.g. fields and particles) are emergent in the form of joint probability amplitudes for position and time of events. Textbook relativistic quantum mechanics and quantum field theory can be recovered by dividing the event Hilbert spaces into space and time (a foliation) and then conditioning the event states onto the time part. Our theory satisfies the full Lorentz symmetry as a ‘geometric’ unitary transformation, and possesses relativistic observables for space (location of an event) and time (position in time of an event).

Quantum Mechanics (QM) relies on time-conditioned quantities: observables conditioned on time in the Heisenberg picture (e.g.X(t) is the position operator at time t) or states conditioned on time in the Schrödinger one (e.g.|ψ(t) is the state at time t).As such, it is inherently incompatible with the Poincaré symmetry of special relativity.Indeed quantum mechanics can be formulated in a relativistic covariant fashion only in very specific circumstances, such as Quantum Field Theory (QFT) when using Heisenberg picture operators acting on a relativistic invariant state such as the field vacuum.In this paper we depart from this approach and introduce a quantum mechanical theory of events, the Geometric Event-Based Quantum Mechanics (GEB), which is based on unconditioned spacetime quantities and hence intrinsically covariant.In QFT one starts from the dynamical equation of motion (either in the Hamiltonian formulation or from a Lagrangian [1]) and quantizes the dynamical solutions imposing equal time commutation relations.We take the opposite track: we start by defining an (unconditioned) purely kinematic Hilbert space H E which is well suited to account for the symmetries of a relativistic theory [2][3][4].A formal correspondence with QM and QFT is then established by showing that the quantum evolutions defined by these theories can be identified as special subset H QM of the distributions of H E which is determined not via dynamical equations, but through purely geometrical constraints [5,6].
The Hilbert space H E of GEB, rather than systems, can be thought to describe events [7].In our approach the event is taken as a primitive notion, i.e. not something that is derived from a pre-existing notion of (say) a preexisting particle that has been detected (as happens in QM).Indeed, in GEB the detection of particle at a particular location in space and time by an inertial measuring device [8] is a way to identify the event itself, and the particle (or, more generically, any quantum system) is a derived notion.A quantum system is then interpreted as a probability amplitude for an event out of a sequence of events, which takes the place of the "sequence of events" [81], which is the customary definition of "physical sys-tem" in relativity.Schrödinger had a similar observation: "it is better to regard a particle not as a permanent entity but as an instantaneous event" [9].Jorge Luis Borges is, obviously, more captivating: "the world is not a concurrence of objects in space, but a heterogeneous series of independent acts" [10].Examples of events (discussed below) are 'a fermion with spin σ is detected at a time t and position x in spacetime', or 'a boson is detected with energy E and momentum p ' (more general events will be discussed elsewhere).While we will be using some of the formalism developed in [2,3], our interpretation of the formulas and the conceptual framework of GEB is fundamentally different.Relativistic versions of constrained quantum mechanics [11][12][13][14][15][16][17][18][19][20] are also explored in [2][3][4][21][22][23][24][25][26][27][28][29][30][31][32][33].A different claim of 'covariant quantum mechanics' is in [34], where the Hilbert space is expressed through position eigenstates of a time-evolved (Heisenberg picture) position operator.While there is some similarity in the notation used, our approach is completely different: we do not use any dynamical assertion (hence, no pictures) in our treatment of time.
The outline follows.We start in Sec.I by introducing the Hilbert space H E for a single-event model, defining its relativistic observables, giving interpretations of the system 4D wave-function as unconditioned properties of events and the representation of Lorentz transformations.In Sec.II we consider multiple events in a first quantization approach which is then embedded it into a second quantization Fock formalism.In Sec.III we provide a formal connection between GEB and QM by introducing a correspondence rule that enables one to map quantum trajectories of the latter into distributions of the former.Conclusions and future perspectives are given in Sec.IV.Technical details are presented in appendices. is introduced.We start in Sec.I A by considering the simplest non trivial scenario, i.e. a universe characterized by a single (spinless) event, which is the building block for the general case presented in Sec.II.In Sec.I B we discuss the covariance properties of the theory under Poincaré transformations, while in Sec.I C we show how to generalize the analysis to include spinor degrees of freedom.
A. A universe with a single event In the formulation of GEB we are guided by the fundamental observation that spacetime is physically meaningful only insofar as it is mapped by clocks and rods (clicks and ticks) which are events [35,36].From this we can infer that there is no localization in time of an event without any energy and energy spread (nothing can happen) and there is no localization in space of an event without momentum and momentum spread (any event would be delocalized over the whole space): so, in addition to being characterized by spacetime coordinates, any given event must also be connected to energy and momentum degrees of freedom.Accordingly the structure of the Hilbert space H E associated with a single event can be identified by declaring that among the linear operators that acts on such space, there must exist (at least) a 4 component vectorial observable X := (X 0 , X 1 , X 2 , X 3 ) that determines the 4-position x := (x 0 = t, x) in spacetime of the event, and an associated 4-momentum operator P := (P 0 , P 1 , P 2 , P 3 ) that instead defines the corresponding energy-momentum values p := (p 0 = E, p) of the event [82].(We use the overbar x to denote contravariant 4-vectors, the underbar x for covariant ones, and the arrow x for spatial 3vectors -see App.A.) The existence of X and P is a minimal assumption of the theory: other observables can in fact be introduced that describe extra (not kinematic) degrees of freedom of the event, something that for instance will be revealed by the internal degree of freedom of an event-defined particle (say its spin) (see Sec.I C).
We now impose the canonical commutation rules [X µ , P ν ] = −iη µν and [X µ , X ν ] = [P µ , P ν ] = 0 , (1) with µ, ν ∈ {0, 1, 2, 3} and η the metric diag(1, −1, −1, −1) (note the minus sign in the 00 commutator).The rationale of this choice is that, on one hand, it allows us to satisfy Poincaré algebra [5]: where M µν := X µ P ν − X ν P µ = (X ∧ P ) µν [37,38] is the relativistic angular momentum tensor (the spatial part M ij with i, j ∈ {1, 2, 3}, containing the angular momentum tensor, the generator of rotations, and the temporal part M 0j with j ∈ {1, 2, 3} containing the generator of boosts) [83].On the other hand, the condition (1) automatically imposes H E to be infinite dimensional and leads to the following spectral decompositions (where a = (a 0 , a), a = ηa = (a 0 , − a) so that a  [39] is a collection of three objects: the Hilbert space itself H E , the set of distributions on the space H + E , such as Dirac deltas identifying eigenvectors of continuous-variable eigenvalue observables, and the nuclear elements of the Hilbert space, i.e. a dense subset H − E on which one can apply the distributions.]As such, similarly to | x and | p in QM, we shall not assign to them a precise physical interpretation.The best we can say is that |x (|p ) represent an (unphysical) absolute localization of the event in spacetime (resp.energy-momentum space) which can be used to characterize the statistical distributions of the elements of H E via decompositions of the form with the amplitudes being square integrable functions (4D wave-functions) [85].Quantum probabilities follow then directly from the Born rule.In particular the probability that the measurement of the observable X µ has result x on a event state associated with the normalized vector |Φ ∈ H E is We interpret this as the probability distribution that the event happens at spacetime position x: this clarifies that in our approach the vectors |Φ are spacetime states, representing (in the position representation) the probability amplitudes of the spacetime locations of the event.It is worth stressing that in QM the Born rule written in the Schrödinger S or Heisenberg H pictures) gives the conditional probability that a particle in state ψ is found at position x given that the time is t.Instead, in GEB, the probability ( 8) is unconditioned: it is a joint probability that the event happens at position x AND that the time is t for x = (t, x).Analogously, in GEB we can also interpret as the probability that the event will carry an energy-momentum p, and as the probability to confuse the event |Φ 2 as the event A direct consequence of Eq. ( 1) are the Heisenberg-Robertson [40] uncertainty relations which for µ = 0, 1, 2, 3 have to be fulfilled by the statistical distributions P (x|Φ) and P (p|Φ) for all choices of the normalized vector |Φ ∈ H E .These equations effectively translate in strict mathematical form our initial observation that no event can be spacetime localized without energy and momentum spread.While for µ = 1, 2, 3, Eq. ( 10) is the usual Heisenberg uncertainty relation [41], for µ = 0 it takes a special role [42,43]: as a matter of fact it cannot [44] be derived in QM, where time is a parameter and not an observable.(Indeed, in the Mandelstamm-Tamm uncertainty relation [45] ∆t takes the role of a time interval (the minimum time interval it takes for a system to evolve to an orthogonal configuration), not of a statistical uncertainty due to quantum stochasticity.)Here, instead, time X 0 is a quantum observable, and we can assign to ∆X 0 ∆P 0 /2 the Heisenberg-Robertson interpretation.(The acute objections [44,46] against this type of interpretation of the time-energy uncertainty were formulated in the framework of QM.In GEB, these objections do not apply.)This interpretation is a purely kinematical statement, not a dynamical one: P 0 is not some system's Hamiltonian, but it is the energy devoted to the event.The connection between the energy and a Hamiltonian (dynamics) is here only enforced as a constraint on the physical states, see below.

