Introduction

When we talk about Lorentz covariance in Quantum Physics, we refer to two different formalisms that must be mutually compatible:

1. a theory which is diffeomorphism-invariant (which includes Poincare invariance) with particular tetrad fields corresponding to a Minkowski space-time;

2. a theory which only needs to be Poincare invariant but without tetrad fields not even a space-time (both could otherwise be helpful implementing the Poincare invariance).

Assuming that we can define1 the diffeomorphism-invariant theory (1), there are still many predictions which involve the Minkowski space-time only or not even any fields (such as in the non-relativistic limit of Quantum Mechanics) and which would be impractical to calculate in the theory (1). For all those predictions, the theory (2) is a useful stepping stone.

There are then two different foundations for the theory (2):

• just define the theory2 and show that it is Poincare invariant, postponing foundational questions to when the diffeomorphism-invariant theory (1) is defined3;

• establish the foundations of the Poincare-invariant theory (2) by itself, without references to a space-time, as we will do in this chapter.

Clearly, the foundations of theory (2) by itself are not indispensable and so this chapter can be skipped in a first reading. Nevertheless, since the foundations of theories (1) and (2) must be mutually consistent, establishing both of these foundations is an important consistency check, especially because not much is known about Quantum Gravity theoretically and experimentally.

Concerning the relation between the Hamiltonian formalism and special relativity, there are two kind of questions we can ask: 1) is the Hamiltonian formalism compatible with Lorentz symmetry? 2) based on the space-time “philosophy”, why should the time-evolution play a distinguished role in the Hamiltonian formalism? The first question is technical, while the second question is conceptual. We do not have an answer to the second question, which is expected given the difficulties with Lorentz-symmetry of other approaches to quantization [1]. But in this section we will answer explicitly and positively to the first question. In short, the fact that time evolution plays a special role allows us to use only Poincare representations with positive squared mass. Considering Poincare representations with positive squared mass is self-consistent and it is in no way in conflict with Lorentz symmetry.

A complete physical system is a free system. If we neglect gravity, the wave-function associated to the free system is a unitary representation of the Poincare group, regardless of the interactions occurring within the free system [2].

When the Hilbert space is the direct sum of irreducible representations of a symmetry group, then these representations will be defined by numbers (e.g. mass and spin) which are invariant under the symmetry group. Thus there will be a set of operators whose diagonal form corresponds to those invariants, we will call them Casimir operators. When the symmetry group is abelian and continuous (e.g. translation in time), then the generator of the group (e.g. the Hamiltonian) is a Casimir operator, and the invariant numbers defining the representations are called the constants of motion. Certainly, when we move from non-relativistic quantum mechanics and consider instead the Poincare group, then the Hamiltonian is no longer a Casimir operator and the notion of constants of motion needs to be reviewed. Nevertheless, the Casimir operators can be chosen arbitrarily (just like the Hamiltonian in non-relativistic quantum mechanics).

For a positive squared mass, the spin and the sign of the Energy are also Poincare invariants. The sign of the Energy times the modulus of the mass is the center-of-mass Energy, while the spin is the center-of-mass angular momentum. Thus, the Casimir operator whose eigenvalues are the center-of-mass Energy may have negative eigenvalues and it will be the analogous operator to the Hamiltonian of the non-relativistic formalism. As will be seen in Section 4, such operator has the formal form of the Hamiltonian action (i.e. it is the difference between the generator of translations in time and the Hamiltonian operator). The Casimir operators necessarily commute with the momentum operator and thus they do not change the 3-momentum eigenstate. Thus we can solve the problem in a basis where the 3-momentum operator is diagonal.

In such a basis, the translations in space-time can be written as $T(x)\Psi(\gamma\vec{v})=e^{i M \tau(\gamma\vec{v}, x)}\Psi(\gamma\vec{v})$, where $\tau(\gamma\vec{v}, x)=\gamma x_0-\gamma \vec{v}\cdot \vec{x}$.

Note that $\gamma=\sqrt{1+\gamma^2\vec{v}^2}$ is a function of $\gamma\vec{v}$. Thus in a basis where the 3-momentum is diagonal, the translations in space-time have the same structure as the time-evolution in non-relativistic space-time, with $M$ playing the role of the non-relativistic Hamiltonian and the numerical factor $\tau(\gamma\vec{v}, x)$ playing the role of the time (it is indeed the proper-time).

For each 3-momentum eigenstate, there is a corresponding inertia referential where the 3-momentum is null, i.e. the referential of the center-of-mass. In such referential, the modulus of the energy is the invariant mass, the signal of the energy is also a Lorentz invariant and the angular momentum is the spin. Thus, the eigenvalues of the Hamiltonian and angular momentum operator in the center-of-mass define three Lorentz invariants which define the Poincare representation completely.

Despite we do not know a priori the diagonal form of the Hamiltonian, we know that it is either continuous or discrete in the neighborhood of the eigenvalue 0 (in the referential of the center-of-mass). If it is continuous then the zero energy has null measure. If it is discrete, we can modify the Hamiltonian adding an appropriate constant such that the zero energy is not one of the eigenvalues. Note that this is only possible in a complex Hilbert space and this is equivalent to adding to the system a free massive particle with null 3-momentum relative to the system. In any case, we can assume without loss of generality that our system is a quantum superposition of massive free systems with null 3-momentum. Then, the Lorentz transformations become known and are given by the Wigner irreducible massive representations of the Poincare group[2]. If the Hamiltonian is bounded from below then the vacuum state is not Lorentz invariant, as it was already suggested[1].

In the center-of-mass, the relevant group is not the Poincare group, but the little group of spatial rotations and the translation in time[2]. Thus the spatial and time coordinates of space-time, become separated. The fields are no longer representations of the Lorentz group, but only of the rotation group and the canonical commutation relations are not in conflict with the little group of spatial rotations.

Note that we use Wigner’s convention for the definition of the 3-momentum of the free complete system: it is the eigenvalue of the generator of the translations in space for the complete system (i.e. all fields defining the phase-space are translated in space). Thus in the center-of-mass, the algebra of operators has a constraint imposing that the operators are translation invariant.

