Generalized unitary evolution for symplectic scalar fermions

The theory of symplectic scalar fermion of LeClair and Neubert is studied. The theory evades the conventional spin-statistics theorem because its Hamiltonian is pseudo Hermitian. Here, we clarify the derivation of the symplectic currents and charges. By demanding the currents and charges to be pseudo Hermitian, the global symmetry of the free Lagrangian density reduces from Sp(2,C) to SU(2). By explicit calculations, we show that the LeClair-Neubert model of N quartic self-interacting scalar fermions admits generalized unitary evolution.


Introduction
In quantum mechanics and quantum field theory, physical operators such as the Hamiltonian, are postulated to be Hermitian because their eigenvalues, which are observables, are real.Transformations generated by Hermitian operators admit unitary evolution of physical systems which preserve transition probabilities.The synthesis of unitarity and Lorentz symmetry inevitably lead to the celebrated spin-statistics theorem [1][2][3].
The postulates in quantum mechanics and the spin-statistics theorem have played indispensable roles in our ongoing efforts to understand the fundamental constituents of matter.And yet, according to the ΛCDM model and measurements from the Planck Collaboration, elementary particles of the Standard Model (SM) accounts only up to 5% of the total energy and matter contents of the observed universe [4].Therefore, it may be premature to suppose the present theoretical framework and their theorems are fixed.In fact, the development of physics from the beginning of the twentieth century has taught us that progresses are made by continual modifications and generalizations of the foundational axioms. 1n this spirit, the works on PT symmetric Hamiltonians by Bender and Boettcher [6,7] and its subsequent generalization to pseudo Hermitian Hamiltonians by Mostafazadeh [8,9], are of central importance.These works represent an expansive program to extend quantum mechanics and quantum field theory beyond the formalism of Hermitian operators.To be precise, a pseudo Hermitian Hamiltonian is defined by the following equation where η is an operator to be determined.While definition (1.1) was already known to Pauli [10], its physical implication was only realized after Mostafazadeh proved two important theorems concerning the spectrum of H and the generalized unitary evolution of states when equipped with the appropriate inner-product.To facilitate ensuing discussions, we now present the two theorems of Mostafazadeh: Theorem 1.Let |α⟩ be an eigenstate of H with eigenvalue E α .From (1.1), the eigenvalue E α is real when η α = ⟨α|η|α⟩ ̸ = 0.
Physically well-defined quantum field theories with pseudo Hermitian Hamiltonians that evade the spin-statistics theorem have been constructed in the spin-zero and spin-half representations [11][12][13][14][15].In the spin-half sector, these constructs have important ramifications for physics beyond the Standard Model (SM).The author and collaborators have, from first principle, constructed spin-half bosonic as well as fermionic fields of mass dimension one and three-half [12][13][14].Due to their mismatch in statistics and mass dimensionality with the SM fermionic fields, they cannot form multiplets with the SM leptons, thus making them natural dark matter candidates.The pseudo Hermiticity of mass dimension three-half bosons and their generalized unitary evolution have been established in [13].The relevant results for mass dimension one fields will be presented elsewhere.
In this paper, we study the theory of symplectic scalar fermion (scalar fermion) constructed by LeClair and Neubert (LN) [11].This theory is local, Lorentz-invariant, admits a positive-definite free Hamiltonian but furnishes fermionic statistics.The point of departure from the bosonic scalar field theory comes from the introduction of pseudo Hermitian adjoint.Due to the fermionic statistics, the theory is shown to have global symplectic symmetry and its β function, in the case of quartic self-interaction has non-trivial fix point [11].The theory has found application in conformal field theory [15], dS/CFT correspondence [16][17][18][19][20][21].
The paper is organized as follows.In sec.2, we review the theory of scalar fermion and clarify the derivation of the global symplectic symmetry.In sec.3, we show that for the model of N quartic self-interacting scalar fermions, the S-matrix satisfies the generalized unitarity relation.