B. Lorentz tranforms
Relativistic QM, e.g.QFT, can be written in a covariant fashion only in very specific situations.Indeed, as discussed above, the Born rule probability when calculated in the Schrödinger, Heisenberg or interaction picture, is a conditional probability, conditioned on time.As such, in contrast to what it is claimed sometimes (e.g.[47]), QM probabilities are not covariant: they refer to a specific spacetime foliation, and a Lorentz transform cannot be simply applied to something defined on a foliation slice.In QFT one can recover covariance if one writes the observables in the interaction picture in terms of creation and annihilation operators that have a spacetime dependence of the form e ±ix•p .But observables by themselves are not sufficient to obtain quantum probabilities or expectation values: they must be applied to states.QM states explicitly depend on a foliation (at time t in the Schrödinger picture, at time t = 0 in the Heisenberg picture, and at a foliation determined by the state-evolution operator in the interaction picture).For this reason, one cannot in general apply a Lorentz transform to a quantum state.There is an important exception, customarily employed in QFT, where one uses states that are eigenstates of the Hamiltonian, e.g. the vacuum state.These are invariant for the dynamics, and hence can be easily Lorentz-transformed as they are left invariant.
In contrast, the GEB formalism we introduce here is fully covariant as the canonical observables of the theory are trivially transformed using a unitary representation of the Poincaré group.To clarify this fact let us consider two inertial observers O and O sitting on different reference frames R and R connected via the transformation Λ of the Lorentz group so that, given x = (t, x) and x = (t , x ) the coordinates that they assign to the same event one has where U Λ is the unitary mapping that represents Λ in H E , i.e. the transformation associated with the generators M µν of Eq. (2).Equation (13) ensures that both observers will assign the same scalar products to any two couples of states, i.e.
which implies that the probabilities (9) are invariant under change of reference frames.This is in line with the fact that the Born rule probabilities in GEB refer to the occurrence of events in spacetime (rather than the conditional probabilities on a foliation at a specific time t): they are unconditioned and hence invariant.As a direct consequence of (12), the spacetime probability distributions (8) Similarly for the energy-momentum distributions we get Φ (p) = Φ(Λp) and hence P (p|Φ ) = P (Λ −1 p|Φ), (these last follow directly from the identity (B7) of App.B).
A direct consequence of these relations is that any statement O and O make on the expectation values of these observables on the states of H E (or on their higher momenta like those appearing in (10)) are automatically covariant under Lorentz (and more generally Poincaré) transformations.This can be made more explicit (App.B) using the 4D "Heisenberg picture" where the mapping (13) results in the following connection between the canonical observables of O and O

II. MULTIPLE EVENTS
The Hilbert space H E spanned by the vectors (19) describes a single spacetime event.The extension to the case of multiple events is obtained by considering tensor products of such space, possibly equipped with proper symmetrisation rules that account for the statistical properties of the particles that are used to reveal them (see below).An important advantage of this construction is that the transformations under the Poincaré group follow directly from those established for those of the singleevent model (13).Indeed given |Φ [n] , |Φ [n] ∈ H ⊗n E the vectors used by the observers O and O to describe the same state of n events, they will be connected via the mapping with U Λ being the unitary operator that represents the matrix Λ of Eq. ( 11) for a single event.Of course with this choice, each individual event is connected to its own time of occurrence, whereas interpretations of the covariant formalism in terms of particles are problematic [2,48] because the "time of a particle" is a meaningless concept [44].In the following paragraphs we discuss the different types of multi-event models that arise from the above formalization and show how this analysis can be lifted to a higher level of complexity by constructing an effective quantum field theory version of GEB, through Fock space (second quantization).