The cyclic vector defining the Hilbert space needs not be translation-invariant (in the center-of-mass), just the operators need to be translation-invariant in the center-of-mass4. This gives us a big freedom to choose the cyclic vector defining the Hilbert space (which is related with the initial state of the system).

We assume that the translations in space of the complete system conserve the Hamiltonian and the constraints equations, such that setting the total 3-momentum to zero in no way conflicts with the constrained Hamiltonian system. Nevertheless, the restriction that there is a referential where the total 3-momentum is null, excludes the free complete system from traveling at the speed of light (e.g. a photon with non-null energy). Then the dynamics determined by the Hamiltonian becomes linked with the time coordinate (for a photon this would not be the case[3]).

Therefore and unlike what it is often claimed in the literature, it is false that (canonical) quantization is incompatible with Lorentz covariance. Note that the phenomenologically successful (but ill defined) path integral formalism based on the Lagrangian is in fact equivalent to a path integral based on the Hamiltonian[4]. In our formalism, the only restriction is that we need to consider representations with positive squared mass, then the dynamics determined by the Hamiltonian becomes linked with the time coordinate[3]. The question why only positive squared masses are relevant is a reformulation of question 2) which will be left open in this paper. Similar assumptions concerning the energy-momentum of the full system are also done in the Källén-Lehmann representation of a non-perturbative two-point correlation function, where it is assumed that the eigenvalues of the 3-momentum squared are not larger than those of the squared energy[5].

The special role of the little group of rotations and the time evolution in our definition of a (special relativistic) Quantum Field Theory may seem to be a step back in the road towards a general relativistic quantum theory. In Chapter /pub/timepiece we show that this is not the case, in fact our rigorous definition of a Quantum Yang-Mills theory marks “a turning point in the mathematical understanding of quantum field theory, with a chance of opening new horizons for its applications” in quantum gravity, as requested in reference[6].

We also make a comment on relativistic causality: the fact that we are considering only Poincare representations with non-negative squared mass leaves us in a good position to guarantee relativistic causality. However, we are working only in the 3-momentum space of the (free) complete quantum system. In order to study relativistic causality we need to make a unitary transformation to the 3-coordinate space, thus we need to define a position operator for a free quantum system.

Motivation

The Poincare group was first introduced as the set of transformations that leave invariant the Maxwell equations for the classical electromagnetic field. The complex representations of the Poincare group were systematically studied[7][8][9][10] and used in the definition of quantum fields[11][12].

The formulation of quantum mechanics with a complex Hilbert space is equivalent to a formulation with a real Hilbert space and particular density matrix and observables[13]. Moreover, for time-independent Hamiltonian, quantum mechanics can be defined as the eigenvalue problem $(H(\vec{x})-E)\Psi(\vec{x})=0$[14], in the relativistic version, the energy may be replaced by the mass squared in the equation $((\eta^{\mu\nu}\partial_\mu\partial_\nu)-m^2)\Psi(x)=0$. Quantum Theory on real Hilbert spaces was investigated before[15][16][17][18][19], the main conclusion was that the formulation of non-relativistic Quantum Mechanics with a real Hilbert space is necessarily equivalent to the complex formulation. We could not find in the literature a systematic study on the real representations of the Poincare group, as it seems to be common assumptions:
1) since non-relativistic Quantum Mechanics is necessarily complex then the relativistic version must also be—it is hard to accept this as relativistic causality requires the existence of anti-particles[20];
2) the energy positivity implies complex Poincare representations—it is not necessarily true as only in a many-particles description the energy positivity is well defined, remember the Feynman–Stueckelberg or the Dirac sea interpretations of anti-particles[21][22];
3) Quantum Field Theory based on the Wightman axioms (which assume complex Poincare representations) is the most general framework incorporating the physics principles of Quantum Mechanics and Poincare covariance— the quantization of gauge fields does not respect Wightman axioms[23], attempts to define non-perturbatively a Quantum Field Theory with gauge interactions involve string theory or space-times with dimensions lower than 4, Euclidean metric or toroidal topology, in this context studying the real Poincare representations cannot be considered a departure from physics principles.

The reasons motivating the study of the real representations of the Poincare group are:

1) The real representations of the Poincare group play an important role in the classical electromagnetism and general relativity[24][25] and in Quantum Theory— e.g. the Higgs boson, Majorana fermion or quantum electromagnetic fields transform as real representations under the action of the Poincare group.

2) The parity—included in the full Poincare group—and charge-parity transformations are not symmetries of the Electroweak interactions[26]. It is not clear why the charge-parity is an apparent symmetry of the Strong interactions[27] or how to explain the matter-antimatter asymmetry[28] through the charge-parity violation. Since the self-conjugate finite-dimensional representations of the identity component of the Lorentz group are also representations of the parity, this work may be useful in future studies of the parity and charge-parity violations.

3) The localization of complex irreducible unitary representations of the Poincare group is incompatible with causality, Poincare covariance and energy positivity[29][30][31][32][33][34], while the complex representation corresponding to the photon is not localizable[35][36]. In contrast to the classical theory, in Quantum Field Theory with gauge interactions it impossible to define the electric charge localization of a large family of charged states in a meaningful way[23]. Since the free Dirac equation is self-conjugate in the Majorana basis, this study may be useful to the definition of a Poincare covariant position operator as a projection-valued measure (which Wightman considered to express the physical idea of localizability[37]).

Systems on real and complex Hilbert spaces

The position operator in Quantum Mechanics is mathematically expressed using a system of imprimitivity: a set of projection operators— associated with the coordinate space—acting on a Hilbert space; a group of symmetries acting both on the Hilbert space and on the coordinate space in a consistent way[36][38].

Many representations of a group—such as the finite-dimensional representations of semisimple Lie groups[39] or the unitary representations of separable locally compact groups[40]—are direct sums (or integrals) of irreducible representations, hence the study of these representations reduces to the study of the irreducible representations.