Scalar fermions
Let ϕ be a complex scalar field and ¬ ϕ be its adjoint.LeClair and Neubert made the crucial observation that ¬ ϕ does not have to be the Hermitian conjugate of ϕ.In fact, the following expansions are also legitimate.The demand of locality, and the minus sign in (2.2) force ϕ and ¬ ϕ to be fermionic rather than bosonic while respecting Lorentz symmetry.That is, at equal time, they anti-commute with each other {ϕ(t, x), ¬ ϕ(t, y)} = 0.In fact, they also anti-commute at the same space-time point {ϕ(x), ¬ ϕ(x)} = 0. Given an arbitrary Lorentz transformation Λ, they transform as where U (Λ) is the unitary representation of Λ in the Hilbert space.Taking into account of the fermionic statistics, the free propagator and Lagrangian density are given by and One can readily verify that the fields and their conjugate momenta satisfy the canonical equal-time anti-commutation relations and that the free Hamiltonian is positive-definite after normal-orering [11] Because ϕ and ¬ ϕ furnish fermionic statistics, we cannot add a Hermitian conjugate term L † to (2.6) to make the Lagrangian density Hermitian as it would lead to non-locality and non-unitarity because ϕ does not anti-commute with ϕ † .Similarly, given a pseudo Hermitian interacting potential, we cannot add a Hermitian conjugate to it.Therefore, the kinematics and dynamics of scalar fermions can only be described by ¬ ϕ and ϕ.The theory is pseudo Hermitian and non-trivial.This is because ¬ ϕ is the pseudo Hermitian conjugate of ϕ in the sense that there exists an η such that [11 To find η, we expand (2.8) using (2.1-2.2) to obtain Equation (2.9) is equivalent to where and we have demanded that the vacuum to be invariant under the action of η.We note, if the scalar field is taken to be real, then there is no non-trivial η.This tells us that a consistent theory of scalar fermion cannot be formulated in terms of real scalar fields.Doing so inevitably leads to non-locality and non-unitarity.Equation (2.10) can now be solved by the ansatz η = exp(iθχ) where χ ≡ d 3 p[b † (p)b(p)] and θ ∈ R is a phase to be determined.Acting η on the particle and anti-particle state using the ansatz, we obtain η † |a⟩ = |a⟩ and η|b⟩ = e iθ |b⟩.Choosing θ = −π, we obtain [22] so the operator is unitary.Next, act η on an arbitrary state |α⟩ successively, we obtain where n α is the number of anti-particle states in |α⟩.Therefore, η2 = η † η = I so η is Hermitian.Using (2.8) and η(t) = e iH 0 t ηe −iH 0 t = η, the free Lagrangian density is pseudo Hermitian.Similarly, interactions of the form ¬ ϕOϕ are Lorentz-invariant and pseudo Hermitian provided that O transforms as a scalar and is pseudo Hermitian.
Next, we examine the symmetry of the free Lagrangian density (2.6).Apart from the global U(1) symmetry, the Lagrangian density also has a global symplectic symmetry Sp(2, C) [11].Since {ϕ(x), where Therefore, the Lagrangian density is invariant under the symplectic transformations 2 where M is a 2 × 2 complex matrix satisfying with M T being the transposition of M .Equation (2.16) is satisfied for all complex matrices of unit determinant.Therefore, they are continuous transformations and can be generated via M = e X .Expand M near the identity we obtain X T Ω = −ΩX.
(2.18) Solving (2.18), we find the general solution Substituting (2.19) into M , the generators are given by from which we obtain Under the infinitesimal transformation the conserved currents are given by Due to fermionic statistics, there is an ambiguity in the ordering of ¬ ϕ and ϕ.To deal with this issue, we introduce the operation N to order ¬ ϕ to the left of ϕ.The results are (2.30) To couple the currents to gauge fields, they must be pseudo Hermitian.Given an arbitrary operator O, it can be made pseudo Hermitian by the linear combination e iϑ O + e −iϑ O # where ϑ ∈ R. Applying this procedure to the above currents, we find for ϑ i ∈ R. We find the following currents, namely, J µ 1 , J µ 3 , and J µ 4 to be linearly-dependent on J µ 2 , J µ 5 , and J µ 6 respectively.Therefore, there are three linearly-independent currents (2.39) The corresponding generators are Choosing the phases to be the resulting generators satisfy the su(2) algebra (2.42) The currents become and the conserved charges are given by (2.48) Substituting (2.1-2.2) into (2.46-2.48),we obtain All three charges are pseudo Hermitian.Additionally, Q 1 and iQ 2,3 are Hermitian, satisfying the su(2) algebra (2.54) After normal ordering Q 1 , it defines the charges of the particle and anti-particle, namely, For Q 2,3 , they map the particle state to the anti-particle state and vice versa3 By demanding the currents to be pseudo Hermitian, the global Sp(2, C) symmetry becomes SU (2).Therefore, the Lagrangian density, including interacting potentials constructed from ¬ ϕ and ϕ has a global SU(2) × U(1) symmetry.Because gauge group is semi-simple and compact, by treating ¬ ϕ and ϕ as doublet (2.14), we can couple them to non-Abelian gauge fields that resemble the electroweak sector of the SM.However, we should also note that the doublet structure presented here is fundamentally different from the SM fermionic doublet because it is bosonic and contains only of one specie of particle.It would be interesting to see if the symmetry is preserved in presence of quantum corrections.We leave this task for future investigations.