A. Distinguishable vs Indistinguishable events
The simplest example of a multi-event model is represented by an universe of n distinguishable spacetime events, i.e. events that define n distinguishable particles.In this case any normalized vector |Φ [n] of H ⊗n E qualifies for a proper GEB state of the model, with the decomposition (19) being replaced by where |x 1 , σ 1 ; • • • ; x n , σ n stands for j |x j , σ j with |x j , σ j j the 4-position and spin eingenstate of the j-th event so that Φ [n] (x 1 , σ 1 ; • • • ; x n , σ n ) is the wave-function which yields the joint probability of revealing the jth event in spacetime position x j with spin σ j , for all j.-similar definitions apply also to Consider next the scenario of n indistinguishable spacetime events, i.e. events that are used to define n identical particles.We describe the states of such models via vectors (22) which induce the Bosonic or Fermionic character of the derived particles through the property of being either completely symmetric or completely antisymmetric under exchange of the ket indexes.Specifically a Bosonic n-event GEB model is described by the completely symmetric linear subset H (n,S) E ⊂ H ⊗n E spanned by the vectors (22) with amplitudes probabilities Φ [n] (x 1 , σ 1 ; • • • ; x n , σ n ) that are invariant under an arbitrary permutation p of the n systems labels, i.e.
(a condition that automatically carries over to the 4D momentum wave-function).A Fermionic n-event GEB model, instead, will be described by the completely antisymmetric linear subset H (n,A) E ⊂ H ⊗n E formed by vectors with amplitudes that fulfill the identity with sign[p] being the sign of the permutation p.

B. Fock space representation
The above construction can only deal with a fixed, predetermined number n of events.To analyze situations where n is, itself, a quantum degree of freedom, we need to escalate to a Fock space.The starting point of this construction is to introduce a "4D-vacuum" state vector |0 4 which represents the state of a universe where there are no events at any spacetime location, and by defining raising and lowering operators [2,4] a † x,σ , a x,σ that act as creators/annihilators of spacetime events in the theory.Properly symmetrized versions of the generalized 4-position eigenstates will now be expressed as reiterated applications of the a † x,σ 's on |0 4 , i.e.
while the generalized 4-momentum eigenstates with a † p,σ and a p,σ connected with a † x,σ , i.e. the operators The consistency of the representation is enforced by assigning proper commutation relations to the raising and lowering operators (App.C).Specifically, the Bosonic/Fermionic character follows by requiring which automatically translate into analogous relations for the a † p,σ , a p,σ , i.e. Bose: Accordingly in the Fock state representation Eqs. ( 22) and ( 23) can be expressed as with retaining the same probabilistic meaning given in Eq. ( 24).
It is important to stress that the 4D-vacuum |0 4 is a distinct state from the 3D-vacuum |0 3 used in QFT.Indeed |0 3 represents a configuration in which there are no particles at a specific time (time t in the Schrödinger picture or t = 0 in the Heisenberg one), whereas |0 4 has no events at any time.The QFT vacuum |0 3 is the ground state of a field.As such, it is the spatial part (in some foliation) of the GEB state relative to a zero 4-momentum event: |0 3 =foliate(a † p=0 |0 4 ).It is not the spatial part of |0 4 .The state |0 4 represents no events, whereas a † p=0 |0 4 represents a zero 4-momentum event which corresponds to a uniform distribution of zero-energy events in all spacetime a † p=0 |0 4 = d 4 x a † x |0 4 : in other words, there is a difference between saying "nothing happens everywhere and everywhen" (i.e. the QFT vacuum |0 3 at all times), and saying "nothing happens anywhere and at any time", the Aristotelian void |0 4 .
Also, the GEB raising and lowering operators are completely different from the QFT ones, and it is not just a matter of adding the temporal degree of freedom: a p = a p 0 ⊗ a p .The QFT ones are obtained from the canonical quantization of the harmonic oscillator, namely starting from the dynamics.Here, instead, we are introducing the raising and lowering operator from the kinematics, namely the ones that applied to the 4-vacuum create the position (or momentum) eigenstates.The QFT operators lose their meaning when one changes the dynamics (e.g. by adding interactions), whereas ours do not.
Thanks to Fock space, we can now have states with superpositions of different numbers of events, i.e. with ) is the joint probability density of having n detection events and that they happen at spacetime locations The Lorentz transformations are a straightforward extension of (21).Indeed, indicating with U Λ the unitary mapping that represents Λ in Fock space, given |Φ and |Φ the states two observers O and O assign to same state event, we can write by requiring that the vacuum state is left invariant by , and by imposing or, equivalently, III. QM/GEB CORRESPONDENCE QM is a physical theory of systems while GEB is a physical theory of events: in this section we shall see that these two different approaches can be connected by associating the dynamical quantum trajectories of QM to elements of the distributions set H + E of GEB.For the sake of simplicity we start in Sec.III A by considering the special case of a single event universe showing that it can be put in correspondence with the QM description of a point-like single particle system.The generalization to multi-event scenario will instead be addressed in Sec.III B and in Sec.III C where we shall make use of the Fock space representation introduced in Sec.II B.