If the set of normal operators commuting with an irreducible real unitary representation of the Poincare group is isomorphic to the quaternions or to the complex numbers, then the most general position operator that the representation space admits is not complex linear, but real linear. Therefore, in this case, the real irreducible representations generalize the complex ones and these in turn generalize the quaternionic ones.

The study of irreducible representations on complex Hilbert spaces is in general easier than on real Hilbert spaces, because the field of complex numbers is the algebraic closure —where any polynomial equation has a root— of the field of real numbers. There is a well studied map, one-to-one or two-to-one and surjective up to equivalence, from the complex to the real linear finite-dimensional irreducible representations of a real Lie algebra[41][42].

Section 2 reviews and extends that map from the complex to the real irreducible representations—finite-dimensional or unitary—of a Lie group on a Hilbert space. Using Mackey’s imprimitivity theorem, we extend the map further to systems of imprimitivity. This section follows closely the reference[41], with the addition that we will also use the Schur’s lemma for unitary representations on a complex Hilbert space[43].

Related studies can be found in the references[44][45].

Finite-dimensional representations of the Lorentz group

The Poincare group, also called inhomogeneous Lorentz group, is the semi-direct product of the translations and Lorentz Lie groups[39]. Whether or not the Lorentz and Poincare groups include the parity and time reversal transformations depends on the context and authors. To be clear, we use the prefixes full/restricted when including/excluding parity and time reversal transformations. The Pin(3,1)/SL(2,C) groups are double covers of the full/restricted Lorentz group[46]. The semi-direct product of the translations with the Pin(3,1)/SL(2,C) groups is called IPin(3,1)/ISL(2,C) Lie group — the letter (I) stands for inhomogeneous.

A projective representation of the Poincare group on a complex/real Hilbert space is an homomorphism, defined up to a complex phase/sign, from the group to the automorphisms of the Hilbert space. Since the IPin(3,1) group is a double cover of the full Poincare group, their projective representations are the same[46]. All finite-dimensional projective representations of a simply connected group, such as SL(2,C), are usual representations[20]. Both SL(2,C) and Pin(3,1) are semi-simple Lie groups, and so all its finite-dimensional representations are direct sums of irreducible representations[39]. Therefore, the study of the finite-dimensional projective representations of the restricted Lorentz group reduces to the study of the finite-dimensional irreducible representations of SL(2,C).

The Dirac spinor is an element of a 4 dimensional complex vector space, while the Majorana spinor is an element of a 4 dimensional real vector space[47][48][49][50]. The complex finite-dimensional irreducible representations of SL(2,C) can be written as linear combinations of tensor products of Dirac spinors.

In Section 2.3 we will review the Pin(3,1) and SL(2,C) semi-simple Lie groups and its relation with the Majorana, Dirac and Pauli matrices. We will obtain all the real finite-dimensional irreducible representations of SL(2,C) as linear combinations of tensor products of Majorana spinors, using the map from Section 2. Then we will check that all these real representations are also projective representations of the full Lorentz group, in contrast with the complex representations which are not all projective representations of the full Lorentz group. We could not find these results explicitly in the literature but they are straightforward to derive and so probably known by some people, the results are derived here for completeness and explicitness.

Unitary representations of the Poincare group

According to Wigner’s theorem, the most general transformations, leaving invariant the modulus of the internal product of a Hilbert space, are: unitary or anti-unitary operators, defined up to a complex phase, for a complex Hilbert space; unitary, defined up to a signal, for a real Hilbert space[51][36]. This motivates the study of the (anti-)unitary projective representations of the full Poincare group.

All (anti-)unitary projective representations of ISL(2,C) are, up to isomorphisms, well defined unitary representations, because ISL(2,C) is simply connected[20]. Both ISL(2,C) and IPin(3,1) are separable locally compact groups and so all its (anti-)unitary projective representations are direct integrals of irreducible representations[40]. Therefore, the study of the (anti-)unitary projective representations of the restricted Poincare group reduces to the study of the unitary irreducible representations of ISL(2,C).

The spinor fields, space-time dependent spinors, are solutions of the free Dirac equation[52]. The real/complex Bargmann-Wigner fields[53][54], space-time dependent linear combinations of tensor products of Majorana/Dirac spinors, are solutions of the free Dirac equation in each tensor index. The complex unitary irreducible projective representations of the Poincare group with discrete spin or helicity can be written as complex Bargmann-Wigner fields.

In Section 2.4, we will obtain all the real unitary irreducible projective representations of the Poincare group, with discrete spin or helicity, as real Bargmann-Wigner fields, using the map from Section 2. For each pair of complex representations (of ISL(2,C)) with positive/negative energy, there is one real representation. We will define the Majorana-Fourier and Majorana-Hankel unitary transforms of the real Bargmann-Wigner fields, relating the coordinate space with the linear and angular momenta spaces. We show that any localizable unitary representation of the Poincare group (ISL(2,C)), compatible with Poincare covariance, verifies: 1) it is a direct sum of irreducible representations which are massive or massless with discrete helicity. 2) it respects causality; 3) if it is complex it contains necessarily both positive and negative energy subrepresentations 4) it is an irreducible representation of the Poincare group (including parity) if and only if it is: a)real and b)massive with spin 1/2 or massless with helicity 1/2. If a) and b) are verified the position operator matches the coordinates of the Dirac equation.

The free Dirac equation is diagonal in the Newton-Wigner representation[35], related to the Dirac representation through a Foldy-Wouthuysen transformation[55][56] of Dirac spinor fields. The Majorana-Fourier transform, when applied on Dirac spinor fields, is related with the Newton-Wigner representation and the Foldy-Wouthuysen transformation. In the context of Clifford Algebras, there are studies on the geometric square roots of -1 [57] and on the generalizations of the Fourier transform[58], with applications to image processing[59]. It was showed before that point localized local quantum fields—operator valued distributions satisfying the Wightman axioms— cannot be a massless infinite spin representation [60].