Generalized unitary evolution
Interacting Hamiltonians constructed as functions of ¬ ϕ and ϕ are pseudo Hermitian and hence complex.As a result, one may suspect the resulting S-matrix to be non-unitary.This concern also apply to the theory of spin-half bosons of mass dimension three-half and it was fully addressed in [13].There, the formalism to compute scattering amplitudes, physical observables associated pseudo Hermitian Hamiltonians were derived.The resulting S-matrix naturally satisfy the generalized unitarity relation.Using this formalism, we will show that the LN-model of N quartic self-interacting scalar fermions [11] admits generalized unitary evolution up to one-loop in perturbation theory.
Let us first present the relevant results in [13].For a given process α → β, the S-matrix and its adjoint have the following expansions where |α⟩ and |β⟩ are free states and η βα = ⟨β|η|α⟩.The S-matrix and its adjoint satisfy the generalized unitarity relation Normalizing the S-matrix as we obtain the generalized optical theorem Setting γ = α, we obtain Physical observables are computed by defining the transition probability to be proportional to ℘ βα M # αβ M βα .Since M # αβ M βα is not guaranteed to be positive-definite, we multiply it by a factor ℘ βα to maintain positivity.Using the definition of pseudo Hermitian conjugation and the solution of η, we find where n α and n β is the number of anti-particles in state |α⟩ and |β⟩ respectively.Therefore, by choosing the factor to be ℘ βα = e iπ(nα−n β ) , we obtain ℘ βα M # αβ M βα = M † αβ M βα so the standard formalism to compute cross-sections and decay rates can be readily applied.

The LN-model
We now consider the LN-model [11] where all the fields have equal mass and anti-commute with each other Due to fermionic statistics, The interacting density simplifies to The model is pseudo Hermitian because there exists an η given by such that L # = L .From (3.11), the following two-body scattering processes are allowed where i, j and ī, j denote particle and anti-particle states created by the ith and jth fields respectively.We will now verify that the S-matrix for these processes satisfy the generalized unitarity relation up to one-loop.For all three processes, as there are equal number of particles and anti-particles in the initial and final states, we have M # αβ = M † αβ .Therefore, the optical theorem (3.7) can be rewritten in terms of two-body cross sections where and In the center of mass frame p 1 = (E, p) and p 2 = (E, −p), so that u α = 4|p|/E CM with E CM = 2E.The cross-section simplifies to [3, sec.3.4] where dΩ is the differential solid angle of the final particle states.For these processes, the matrix elements of η βα are given by so we obtain where we have used |p| = 1 − 4m 2 /E 2 CM 1/2 .At tree-level, the cross-sections are given by (3.27) Substituting the cross-sections into the right-hand side of (3.23-3.24)and compare them with the imaginary part of the amplitudes given by (A.15-A.17),we find that the generalized optical theorem is satisfied.

Conclusions
According to the conventional spin-statistics theorem, scalar fields must furnish bosonic statistics.When phrased as a no-go theorem, it means that anti-commuting scalar fields violate locality and unitarity.The theory of scalar fermion constructed by LN showed that once the Hamiltonian is allowed to be pseudo Hermitian, the no-go theorem no longer applies.We believe this construct to be of fundamental importance because it represents an extension to the spin-statistics theorem.We have shown in this paper, not only does the theory respect Lorentz symmetry, it also admits generalized unitary evolution.It would be gratifying, if scalar fermion is to find applications in elementary particles physics.We shall now list, what is in our opinion, topics worthy of future investigations.An interesting feature of scalar fermion is that when the currents are pseudo Hermitian, the Lagrangian density has a global SU(2) × U(1) symmetry.This allows us to investigate their interactions with non-Abelian gauge fields having similar structure to the electroweak sector of the SM.Of course, this similarity may be purely coincidental as the bosonic doublet considered here is fundamentally different from the lepton doublet in the SM and cannot be associated with the notion of weak isopin and hypercharge.Regardless of its relevance to the SM, whether the theory is free of quantum anomalies should to be ascertained.If there are anomalies, one can attempt to look for possible ways to cancel them.Should anomalies be absent or can be cancelled, then in addition to the U(1) charge, it would be interesting to study how the symplectic charges contribute to the partition function at finite-temperature.
Besides possible gauge and gravitational interactions with the SM particles, interactions between the SM fermions and the scalar fermions are limited.As the scalar fermionic fields are complex, direct interactions with Dirac fermions would be of the form ψψ ¬ ϕϕ which has mass dimension five and is therefore suppressed.If we consider the LN model as an extension to the SM, it would potentially be a model of self-interacting fermionic dark matter of spin-zero.
Concerning the quartic self-interaction involving N scalar fermions, LeClair and Neubert have shown that when N > 4, the theory exhibits asymptotic freedom [11].This feature is reminiscent of the Gross-Neveu model for quartic self-interacting fermions in two space-time dimensions [23].Except here, the scalar fermions have renormalizable quartic self-interaction in four space-time dimensions.Therefore, it would be interesting to determine the behaviour of the β function for scalar fermions in the large N limit and investigate the possibility of dynamical mass generation as in the case of Gross-Neveu model.

A Amplitudes
We compute the one-loop amplitudes and their imaginary part that are relevant for verifying the generalized optical theorem in sec.3.Because of the fermionic statistics and the nontrivial adjoint, caution must be exercised when contracting the fields among themselves and with the states.The amplitudes are given by

AcknowledgmentsI
am grateful to Dharam Vir Ahluwalia for constant discussions and encouragements.I would like to thank James Brister, Ali Mostafazadeh, Zheng Sun, and Cong Zhang discussions.This work supported by The Sichuan University Post-doctoral Research Fund No. 2022SCU12119.