A. Single-particle/single-event correspondence
In QM the temporal evolution of a single particle described by an observer O sitting in his reference frame R, is obtained by assigning a 3D+1 spinor wave-function Ψ QM ( x, σ|t) which extends both in time and in space.The unitary character of the dynamics ensures that 3D norm of this function is a constant of motion.Accordingly setting the function Ψ QM ( x, σ|t) can be interpreted as the conditional probability amplitude that the observer O will find the particle at position x with spin σ, given that time is t.A natural correspondence between the single particle states of QM and the single event states of the GEB formalism can hence be established by interpreting Ψ QM ( x, σ|t) as a 4D spinor wave-function and then using the following mapping The exclamation mark is a reminder that, with the normalization (38), the vector |Ψ QM has an infinite norm, in contrast to |Φ of (6).Indeed the square integrability of the GEB wave-functions Φ(x, σ) is incompatible with (38) obeyed by the QM wave-function Ψ QM (x, σ): in general an element |Φ of H E will exhibit modulations with respect to t that in QM would be interpreted as unphysical losses and gains of probability during the temporal evolution of the particle but which are perfectly allowed at the kinematic level in the GEB formalism (and they can then be removed at the dynamical level, see below).Because of their infinite norm, the vectors |Ψ QM introduced above are not elements of H E and cannot be interpreted as proper event states of GEB.The mapping (40) associates the QM wave-functions Ψ QM ( x, σ|t) of O to distributions of GEB.This fact is explicitly shown in App.D: here we point out that Eq. ( 40) identifies only a proper subset Examples of elements of H + E which are not in H QM are provided for instance by the generalized position and momentum eigenvectors |x and |p which clearly cannot be expressed as in (40) with 3D normalized QM solutions Ψ QM,σ ( x|t).We also notice H QM can be identified via geometric constraints analogous to those adopted in [2,3,11,21,25,38,[49][50][51][52].Specifically one has where K is a constraint operator that encodes the QM dynamics (as discussed in App.E we can also add extra constrains that force |Ψ QM to represent 3D+1 spinor wave-functions that fulfill assigned initial conditions for a given observer O).We stress that in contrast to previous literature [2,3,29,38] where the solutions of constraint equations are interpreted as history states for systems, in GEB they are used to identify distributions which define event states.Another difference with previous approaches is that for the purpose to generalizing the analysis to the multi-event scenarios, in our construction we find it convenient to work with constraint operators which are explicitly self-adjoint and semidefinite-positive, i.e.K ≥ 0. Of course this choice can be enforced without loss of generality since given J a generic operator fulfilling (41) we can always identify a positive semi-definite one that does exactly the same e.g. by taking K = J † J exploiting the fact that For non-relativistic models, the QM dynamics takes always the form of a Schrödinger equation which can be cast in the form (41) along the lines detailed e.g. in in Ref. [14].Unfortunately, there does not appear to be a similarly general method to describe the relativistic dynamics [86], but one should use covariant constraints to avoid ruining the theory's covariance.Indicating with := ∂ 2 t − ∇ 2 the D'Alambert operator, in the case of spinless particle of mass m this can be done for instance by invoking the Klein-Gordon (KG) equation filtering out its positive energy solutions (see App. F).
For the case of a massive spin 1/2 particle instead one can use the Dirac equation with ∂ := (∂/∂t, − ∇) and γ := (γ 0 , γ 1 , γ 2 , γ 3 ) the Dirac matrices (see Eq. (A11)).Adopting the position representation P µ → i∂ µ , both Eqs. ( 43) and ( 44) can be turned into a constraint of the form (41) for the associated distributions |Ψ QM [87].Specifically, in the case of the positive-energy KG equation ( 43) one can identify the constraint operator K of ( 41) with the self-adjoint operator where Θ(x) is the Heaviside step function (see App. F) or with its positive definite counterpart Similarly, for the Dirac equation: we can directly translate ( 44) into (41) by identifying K with the operator (which is not self-adjoint), or with its associated positive semi-definite counterpart with |λ σ (p)| being the singular eigenvalues of J D and the generalized vectors |φ σ (p) forming an orthonormal set (see App. G).One of the advantages of adopting the above definitions for the constraint operator K is that all and O assign to an n-particle state in their own reference frames: thanks to the QM/GEB correspondence (40) such connection can be expressed via the unitary mapping (50) (green arrow) that links the associated GEB distributions (54).The blue arrow elements represent the association of 3D+1 spinor wave-functions of QM with their 4D spinor GEB counterparts, given in (39) for n = 1 and (52) for n > 1; the red arrow elements instead represent the connection between 4D spinor wave-functions and the distributions of HE.
of them are Lorentz invariant quantities (this is clearly evident for (47), while for (45) an explicitly proof is given in App.F).Accordingly, the elements of H QM identified by one observer O via Eq.( 41) will be related with those assigned by the observer O via the same unitary transformation (13) that links their state event descriptions, i.e.
or equivalently (the spinless case being obtained by simply removing S and neglecting the σ terms), which via (39) properly describes how to relate the QM 3D+1 spinor wave-functions Ψ QM ( x, σ|t) and Ψ QM ( x, σ|t) the observers assign to the same single-particle trajectory (see Fig. 1).

B. Multi-event QM/GEB correspondence
To generalize the correspondence (40) to the multievent case we need to address the problem that in QM the wave-function Ψ [n] QM ( x 1 , σ 1 ; • • • ; x n , σ n |t) of a n particle system is associated with n independent 3D spatial coordinates (plus possibly n spinor components) but with a single time-coordinate.It is hence not at all clear how to map such terms into elements (or distributions) of H ⊗n E which instead possess n independent time coordinate values.In the case where the n particles are not interacting, we can use the fact that Ψ [n] QM ( x 1 , σ 1 ; • • • ; x n , σ n |t) can always be expressed as linear combinations of products of time-dependent single-particles, i.e.
where given = ( 1 , • • • , n ), α are time-independent probability amplitudes, and where for j = 1, QM ( x j , σ j |t) is the 3D+1 wave-function describing the evolution of the j-th particle of the system.Equation (52) is the key to generalize (40) as it allows us to formally associate Ψ spinor wave-function with n distinct time coordinates via the construction and then using such term to identify the distribution |Ψ QM via the identity × Ψ [n] (where, again, "!" is a reminder of the nonnormalization).While the choice of the single-particles QM spinor wave-functions Ψ QM ( x j , σ j |t) and of the coefficients α entering in (52) are in general not unique, the vector (54) does not depend on such freedom ensuring that the connection between Ψ QM is one-to-one (see App. H 1). Viceversa, given |Ψ [n] QM one can recover the QM spinor 3D wave-function Ψ As in the single-event case we can identify the distributions (54) by means of a geometric constraint (41) induced by an n-body operator K [n] .To identify such a term we start form individual single particle constraint terms K j that are explicitly positive semidefinite (i.e.K j ≥ 0), and take K [n] as their sum The positivity requirement on the individual K j is an important ingredient as it ensures that the kernel of K [n]  coincides with the intersection of all the kernels of the single-particle constraints, i.e.
which automatically implies that the only acceptable solutions to (57) must have each individual particle evolving according to its own dynamical constraint (the model being interaction free for now).Observe also that as Eq. ( 56) is symmetric under exchange of the particle indexes it has no problem to act as constraint operator also in the case the particles are indistinguishable (in particular it does not mix the complete symmetric part H (n,S) E of H ⊗n E with the complete anti-symmetric part H (n,A) E ).For instance, in the case of a Bosonic model governed by the positive energy KG equation ( 43), the positivity requirement on the K j forces us to select (46) (instead of ( 45)) as the proper single particles terms: accordingly for this model the n-body constraint operator K [n] can be identified with Similarly for Fermionic models we should identify the single-particle terms K j with the operator (48) instead of (47).Accordingly in this case n-body constraint operator K [n] becomes If the QM dynamical equations that rule the equation of motion of the particles are relativistically covariant as in the cases of Eqs. ( 58) and ( 59), then the identities ( 50) and ( 51) that in the single particle case allows us to connect the distributions of the observers O and O , translate into and respectively.Notice also that setting t 1 = • • • = t n = t in the last one, invoking Eqs. ( 52) and ( 53) we obtain the connection between the spinor 3D wave-functions of QM , that O and O would assign to the same quantum trajectory of the n particles -see Fig. 1 for a schematic representation of this identity and App.H 2 for a technical discussion.