The current literature related with the position operator of representations of the Poincare group include: modular and string-like localization in the context of local quantum field theory [61][62][63][64]; non-commutative coordinates [65][66]; coordinates based on equations [67][68]; unsharp (fuzzy) localization using positive operator valued measures [69][70][71][72][73]; localization of the energy density [74][75][76][77][78]; two dimensional[79] or axial symmetric[80] or space-time[81] localization of photons; pseudo-hermitian representations of Quantum Mechanics[82]. All the above mentioned approaches departure, in one way or another, from using the system of imprimitivity to implement the position operator for a unitary representation of the Poincare group. Finally taking the Newton-Wigner position seriously has the problem that for spin one-half the position does not coincide with the coordinates appearing in the Dirac equation, with all the phenomenological consequences that it implies[83][84]. The results presented in Section 2.4 are a motivation to not departure from the systems of imprimitivity to describe the position of relativistic systems.

Energy Positivity

While it seems that the localization of particles in either relativistic quantum mechanics[61] or relativistic quantum field theory[85] is not possible (at least with the properties we would expect), it is not clear whether this is a limitation of the relativistic Quantum framework itself, or due to some properties of our definition of particles which are incompatible with a proper definition of localization. For instance, it is expected that by specifying the energy-momentum properties of the vacuum we then may have troubles to define the localization of all the states related with the vacuum—a consequence of the Reeh-Schlieder theorem[86]—, since momentum and position do not commute; that does not imply that the localization cannot be defined at all within the relativistic Quantum framework once we relax the energy-momentum properties of the vacuum.

In non-relativistic Quantum Mechanics the time is invariant under the Galilean transformations —excluding the time reversal transformation—and so the generator of translations in time is also invariant. Therefore, the positivity of the Energy and the localization in space of a state can be defined simultaneously. In relativistic Quantum Mechanics, the time is not invariant under Lorentz transformations, as a consequence the positivity of the Energy and the localization in space of a state cannot be defined simultaneously— the corresponding projection operators do not commute.

In the framework of Algebraic QFT(related with the Wightman axioms), there is a definition for Quantum Field Theories with interactions in terms of formal power series in the coupling constants[61], which is based on the canonical quantization where the positivity of Energy is well defined by construction and the localization problem is handled by introducing anti-particles—causality implies the existence of anti-particles[20], a related approach led Dirac to predict the positron[21]. Yet, it is also possible to build a description of a many particles system where the localization in space of a state is well defined by construction and the Energy positivity problem can be handled with the Feynman–Stueckelberg interpretation for anti-particles, as Energy positivity and localization are complementary. Dirac himself was the first to consider an approach which do not assume the positivity of Energy by construction[87] and quantization in de Sitter space-time may be achieved in a related approach[88].

If we need to revisit known results at a deep level to non-perturbatively define a Quantum Field Theory with interactions[6], defining the position operator with a projection-valued measure(which Wightman considered to represent the physical idea of localizability) in a many-particles system seems to be useful. In the known formulations of Algebraic QFT there are no pure states (potentially preferring an ensemble interpretation of Quantum Mechanics)[61] and so the position operator is not defined with a projection-valued measure.

The description of a many-particles system based on the definition of the position operator with a projection-valued measure will be discussed in the section5.

Systems on real and complex Hilbert spaces

Definition 1 (System). A system $(M,V)$ is defined by:
1) the (real or complex) Hilbert space $V$;
2) a set $M$ of bounded endomorphisms on $V$.

The representation of a symmetry is an example of a system: a representation space plus a set of operators representing the action of the symmetry group in the representation space[89].

Definition 2 (Complexification). Consider a system $(M,W)$ on a real Hilbert space. The system $(M,W^c)$ is the complexification of the system $(M,W)$, defined as $W^c\equiv \mathbb{C}\otimes W$, with the multiplication by scalars such that $a(b w)\equiv (ab)w$ for $a,b\in \mathbb{C}$ and $w\in W$. The internal product of $W^c$ is defined—for $u_r,u_i,v_r,v_i\in W$ and $<v_r,u_r>$ the internal product of $W$—as:

Definition 3 (Realification). Consider a system $(M,V)$ on a complex Hilbert space. The system$(M,V^r)$ is the realification of the system $(M,V)$, defined as $V^r\equiv V$ is a real Hilbert space with the multiplication by scalars restricted to reals such that $a(v)\equiv (a+i0)v$ for $a\in \mathbb{R}$ and $v\in V$. The internal product of $V^r$ is defined—for $u,v\in V$ and $<v,u>$ is the internal product of $V$—as:

Note 4. Let $H_n$, with $n\in\{1,2\}$, be two Hilbert spaces with internal products $<,>:H_n\times H_n\to \mathbb{F}$,($\mathbb{F}=\mathbb{R},\mathbb{C}$). A (anti-)linear operator $U:H_1\to H_2$ is (anti-)unitary iff:
1) it is surjective;
2) for all $x\in H_1$, $<U(x) , U(x)>=<x, x>$.

Proposition 5. Let $H_n$, with $n\in\{1,2\}$, be two complex Hilbert spaces and $H^r_n$ its complexification. The following two statements are equivalent:

1) The operator $U:H_1\to H_2$ is (anti-)unitary;

2) The operator $U^r:H_1^r\to H_2^r$ is (anti-)unitary, where $U^r(h)\equiv U(h)$, for $h\in H_1$.

Proof. Since $<h,h>=<h,h>_r$ and $U^r(h)=U(h)$, for $h\in H_1$, we get the result. ◻

Definition 6 (Equivalence). Consider the systems $(M,V)$ and $(N,W)$:
1) A normal endomorphism of $(M,V)$ is a bounded endomorphism $S:V\to V$ commuting with $S^\dagger$ and $m$, for all $m\in M$; an anti-endomorphism in a complex Hilbert space is an anti-linear endomorphism;
2) An isometry of $(M,V)$ is a unitary operator $S:V\to V$ commuting with $m$, for all $m\in M$;
3) The systems $(M,V)$ and $(N,W)$ are unitary equivalent iff there is a isometry $\alpha:V\to W$ such that $N=\{\alpha m\alpha^{\dagger}: m\in M\}$.

We use the trivial extension of the definition of irreducibility from representations to systems.

Definition 7 (Irreducibility). Consider the system $(M,V)$ and let $W$ be a linear subspace of $V$:
1) $(M,W)$ is a (topological) subsystem of $(M,V)$ iff $W$ is closed and invariant under the system action, that is, for all $w\in W$:$(m w)\in W$, for all $m\in M$;
2) A system $(M,V)$ is (topologically) irreducible iff their only sub-systems are the non-proper $(M,V)$ or trivial $(M,\{0\})$ sub-systems, where $\{0\}$ is the null space.

Definition 8 (Structures). 1) Consider a system $(M,V)$ on a complex Hilbert space. A C-conjugation operator of $(M,V)$ is an anti-unitary involution of $V$ commuting with $m$, for all $m\in M$;
2) Consider a system $(M,W)$ on a real Hilbert space. A R-imaginary operator of $(M,W)$, $J$, is an isometry of $(M,W)$ verifying $J^2=-1$.

The map from the complex to the real systems

Definition 9. Consider an irreducible system $(M,V)$ on a complex Hilbert space:
1) The system is C-real iff there is a C-conjugation operator;
2) The system is C-pseudoreal iff there is no C-conjugation operator but there is an anti-unitary operator of $(M,V)$;
3) The system is C-complex iff there is no anti-unitary operator of $(M,V)$.

Definition 10. Consider the system $(M,W)$ on a real Hilbert space and let $(M,W^c)$ be its complexification: 1) $(M,W)$ is R-real iff $(M,W^c)$ is C-real irreducible;
2) $(M,W)$ is R-pseudoreal iff $(M,V)$ is C-pseudoreal irreducible, with $W^c=V\oplus \bar V$; 3) $(M,W)$ is R-complex iff $(M,V)$ is C-complex irreducible, with $W^c=V\oplus \bar V$.

Proposition 11. Any irreducible real system is R-real or R-pseudoreal or R-complex.

Proof. Consider an irreducible system $(M,W)$ on a real Hilbert space. There is a C-conjugation operator of $(M,W^c)$, $\theta$, defined by $\theta(u+iv)\equiv (u-iv)$ for $u,v\in W$, verifying $(W^c)_\theta=W$.

Let $(M,X^c)$ be a proper non-trivial subsystem of $(M,W^c)$. Then $\theta$ is a C-conjugation operator of the subsystems $(M,Y^c)$ and $(M,Z^c)$, where $Y^c\equiv \{u+\theta v: u,v\in X^c\}$ and $Z^c\equiv \{u: u,\theta u\in X^c\}$. Therefore, $Y^c=\{u+iv: u,v\in Y\}$ and $Z^c=\{u+iv: u,v\in Z\}$, where $Y\equiv\{\frac{1+\theta}{2}u: u\in Y^c\}$ and $Z\equiv\{\frac{1+\theta}{2}u: u\in Z^c\}$, are invariant closed subspaces of $W$. If $Y=\{0\}$ then $Z=\{0\}$ and $Y^c=X^c=\{0\}$, in contradiction with $X^c$ being non-trivial. If $Z=W$ then $Y=W$ and $Z^c=X^c=W^c$, in contradiction with $X^c$ being proper. Therefore $Z=\{0\}$ and $Y=W$, which implies $Z^c=\{0\}$ and $Y^c=W^c$.

So, $(M,W)$ is equivalent to $(M,(X^c)^r)$, due to the existence of the bijective linear map $\alpha:(X^c)^r\to W$, $\alpha(u)=u+\theta u$, $\alpha^{-1}(u+\theta u)=u$, for $u\in(X^c)^r$. Suppose that there is a C-conjugation operator of $(M,X^c)$, $\theta'$. Then $(M,W_{\pm})$ is a proper non-trivial subsystem of $(M,W)$, where $W_\pm\equiv\{\frac{1\pm\theta'}{2}w: w\in W\}$, in contradiction with $(M,W)$ being irreducible. ◻

Proposition 12. Any real system which is R-real or R-pseudoreal or R-complex is irreducible.

Proof. Consider an irreducible system on a complex Hilbert space $(M,V)$. There is a R-imaginary operator $J$ of the system $(M ,V^{r})$, defined by $J(u)\equiv i u$, for $u\in V^r$.

Let $(M,X^r)$ be a proper non-trivial subsystem of $(M,V^{r})$. Then $J$ is an R-imaginary operator of $(M,Y^r)$ and $(M^r,Z^r)$, where $Y^r\equiv \{u+J v: u, v \in X^r\}$ and $Z^r\equiv \{u: u,Ju\in X^r\}$. Then $(M,Y)$ and $(M,Z)$ are subsystems of $(M,V)$, where the complex Hilbert spaces $Y\equiv Y^r$ and $Z\equiv Z^r$ have the scalar multiplication such that $(a+ib)(y)=ay+bJy$, for $a,b\in\mathbb{R}$ and $y\in Y$ or $y\in Z$. If $Y=\{0\}$, then $Z=X^r=\{0\}$ which is in contradiction with $X^r$ being non-trivial. If $Z=V$, then $Y=V$ and $X^r=V^r$ which is in contradiction with $X^r$ being non-trivial. So $Z=\{0\}$ and $Y=V$, which implies that $V=(X^r)^c$.

Then there is a C-conjugation operator of $(M,V)$, $\theta$, defined by $\theta(u+iv)\equiv u-iv$, for $u,v\in X^r$. We have $X^r=V_\theta$. Suppose there is a R-imaginary operator of $(M,V_\theta)$, $J'$. Then $(M,V_{\pm})$, where $V_{\pm}\equiv\{\frac{1\pm iJ'}{2}v: v\in V\}$, are proper non-trivial subsystems of $(M,V)$, in contradiction with $(M,V)$ being irreducible.