C. QM/GEB correspondence in Fock space
In this section we generalize the correspondence (40) to the Fock space representation of GEB discussed in Sec.II B. At variance with what we did in Secs.III A and III B, here we start by first introducing the constraint operator (41), and then show that the associated solutions can be directly connected to those of QFT.

Constraint operators
To connect GEB to QFT, start by considering the case of a Bosonic model where each individual particle evolves according to the positive energy KG equation ( 43).As we have seen in the previous section, in the first quantization formalism the constraint operator of the model is provided by (58).When the total number of particles is fixed to n, the first quantization version of the constraint operator (58) assigns a contribution Θ(p 0 ) p • p − m 2 2 to each particle with 4-momentum p, specifically with |S(p 1 ; • • • ; p n ) the symmetric version of |p 1 ; • • • ; p n (see App. C).Exploiting the correspondence (C5), Eq. ( 62) can now be turned into its second quantization form by identifying K [n] KG + with the Fock operator with a † p , a p the creation and annihilation operators that obey the canonical commutation rules (30).
In the Fermionic case we proceed in similar fashion.In this case, from (59), we see we need to introduce a Fock number operator that counts how many particles of the system are in single particle states described by the vectors |φ σ (p) .To construct such a term we introduce a new collection of annihilation operators with u σ,σ (p) the unitary matrices that connects the vectors |φ σ (p) with the vectors |p, σ (see App. G).By construction they fulfill the same anti-commutation rules of a † p,σ and a p,σ , i.e.

Connection with the QFT solutions
Here we analyze the solutions of the geometric constraint (41) that follow from the definitions of K (Fock) KG + and K (Fock) D given in the previous section, i.e.
for the Bosonic model, and for the Dirac one.It is clear that in both scenarios the no-event state, that in the theory is represented by 4D vacuum state |Ψ QM = |0 4 is an allowed solution.It corresponds to the trivial case of no particles (Bosons for (69) or Fermions for (70)) at all times.To discuss the other solutions in what follow we shall address first the Bosonic case that allows for some simplification due to the absence of spinor components.
a. Bosonic model:-Express the vector |Ψ QM that appears on the r.h.s. of Eq. ( 69) as the one given in (34) (with no spin), namely The functional dependence of the operator K (Fock) KG + upon the number operator a † p a p suggests to analyze Eq. ( 69) in the 4-momentum representation (71) instead of the position representation (72).Indeed when acting on p a p generates a multiplicative factor n j=1 δ(p j − p) that allows us to translate the constraint (69) into a constraint on the momentum-representation (33) of the wavefunction as

Such equation forces Ψ[n]
QM (p 1 ; • • • ; p n ) to have support only for values of the p j momenta that satisfy the onshell condition p j • p j = m 2 with p 0 j ≥ 0. Specifically using we can express the most general solution of (73) as and f [n] (p 1 ; • • • ; p n ) an arbitrary function which nullifies for p 0 j < 0 and that, in virtue of the implicit symmetry of ( 72), can always be forced to be completely symmetric under exchange of the indexes.The position representation (71) of the solution can now be recovered replacing Eq. ( 76) into Eq.( 72), i.e.
To put these solutions in correspondence with the QFT solutions of the corresponding Bosonic KG field equation we observe that in the Schrödinger picture, the general QFT solutions of a (positive-energy) Bosonic KG field equation writes as [53] |ψ where β n are normalized amplitude probabilities, |0 3 is the 3D vacuum state of the field (not to be confused with the 4-vacuum state |0 4 of GEB) and the c † x 's are Bosonic creation operators fulfilling the equal-time canonical commutation rules In the above expression Ψ [n] QM ( x 1 , • • • , x n |t) are (observer dependent) 3D+1 wave-functions that (under proper normalization conditions) define the joint probabilities of finding at time t, n particles in x 1 , • • • , x n : their temporal dependence is fixed by the single-particle dispersion relation defined in Eq. ( 75) and is computed in Eq. (H6).Our goal is to show that the GEB solutions (72) with Ψ[n] QM (p 1 ; • • • ; p n ) as in Eq. ( 76) can be put in correspondence with (79) by taking β n = α n and setting in Eq. (H6).To verify this fact notice that for fixed value of n ≥ 1 one can invoke Eqs.(C3) -(C5) to map the 4D wave-function Ψ 78) onto a QM 3D+1 wave-function of n Bosonic particles via Eq.( 55): this exactly reproduces the QFT solution Ψ in terms of the creation operators a † φσ(p) , i.e.
with the spinor wave-functions Ψ [n] QM (p 1 , σ 1 ; • • • ; p n , σ n ) that are connected with those of the 4-position representation via the identity Replacing ( 83) into ( 70) we get which due to the positivity of the terms λ 2 σj (p j ) has solutions of the form where E | p| 2 + m 2 , see Eq. (G7), and where f [n] (p 1 , σ 1 ; • • • ; p n , σ n ) are arbitrary functions that can always be assumed to completely anti-symmetric under particle indexes exchange, and where finally .
(86) Substituting this into (84) we hence obtain To establish a formal correspondence between (82) and the solutions of QFT we observe that the 3D+1 spinor wave-function of n Fermionic particles that obey the Dirac equation is given by vectors of the form with creation operators that obey equal-time anticommutation rules, i.e.
Invoking once more Eqs.(C3) -(C5) we can hence conclude that Eq. (84) corresponds to (88) by setting Note that here we also consider possible entanglement between different spinor components, whence the n integrals in (88), which are not usually included in QFT treatments.Superpositions of a particle and an antiparticle are typically considered unphysical because one supposes that superselection rules will prevent them.However, such states have been proposed [54,55], suggesting that superselection rules are never fundamental, but only practical limitations.