Therefore, if $(M,V)$ is C-real, then $(M,V_\theta)$ is R-real irreducible. If $(M,V)$ is C-pseudoreal or C-complex, then $(M,V_\theta^r)$ is R-pseudoreal or R-complex, irreducible. ◻

Schur Systems

Definition 13 (Schur System). A system $(M,V)$, on a complex Hilbert space $V$, is a Schur system if the set of normal operators of $(M,V)$ is isomorphic to $\mathbb{C}$.
Consider an irreducible system $(M,W)$, on a real Hilbert space $W$ and let $(M,W^c)$ be its complexification: 1) $(M,W)$ is Schur R-real iff $(M,W^c)$ is Schur C-real;
2) $(M,W)$ is Schur R-pseudoreal iff $(M,V)$ is Schur C-pseudoreal, with $W^c=V\oplus \bar V$;
3) $(M,W)$ is Schur R-complex iff $(M,V)$ is Schur C-complex, with $W^c=V\oplus \bar V$.

Lemma 14. Consider a Schur system $(M,V)$ on a complex Hilbert space. An anti-isometry of $(M,V)$, if it exists, is unique up to a complex phase.

Proof. Let $\theta_1$,$\theta_2$ be two anti-isometries of $(M,V)$. The product $(\theta_2\theta_1)$ is an isometry of $(M,V)$; since $(M,V)$ is irreducible, $(\theta_2\theta_1)=e^{i\phi}$; with $\phi\in \mathbb{R}$.

Therefore $\theta_2=\alpha \theta_1\alpha^{-1}$; where $\alpha\equiv e^{i\frac{\phi}{2}}$ is a complex phase. ◻

Proposition 15. Two R-real Schur systems are isometric iff their complexifications are isometric.

Proof. Let $(M,V)$ and $(N,W)$ be C-real Schur systems, with $\theta_M$ and $\theta_N$ the respective C-conjugation operators. If there is an isometry $\alpha:V\to W$ such that $\alpha M=N\alpha$, then $\vartheta\equiv \alpha\theta_M\alpha^{-1}$ is an anti-isometry of $(N,W)$. Since it is unique up to a phase, then $\theta_N=e^{i\phi}\vartheta$. Therefore $e^{i\frac{\phi}{2}}\alpha$ is an isometry between $(M,V_\theta)$ and $(N,W_\theta)$, where $V_{\theta_M}\equiv\{(1+\theta_M)v: v\in V\}$. ◻

Proposition 16. Two C-complex or C-pseudoreal Schur systems are isometric or anti-isometric iff their realifications are isometric.

Proof. Let $(M,V)$ and $(N,W)$ be R-complex or R-pseudoreal Schur systems, with $J_M$ and $J_N$ the respective R-imaginary operators. If there is an isometry $\alpha:V\to W$ such that $\alpha M=N\alpha$, then $K\equiv \alpha J_M\alpha^{-1}$ is a R-imaginary operator of $(N,W)$. When considering $(N,W_{J_N})$ and $(M,V_{J_M})$, where $W_{J_N}\equiv \{(1-iJ_N) w: w\in W\}$, we get that $(1-J_N K)(1-K J_N)=r$ as an operator of $W_{J_N}$, where $r$ is a non-negative null real scalar. If $c=0$ then $K=-J_N$ and $\alpha$ defines an anti-isometry between $(M,V_{J_M})$ and $(N,W_{J_N})$. If $c\neq 0$ then $(1-J_N K)\alpha c^{-\frac{1}{2}}$ is an isometry between $(M,V_{J_M})$ and $(N,W_{J_N})$. ◻

Proposition 17. The space of normal operators of a R-real Schur system is isomorphic to $\mathbb{R}$.

Proof. Let $(M,V)$ be a C-real Schur system, with $\theta$ the C-conjugation operator. If there is an endomorphism $\alpha:V\to V$ such that $\alpha M=M\alpha$, we know that $\alpha=re^{i\varphi}$. Then the endomorphism of $V_\theta$ is a real number. ◻

Proposition 18. The space of normal operators of a R-complex Schur system is isomorphic to $\mathbb{C}$.

Proof. Let $(M,V)$ be a R-complex Schur system, with $J$ the R-imaginary operator. If there is a normal operator $\alpha$ of $(M,V)$, then $KK^\dagger$ is a normal operator of the C-complex Schur system $(M,V_{J})$, where $K\equiv (\alpha+J\alpha J)$ and $V_{J}\equiv \{(1-iJ) v: v\in V\}$. If $KK^\dagger=r>0$, then $\frac{K}{\sqrt{r}}$ is unitary and $V_J$ is equivalent to $\overline{V}_{J}$ which would imply that $(M,V)$ is C-pseudoreal. Therefore $K=0$ and hence $\alpha$ is a normal operator of $(M,V_{J})$, so $\alpha=r e^{J\theta}$. ◻

Proposition 19. The space of normal operators of a R-pseudoreal Schur system is isomorphic to $\mathbb{H}$ (quaternions).

Proof. Let $(M,V)$ be a R-pseudoreal Schur system, with $J$ the R-imaginary operator. If there is an endomorphism $\alpha$ of $(M,V)$, then $SS^\dagger$ and $TT^\dagger$ are a self-adjoint endomorphisms of the C-complex Schur system $(M,V_{J})$, where $S\equiv(\alpha-J\alpha J)/2$, $T\equiv(\alpha+J\alpha J)/2$ and $V_{J}\equiv\{(1-iJ) v: v\in V\}$. Let $K$ be an unitary operator of $(M,V)$ and anti-commuting with $J$, then $K^2=e^{J\theta}$ and $Ke^{J\theta}=K(K^2)=(K^2)K=e^{J\theta}K$, therefore $K^2=-1$. If $TT^\dagger=t>0$, then $\frac{T}{\sqrt{t}}$ is unitary and anti-commutes with $J$, $TK$ is a normal endomorphism of $(M,V_{J})$ and therefore $T=Kc+KJd$; if $TT^\dagger=0$ then $c=d=0$. If $SS^\dagger=s>0$, then $\frac{S}{\sqrt{s}}$ is unitary and commutes with $J$, $S$ is a normal endomorphism of $(M,V_{J})$ and therefore $S=a+Jb$; if $SS^\dagger=0$ then $a=b=0$.

Therefore $\alpha=S+T=a+Jb+Kc+KJd$, which is isomorphic to the quaternions. ◻

Finite-dimensional representations

Lemma 20 (Schur’s lemma for finite-dimensional representations[43]). Consider an irreducible finite-dimensional representation $(M_G,V)$ of a Lie group $G$ on a complex Hilbert space $V$. If the representation $(M_G,V)$ is irreducible then any endomorphism $S$ of $(M_G,V)$ is a complex scalar.

Lemma 21. Consider an irreducible complex finite-dimensional representation $(M,V)$ on a complex Hilbert space. Then there is internal product such that: 1) The system is C-real iff there is an anti-linear involution of $(M,V)$;
2) The system is C-pseudoreal iff there is not an anti-linear bounded involution of $(M,V)$, but there is an anti-isomorphism of $(M,V)$;
3) The system is C-complex iff there is no anti-isomorphism of $(M,V)$.

Proof. Let $S$ be an anti-isomorphism of an irreducible representation $(M,V)$. Then $S^2=re^{i\varphi}$. But $S^2$ commutes with $S$ which is anti-linear, so $S^2=\pm r$. So, there is an internal product such that $S$ is anti-unitary. ◻

Definition 22. A finite-dimensional system is completely reducible iff it can be expressed as a direct sum of irreducible systems.

Note 23 (Weyl theorem). All finite-dimensional representations of a semi-simple Lie group (such as SL(2,C)) are completely reducible.

Unitary representations and Systems of Imprimitivity

Definition 24 (Normal System). A System $(M,V)$ is normal iff $M$ is a set $M$ of normal operators on $V$ closed under Hermitian conjugation—for all $m\in M$ there is $n\in M$ such that $n=m^\dagger$.

A unitary representation or a System of Imprimitivity are examples of a normal System.

Note 25. $W^\bot$ is the orthogonal complement of the subspace $W$ of the Hilbert space $V$ if:
1) $V=W \oplus W^\bot$, that is, all $v\in V$ can be expressed as $v=w+x$, where $w\in W$ and $x\in W^\bot$;
2) if $w\in W$ and $x\in W^\bot$, then $x^\dagger w=0$.

Lemma 26. Consider a normal system $(M,V)$. Then, for all subsystem $(M,W)$ of $(M_G,V)$, $(M_G,W^\bot)$ is also a subsystem of $(M,V)$, where $W^\bot$ is the orthogonal complement of the subspace $W$.

Proof. Let $(M,W)$ be a subsystem of $(M,V)$. $W^\bot$ is the orthogonal complement of $W$.

For all $x\in W^\bot$, $w\in W$ and $m\in M$, $<m x, w>=<x,m^\dagger w>$.

Since $W$ is invariant and there is $n\in M$, such that $n=m^\dagger$, then $w'\equiv (m^\dagger w)\in W$.

Since $x\in W^\bot$ and $w'\in W$, then $<x, w'>=0$.

This implies that if $x\in W^\bot$), also $(m x)\in W^\bot$, for all $m\in M$. ◻

Lemma 27. Any Schur normal system on a complex Hilbert space is irreducible.

Proof. Let $(M,W)$ and $(M,W^\bot)$ be sub-systems of the complex Schur system $(M,V)$, where $W^\bot$ is the orthogonal complement of $W$.

There is a bounded endomorphism $P: V\to V$, such that, for $w,w'\in W$, $x,x'\in W^\bot$, $P(w+x)=w$. $P^2=P$ and $P$ is hermitian:

Let $w'\equiv m w\in W$ and $x'\equiv m x\in W^\bot$:

Which implies that $P$ commutes with all $m\in M$, so $P\in\{0,1\}$. If $P=1$, then $W=V$, if $P=0$, then $W$ is the null space. ◻

So a complex Schur normal system is irreducible, and hence, from Defns.Definition 10,Definition 13 and Prop.Proposition 12, a real Schur normal system is also irreducible.

Lemma 28 (Schur’s lemma for unitary representations[43]). Consider an irreducible unitary representation $(M,V)$ of a Lie group $G$ on a complex Hilbert space $V$. If the representation $(M,V)$ is irreducible then any normal operator $N$ of $(M,V)$ is a scalar.

Definition 29. A unitary system is completely reducible iff it can be expressed as a direct integral of irreducible systems.

Note 30. All unitary representations of a separable locally compact group (such as the Poincare group) are completely reducible.

Systems of Imprimitivity

Definition 31. Consider a measurable space $(X, M)$, where $M$ is a $\sigma$-algebra of subsets of $X$. A projection-valued-measure, $\pi$, is a map from $M$ to the set of self-adjoint projections on a Hilbert space $H$ such that $\pi(X)$ is the identity operator on $H$ and the function $<\psi,\pi(A)\psi>$, with $A\in M$ is a measure on $M$, for all $\psi\in H$.

Definition 32. Suppose now that $X$ is a representation of $G$. Then, a system of imprimitivity is a pair $(U,\pi)$, where $\pi$ is a projection valued measure and $U$ an unitary representation of $G$ on the Hilbert space $H$, such that $U(g)\pi(A) U^{-1}(g)=\pi(gA)$.

Note 33 (Imprimitivity Theorem (thrm 6.12 [90][91][92][93][36])). Let $G$ be a Lie group, $H$ its closed subgroup. Let a pair $(V, E)$ be a system of imprimitivity for $G$ based on $G/H$ on a separable complex Hilbert space. Then there exists a representation $L$ of $H$ such that $(V, E)$ is equivalent to the canonical system of imprimitivity $(V_L,E_L)$. For any two representations $L$, $L'$ of the subgroup $H$ the corresponding canonical systems of imprimitivity are equivalent if and only if $L$, $L'$ are equivalent. The sets of normal operators commuting with $(V_L, E_L )$ and $L$ are isomorphic.