Lorentz transform
We conclude the section stressing that also in the Fock formalization of the model, the constrained operators are Lorentz invariant quantities, allowing us to extend the identity ( 35) also to the elements |Ψ QM of H QM , i.e.
indicating that in GEB Lorentz transforms can be done entirely using unitary representations of the Lorentz group as described above, according to Wigner's prescription for symmetry transformations, entirely at the kinematic level.This is clearly different to what happens in QFT where we quantize "on shell", namely, the quantization procedure contains the dynamics.This implies for instance that the state c † p |0 3 lives in a L 2 (R 3 ) space of on-shell states, namely states whose energy is E p .In order to Lorentz transform such state, one must first derive the new hyperboloid that satisfies E 2 p − p 2 = m 2 in the new frame and then quantize in the new frame obtaining c † p |0 3 in the new frame (the vacuum being Lorentz invariant).

IV. CONCLUSIONS
In conclusion we presented an alternative framework (GEB) for special relativistic quantum mechanics.The full axiomatic structure of quantum mechanics (e.g. its statistical interpretation through the Born rule) is applied covariantly.The quantization is performed axiomatically in GEB, constructing a Hilbert space for events, rather than the customary QFT approach of quantizing the solutions of the dynamical equations.The usual textbook relativistic QM and QFT are obtained by conditioning over the temporal degrees of freedom of the GEB event states.
We have not considered interactions here: as in relativistic QM and QFT, interactions pose significant additional challenges (understatement!) that will be tackled in future work.Other covariant approaches that derive from Dirac forms [5] typically work only for free particles (since the Hamiltonian ends up in the boost generators): the "no-interaction theorem" [56][57][58].Our approach might, instead, be able to consider interactions, since we impose the dynamics only through a constraint, which is a procedure known to bypass the no-interaction theorem [23,24,38,50,59].Moreover, GEB does not employ a quantization on the free-field dynamical equation solutions, so it might perhaps be able to describe interacting fields without the usual perturbative approach, if we will ever be able to devise appropriate, solvable, constraint equations.GEB replies affirmatively to a question raised by Kuchaȓ [60] on whether the constraint formalism is able to describe localized relativistic particles (a completely different solution, based on the Newton-Wigner mechanism, is in [61]).Finally, it can treat situations that do not admit a Hamiltonian formulation [62] (such as solutions to the KG equation without positiveenergy restriction or generic solutions of Einstein's field equations [6,49,63,64]), since, as shown above, the constraint procedure does not require Hamiltonians to describe the dynamics.
Of course, we do not claim that QFT is inadequate: the formulation provided here is, as shown above, a (slight) extension of it and in all situations considered in this paper an QFT description exists (mutatis mutandis).It may perhaps be used to clarify some longstanding problems, such as Haag's theorem [65] or particle localization [65,66] by recognizing that a localized particle (that stays localized for a period of time) is not a physical state (it does not satisfy the constraints), but it can be connected to a kinematic state that can be used as an eigenstate of an observable.
and is re-identifiable in time) is highly problematic [68]: the GEB formalism reflects this fully, as the system persistence and re-identifiability is only enforced as a probability amplitude.
[82] Notice that at this stage, as it happens in the axiomatic definition of QM, we do not provide any physical realization of X or P , i.e. a way to detect the spacetime location and momentum of the event.It is intuitively clear however that these observables are related with the conventional definitions of position and momentum of particles in QM: e.g. on one hand if one conditions on the time t of an external clock, X conditioned on t is just the position measurement of a particle happening at some time t, the conventional position operator of QM that describes a screen that is turned on at a certain (externally controlled) time; on the other hand, if one conditions on the position x of a screen, then X conditioned on x is the time of arrival measurement of the particle at an active screen at position x [69,70].In the non-relativistic case, time observables were studied in, e.g., [14,71,72].
[83] This is the simplest choice to satisfy the Poincaré algebra, but it is not unique: one can still satisfy the commutation relations with appropriate redefinitions of Pµ and Mµν .For example, the instant form [5] arises from the requirement that the position and momentum are referred to some instant of time, as in the conventional (conditioned) formulation of quantum mechanics [56]: in the Schrödinger picture the states are conditioned to time being t and the operators to t = 0, viceversa in the Heisenberg picture.
[84] Notice that one can divide the four degrees of freedom (1 temporal and 3 spatial) of |x using tensor products |x = |t | x , as is done in the nonrelativistic Page and Wootters formalism [11,14] (and equivalent ones [21]) but, this tensor product structure is not absolute to preserve Poincaré invariance: it is observable-induced [73][74][75], since observers in different reference frames will tensorize it differently.
[85] Formally speaking the decomposition (6) holds for the dense subset H − E of H E which together with H + E define the Gelfand triple.
[86] A guideline, suggested by Wigner and Bargmann, is to consider as physical fields the ones that correspond to irreducible representations of the Poincaré group [37].
[87] An interaction with an external electromagnetic field can be described through the minimal coupling substitution of P µ with P µ + eA µ , with e the particle charge and A µ the em 4potential.)

Appendix B: Connecting different reference frames
Consider first the simple case of a single spin-less event space.Let O and O be two inertial observer whose coordinates are linked as in Eq. ( 11) of the main text with the 4×4 matrix Λ representing an element of the Lorentz group.Let Φ(x) the wave-function of a state event S as described by O.To show that the observer O in his reference frame will describe it as the function Φ (x) of ( 12) assume that Φ(x) gets its maximum value Φ max for x = x 0 , i.e.Φ max = Φ(x 0 ).The observer O will assign to such point the coordinate x 0 = Λx 0 that represents the value at which Φ (x) reaches its maximum, i.e.
FIG. 2: Figurative representation of the state |Ψ , its timeconditioning at time t0 that gives the state |ψ(t0) , and its relativistic boost |Ψ to the reference R (dashed lines).The boosted reference (x , t ) is obtained through a hyperbolic transformation (Lorentz transform) from the (x, t) reference, pictorially represented with the dashed lines.The foliation in the (x, t) reference gives the usual (conditioned) state |ψ(t0) of textbook quantum mechanics.A similar foliation in the (x , t ) reference (not pictured) is required for the quantum state at time t in the new reference.
Let us now introduce the vectors |Φ and |Φ of H E that O and O will assign to the state S, i.e.
(Fig. 2).By direct inspection one can be easily verify that these vectors fulfill the identity (13) of the main text by identifying the unitary transformation U Λ with the operator where the second identity in the first line follows by a simple chance of integration variables, while the second line is obtained by taking the adjoint of the first.Notice that U Λ and U † Λ verify the conditions which represent the counterparts of ( 11) at the level of the generalized eigenstates of the position operator X.
Analogously for the generalized eigenvectors of the momentum operator P we get The first for instance can be derived recalling Eq. ( 5) and observing that which of course needs not to be the same as Θ .Notice that we can also rewrite Θ = Φ|Θ |Φ where now is a sort of 4D "Heisenberg-picture" that allows us to transform the operators instead of the states in moving from the reference frame of O to the one by O .In particular we shall say that Θ is invariant under Lorentz transformations if Θ = Θ for all choices of Λ, i.e.
while, given a collection of operators A 0 , A 1 , A 2 , A 3 we shall call A = (A 0 , A 1 , A 2 , A 3 ) a vectorial operator if From (A6) it then follows that given A and B arbitrary vectorial operators the operator A • B is invariant.Important examples of vectorial operators are provided by the canonical operators X and P of the theory, as anticipated in Eqs. ( 16) and ( 17) of the main text.To see this explicit observe for instance that from ( 4) and (B12) we get which leads to (16).We notice that the above expressions can be used to show that the U Λ 's admit as generators operators M µ,ν entering in (2): for example, a y-axis rotation by an angle θ is generated by U Λ = e −iθM 13 with M 13 = X 1 P 3 − X 3 P 1 so that U † Λ X 3 U Λ = X 3 cos θ + X 1 sin θ ; a x-axis directed boost by a rapidity v is generated by U Λ = e −ivM 01 with M 01 = X 1 P 0 −X 0 P 1 so that U X 1 U † = X 1 cosh v + X 0 sinh v (the hyperbolic trigonometric functions appear because of the extra minus sign in [X µ , P ν ] = −iη µν ).We stress also that ( 16) and ( 17) are fully consistent with the setting of the problem.In fact indicating with X and X the mean position that O and O assign to the same event state we notice that they are connected via the identity (B16) which is exactly what you would aspect from Eq. (11).Of course the same result can be obtained by working in the Schrödinger picture: in this case in fact we get where in the third identity we used (12).