Lemma 34 (Schur’s lemma for systems of imprimitivity[43]). Let a pair $(V, E)$ be a system of imprimitivity for $G$ based on $G/H$ on a separable complex Hilbert space. If $(V, E)$ is irreducible then then any normal operator $N$ commuting with $(V,E)$ is a scalar.

Proof. Consider a representation $L$ of $H$ such that $(V, E)$ is equivalent to the canonical system of imprimitivity $(V_L,E_L)$. If $L$ would be reducible then there would be a non-trivial normal projection operator commuting with $L$, but then the imprimitivity theorem implies that there would also be a non-trivial normal projection operator commuting with $(V,E)$ which is in contradiction with the irreducibility of $(V,E)$, therefore $L$ is irreducible. The Schur’s lemma for unitary representations then implies that any normal operator commuting with $L$ is a scalar, the imprimitivity theorem then implies the result. ◻

So we can define a map from the real to the complex systems of imprimitivity—analogous to the one for unitary representations. So we extended an existing map from the complex to the real linear finite-dimensional irreducible representations of a real Lie algebra[41][42] to the infinite-dimensional (unitary) case.

Finite-dimensional representations of the Lorentz group

We could not find the following results explicitly in the literature but they are straightforward to derive and so probably known by some people, the results are derived here for completeness and explicitness.

Majorana, Dirac and Pauli Matrices and Spinors

Definition 35. $\mathbb{F}^{m\times n}$ is the vector space of $m\times n$ matrices whose entries are elements of the field $\mathbb{F}$.

In the next remark we state the Pauli’s fundamental theorem of gamma matrices. The proof can be found in the reference[94].

Note 36 (Pauli’s fundamental theorem). Let $A^\mu$, $B^\mu$, $\mu\in\{0,1,2,3\}$, be two sets of $4\times 4$ complex matrices verifying:

Where $\eta^{\mu\nu}\equiv diag(+1,-1,-1-1)$ is the Minkowski metric.

1) There is an invertible complex matrix $S$ such that $B^\mu=S A^\mu S^{-1}$, for all $\mu\in\{0,1,2,3\}$. $S$ is unique up to a non-null scalar.

2) If $A^\mu$ and $B^\mu$ are all unitary, then $S$ is unitary.

Proposition 37. Let $\alpha^\mu$, $\beta^\mu$, $\mu\in\{0,1,2,3\}$, be two sets of $4\times 4$ real matrices verifying:

Then there is a real matrix $S$, with $|det S|=1$, such that $\beta^\mu=S\alpha^\mu S^{-1}$, for all $\mu\in\{0,1,2,3\}$. $S$ is unique up to a signal.

Proof. From remark Note 36, we know that there is an invertible matrix $T'$, unique up to a non-null scalar, such that $\beta^\mu=T'\alpha^\mu T^{'-1}$. Then $T\equiv T'/|det(T')|$ has $|det T|=1$ and it is unique up to a complex phase.

Conjugating the previous equation, we get $\beta^\mu=T^*\alpha^\mu T^{*-1}$. Then $T^*=e^{i 2 \theta} T$ for some real number $\theta$. Therefore $S\equiv e^{i \theta}T$ is a real matrix, with $|det S|=1$, unique up to a signal. ◻

Definition 38. The Majorana matrices, $i\gamma^\mu$, $\mu\in\{0,1,2,3\}$, are $4\times 4$ complex unitary matrices verifying:

The Dirac matrices are $\gamma^\mu\equiv -i(i\gamma^\mu)$.

In the Majorana bases, the Majorana matrices are $4\times 4$ real orthogonal matrices. An example of the Majorana matrices in a particular Majorana basis is:

In reference [95] it is proved that the set of five anti-commuting $4\times 4$ real matrices is unique up to isomorphisms. So, for instance, with $4\times 4$ real matrices it is not possible to obtain the euclidean signature for the metric.

Definition 39. The Dirac spinor is a $4\times 1$ complex column matrix, $\mathbb{C}^{4\times 1}$.

The space of Dirac spinors is a 4 dimensional complex vector space.

Lemma 40. The charge conjugation operator $\Theta$, is an anti-linear involution commuting with the Majorana matrices $i\gamma^\mu$. It is unique up to a complex phase.

Proof. In the Majorana bases, the complex conjugation is a charge conjugation operator. Let $\Theta$ and $\Theta'$ be two charge conjugation operators operators. Then, $\Theta\Theta'$ is a complex invertible matrix commuting with $i\gamma^\mu$, therefore, from Pauli’s fundamental theorem, $\Theta\Theta'=c$, where $c$ is a non-null complex scalar. Therefore $\Theta'=c^*\Theta$ and from $\Theta'\Theta'=1$, we get that $c^* c=1$. ◻

Definition 41. Let $\Theta$ be a charge conjugation operator.

The set of Majorana spinors, denoted here by $Pinor$, is the set of Dirac spinors verifying the Majorana condition (defined up to a complex phase):

The set of Majorana spinors is a 4 dimensional real vector space. Note that the linear combinations of Majorana spinors with complex scalars do not verify the Majorana condition.

There are 16 linear independent products of Majorana matrices. These form a basis of the real vector space of endomorphisms of Majorana spinors, $End(Pinor)$. In the Majorana bases, $End(Pinor)$ is the vector space of $4\times 4$ real matrices.

Definition 42. The Pauli matrices $\sigma^k,\ k\in\{1,2,3\}$ are $2\times 2$ hermitian, unitary, anti-commuting, complex matrices. The Pauli spinor is a $2\times 1$ complex column matrix. The space of Pauli spinors is denoted by $Pauli$.

The space of Pauli spinors, denoted here by $Pauli$, is a 2 dimensional complex vector space and a 4 dimensional real vector space. The realification of the space of Pauli spinors is isomorphic to the space of Majorana spinors.

On the Lorentz, SL(2,C) and Pin(3,1) groups

Note 43. The Lorentz group, $O(1,3)\equiv\{\lambda \in \mathbb{R}^{4\times 4}: \lambda^T \eta \lambda=\eta \}$, is the set of real matrices that leave the metric, $\eta=diag(1,-1,-1,-1)$, invariant.