Spinors
In the presence of spinorial degree of freedom, the 4D spinor wave-functions Φ (x, σ ) and Φ(x, σ) assigned by the observers O and O , will be connected as in Eq. ( 20).This implies that Eqs.(B3) and (B4) are replaced by and Appendix C: Multi-event tensor representation and Fock representation Recall that the projectors Π (n,S) and Π (n,A) associated with the completely symmetric H (n,S) E and the completely anti-symmetric H (n,A) E subspaces of H ⊗n E , can be expressed as where the sums over p run on the set of permutations of n elements, and V p is the unitary operator which represents p on H ⊗n E .As mentioned in the main text the n event states of Bosonic QM/GEB are by vectors |Φ [n] of ( 22) with 4D spinor wave-functions Φ [n] (x 1 , σ 1 ; • • • ; x n , σ n ) obeying the symmetry condition (25) and normalization condition (C2) Since these vectors belong to the completely symmetric H (n,S) E subspace of H ⊗n E we have |Φ = Π (n,S) |Φ which exploiting (C1) allows one to equivalently rewrite Eq. (22) as In the Fock space representation |S(x 1 , σ 1 ; is formally expressed as the application of sequences of the Bosonic creation operators a † x,σ 's to the 4-vacuum state, i.e. C5) which replaced into (C3) leads to (32) (to justify (C5) notice that due to the commutation rules (30) the two family of states on the l.h.s. and the r.h.s. of the above equation have the same symmetry under permutation of indexes and the same scalar products).
Similar considerations apply for the Fermionic case where the 4D spinor wave-function appearing in (22) fulfill the anti-symmetric relation (26).Invoking hence the fact that they are elements of the completely antisymmetric H (n,A) E subspace of H ⊗n E we have now |Φ = Π (n,A) |Φ , which allows one to replace Eq. (C3) with the vector that is now identified by sequences of Fermionic creation operator a † x,σ 's to the 4D-vacuum state, i.e.
C7) leading once more to (32).Here we analyze in detail the technical aspects of the QM/GEB correspondence introduced in Sec.III.Specifically we shall show that the vectors |Ψ QM introduced in Eq. (40), while not being elements of H E , form a special subset H QM of the distributions set H + E of the theory, i.e. the rigged-extended version of H E which we introduce when discussing the generalized position and momentum eigenvectors of GEB.
We have already commented the fact that the normalization condition (38) implies that the |Ψ QM 's of (40) have a divergent norm.This automatically excludes them from the Hilbert space H E .To prove that they are distributions, we need to show that there exists a dense subset D of H E formed by (normalized) vectors |Φ such that the quantity Ψ QM |Φ exists and is finite.To exhibit such subset let first introduce the spectral decomposition of the QM Hamiltonian H which is ruling the dynamical evolution of the single-particle of the problem (i.e. the generator which is responsible for the time evolution of the 3D wave-function Ψ QM ( x|t)).We will consider explicitly the case where H has a (possibly degenerate) continuous spectrum but the analysis can be easily applied to the cases of discrete spectra (or even mixed discrete/continuous spectra).Accordingly we write with the discrete variable k accounting for the degeneracy of the E-energy level, and where {|E, k } E,k are the generalized orthonormal eigenvectors that fulfill with δ k,k the Kronecker delta symbol.Similarly to [11,14], we now adopt a spacetime foliation that separate the temporal coordinate of H E vs the spatial ones via a tensor product, writing (notice that while this choice breaks the covariance of the theory, this is not a problem as in our case we shall compute scalar products between vectors which are explicitly invariant quantities).We then expand a generic normalized element of H E in the following form where we introduced a discrete complete orthonormal set {|n } n for the temporal axis while we adopted the generalized eigenstates {|E, k } E,k of the QM Hamiltonian H to expand the spatial degree of freedom of the system.In the above equation c n,k (E) are probability amplitudes fulfilling the normalization condition Now we define D to be set of vectors of H E which admits a decomposition (D4) with coefficients c n,k (E) that, besides (D5), fulfill also the extra constraint (to see that D is dense observe that such space contains all the vectors |Φ with c n,k (E) = 0 only for a finite set of values of n).Expressing now |Ψ QM of Eq. ( 40) in terms of the same spacetime foliation used in (D4), i.e.
we notice that where in the second identity we introduced the probability amplitudes of the state |ψ(t) with e −iEt being their associated dynamical phase (remember that {|E, k } E,k are eigenvectors of the system Hamiltonian), and where in the third identity we introduce the vectors Observe that this last is a distribution for the temporal coordinate (indeed it is the Fourier transform of "position" coordinates), that fulfills the orthonormalization rule As a matter of fact we can identify |π(E) as a generalized eigenstate of the canonical momentum of the temporal position axis.Accordingly we can interpret π(E)|n as the momentum amplitude probability distribution of |n evaluated at momentum E. Remember next that {|n } n is a basis that we can choose freely.We now take such basis as the orthonormal set of the spectrum of the Harmonic oscillator which allows us to explicitly compute the value of π(E)|n as where for the sake of simplicity we are expressing here the function in renormalized units where all the physical constants are set equal to 1, and where n (x) are the Hermite polynomials.Now the only fundamental aspect of the problem here is that we can put an upper bound on such terms, independently of the choice of n and E.
In particular we can show that Hence invoking the Cauchy-Schwarz inequality, we can now bound the term (D9) as follows: which is finite due to Eq. (D6).
We now briefly comment on the physical significance of the negative energy solutions of the Klein-Gordon equation.Remember that the wave equation ( − m 2 )f (t, r) = 0 has solutions with spacetime dependence f = g( r − vt) + h( r + vt), with v the propagation velocity (both signs of the velocity must appear in the general solution as the wave equation contains only v 2 ).One can expand g and h in terms of plane waves e i k•( r± vt) ≡ e i( k• r−ωt) , where the frequency ω ≡ ∓ k • v can be positive or negative depending on the propagation direction of the wave with respect to the wave vector k.With an appropriate choice of sign in the definition of ω, one can consider a negative-frequency wave as an advanced solution to the wave equation and a positivefrequency wave as a retarded solution, since these solutions can be obtained from one another by time reversal.Usually the advanced solution is discarded (set to zero) appealing to some vague notion of causality, e.g.[76], but more careful analyses [77,78] interpret the retarded solutions as a prediction based on past boundary conditions and the advanced solutions as a retrodiction based on future boundary conditions.Then the choice of which frequency sign to choose (or even a combination of the two [78]) is dictated purely by the available boundary conditions.Clearly, past boundary conditions are more useful in general.One can discard the negative frequency solutions by imposing, in addition to the Klein-Gordon equation of motion, an additional physical condition of positive-energy (as was done in the main text).
In closing we comment on the "negative probability densities" that historically have plagued the acceptance of the Klein-Gordon equation (notoriously, it was discovered, but then discarded, by Schrödinger [79,80]).This problem ensues from the observation that, if one defines a four-current for the Klein-Gordon wave-function ψ 1 as j µ = ψ * 1 ∂ µ ψ 1 − ψ 1 ∂ µ ψ * 1 , it does satisfy a conservation equation ∂ µ j µ = 0, but the density j 0 (representing a putative probability density) is not positive definite (and should be interpreted as a charge density).It is not such j 0 that should take the role of a probability density of the particle position at a certain time, but rather |ψ 1 (x)| 2 that is the probability density of finding a particle-detection event at spacetime position x = (t, x): a joint probability for both the position and for time, rather than a conditioned probability for the position, given the time.As such, |ψ 1 (x)| 2 is a scalar quantity, not the temporal component of a 4-current, and needs not satisfy any current conservation.Moreover, it is obviously always positive definite.In contrast, in the case of the Dirac field, one can build also a (conserved) probability current (see below).

Appendix G: Constraint operator for the Dirac model
The Dirac equation for the spinor wave-function Ψ QM ( x, σ|t) of single particle is a collection of the four differential equations reported in Eq. (44).By taking the 4D Fourier transform we can turn them into the equivalent form Contracting the index σ of (G1) with the matrix elements of the invertible matrix γ 0 , we can further modify Eq. ( 44) into the identity Expressed as in Eq. (G1) it is easy to verify that, at the level of the GEB distribution |Ψ QM = 4 σ=1 d 4 x Ψ QM ( x, σ|t)|x, σ , the Dirac equation ( 44) corresponds to the identity J D |Ψ QM = 0 with J D as in Eq. (47).Indeed, to show this, we need the fact that thanks to (5) ΨQM (p, σ) provides the 4D-momentum spinor wave-functions expansion of |Ψ QM , i.e. |Ψ QM = 4 σ=1 d 4 p ΨQM (p, σ)|p, σ .As mentioned in the main text the operator J D is not a self-adjoint: this is a direct consequence of the fact that for all i = 1, 2, 3 the matrices γ i are anti-Hermitian (indeed (γ i ) † = −γ i = γ i ), while γ 0 is Hermitian, so that J † D = 4 µ=1 (γ µ ) † P µ − m = 4 µ=1 γ µ P µ − m = J D .Notice however that exploiting the fact that γ 0 γ 0 = 1 1, and γ 0 γ i = 0 σ i σ i 0 , we can write where given the matrix elements M σ ,σ (p) of Eq. (G4) J

QM ( x 1 ,
• • • , x n |t) of Eq. (H6) when we impose (81).b.Fermionic model:-Similar considerations apply to the Fermionic case.Here the functional dependence of the constraint operator K (Fock) D upon the number operator a † φσ(p) a φσ(p) suggests to expand the general solution (34), namely

=
−E p for σ = 1, 3 and E (σ) p = E p for σ = 2, 4 with E p = B9) where in the last identity we exploit the invariance of the product (A7) under Lorentz transform, i.e. a • b = a • b , for a = Λa and b = Λb.Consider next the expectation values of a generic operator Θ on S. The observer O will compute this as Θ = Φ|Θ|Φ , (B10) while O will see this as Appendix D: More on the QM/GEB correspondence