_{1}

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We study the quantization of systems with local particle-ghost symmetries. The systems contain ordinary particles including gauge bosons and their counterparts obeying different statistics. The particle-ghost symmetries are new type of fermionic symmetries between ordinary particles and their ghost partners, different from the space-time supersymmetry and the BRST symmetry. There is a possibility that they are useful to explain phenomena of elementary particles at a more fundamental level, by extension of our systems. We show that our systems are formulated consistently or subsidiary conditions on states guarantee the unitarity of systems, as the first step towards the construction of a realistic fundamental theory.

Graded Lie algebras or Lie superalgebras have been frequently used to formulate theories and construct models in particle physics. Typical examples are supersymmetry (SUSY) [

The space-time SUSY [

The BRST symmetry is a symmetry concerning unphysical modes in gauge fields and abnormal fields called Faddeev-Popov ghost fields [

Recently, models that contain both ordinary particles with a positive norm and their counterparts obeying different statistics have been constructed and those features have been studied [

The particle-ghost symmetries have been introduced as global symmetries, but we do not need to restrict them to the global ones. Rather, it would be meaningful to examine systems with local particle-ghost symmetries from following reasons. It is known that any global continuous symmetries can be broken down by the effect of quantum gravity such as a wormhole [

In this paper, we study the quantization of systems with local particle-ghost symmetries. The systems contain ordinary particles including gauge bosons and their counterparts obeying different statistics. We show that our systems are formulated consistently or subsidiary conditions on states guarantee the unitarity of systems, as the first step towards the construction of a realistic fundamental theory. The conditions can be originated from constraints in case that gauge fields have no dynamical degrees of freedom.

The contents of this paper are as follows. We construct models with local fermionic symmetries in Section 2, and carry out the quantization of the system containing scalar and gauge fields in Section 3. Section 4 is devoted to conclusions and discussions on applications of particle-ghost symmetries. In the Appendix Section, we study the system that gauge fields are auxiliary ones.

Recently, the system described by the following Lagrangian density has been studied [

where

Starting from (1), the model with the local

where

The

and the local fermionic transformations,

where

The

respectively. In

respectively. In

The

where

Under the transformations (4)-(7), the field strengths are transformed as

Using the global fermionic transformations,

where

For spinor fields, we consider the Lagrangian density,

where

Starting from (24), the Lagrangian density with local symmetries is constructed as

where

where

respectively.

The

and the local fermionic transformations,

where

Using the global fermionic transformations,

where

We carry out the quantization of the system with scalar and gauge fields described by

Based on the formulation with the property that the hermitian conjugate of canonical momentum for a variable is just the canonical momentum for the hermitian conjugate of the variable [

where

Using the Legendre transformation, the Hamiltonian density is obtained as

where Roman indices i and j denote the spatial components and run from 1 to 3,

Secondary constraints are obtained as follows,

where H is the Hamiltonian

system with canonical variables

where

We take the gauge fixing conditions,

The system is quantized by regarding variables as operators and imposing the following relations on the canonical pairs,

where

On the reduced phase space, the conserved U(1) charges

The following algebraic relations hold:

The above charges are generators of global U(1) and fermionic transformations such that

where

in (20) and (21) are related to

The system contains negative norm states originated from

In the Appendix, we point out that subsidiary conditions corresponding to (62) can be realized as remnants of local symmetries in a specific case.

Let us study the unitarity of physical S matrix in our system, using the Lagrangian density of free fields,

where the gauge fixing conditions (49) are imposed on. The

From (63), free field equations for

where

In the same way, by solving the free Maxwell equations, we obtain the solutions,

where

The index

By imposing the same type of relations as (50) - (55), we have the relations,

and others are zero.

The states in the Fock space are constructed by acting the creation operators

We impose the following subsidiary conditions on states to select physical states,

Note that

where

where

From (88), we find that any state with

The system is also formulated using hermitian fermionic charges defined by _{1}, Q_{2} and

It is also understood that our fermionic symmetries are different from the space-time SUSY, from the fact that Q_{1} and Q_{2} are scalar charges. They are also different from the BRST symmetry, as seen from the algebraic relations among charges.

The system with spinor and gauge fields described by

Our system has local symmetries, and it is quantized by the Faddeev-Popov (FP) method. In order to add the gauge fixing conditions to the Lagrangian, several fields corresponding to FP ghost and anti-ghost fields and auxiliary fields called Nakanishi-Lautrup (NL) fields are introduced. Then, the system is described on the extended phase space and has a global symmetry called the BRST symmetry. We present the gauge-fixed Lagrangian density and study the BRST transformation properties.

According to the usual procedure, the Lagrangian density containing the gauge fixing terms and FP ghost terms is constructed as

where

The

where the transformations for

The sum of the gauge fixing terms and FP ghost terms is simply written as

According to the Noether procedure, the BRST current

and

respectively. Here we use the field equations. The BRST charge is a conserved charge (

By imposing the following subsidiary condition on states,

it is shown that any negative norm states originated from time and longitudinal components of gauge fields as well as FP ghost and anti-ghost fields and NL fields do not appear on the physical subspace, through the quartet mechanism. There still exist negative norm states come from

We have studied the quantization of systems with local particle-ghost symmetries. The systems contain ordinary particles including gauge bosons and their counterparts obeying different statistics. There exist negative norm states come from fermionic scalar fields (or bosonic spinor fields) and transverse components of fermionic gauge fields, even after reducing the phase space due to the first class constraints and the gauge fixing conditions or imposing the subsidiary condition concerning the BRST charge on states. By imposing additional subsidiary conditions on states, such negative norm states are projected out on the physical subspace and the unitarity of systems hold. The additional conditions can be originated from constraints in case that gauge fields have no dynamical degrees of freedom.

The systems considered are unrealistic if this goes on, because they are empty leaving the vacuum state alone as the physical state. Then, one might think that it is better not to get deeply involved them. Although they are still up in the air at present, there is a possibility that a formalism or concept itself is basically correct and is useful to explain phenomena of elementary particles at a more fundamental level. It is necessary to fully understand features of our particle-ghost symmetries, in order to appropriately apply them on a more microscopic system.

We make conjectures on some applications. We suppose that particle-ghost symmetries exist and the system contains only a few states including the vacuum as physical states at an ultimate level. Most physical particles might be released from unphysical doublets that consist of particles and their ghost partners. A release mechanism has been proposed based on the dimensional reduction by orbifolding [

After the appearance of physical fields, _{F}-singlets, Q_{F}-doublets and interactions between Q_{F}-singlets and Q_{F}-doublets. Under the subsidiary conditions_{F} doublets, the theory is free from the gauge hierarchy problem if all heavy fields form Q_{F} doublets [

The system seems to be same as that described by _{F}-doublets. However, in a very special case, an indirect proof would be possible through fingerprints left by symmetries in a fundamental theory. The fingerprints are specific relations among parameters such as a unification of coupling constants, reflecting on underlying symmetries [

In most cases, our ghost fields require non-local interactions [

The author thanks the Editor and the referee for their comments. This work was supported in part by scientific grants from the Ministry of Education, Culture, Sports, Science and Technology under Grant No. 22540272.

YoshiharuKawamura, (2015) Local Particle-Ghost Symmetry. Journal of Modern Physics,06,1721-1736. doi: 10.4236/jmp.2015.612174

Let us study the system without

In this case, gauge fields do not have any dynamical degrees of freedom, and are regarded as auxiliary fields. The conjugate momenta of

Using the Legendre transformation, the Hamiltonian density is obtained as

where

Secondary constraints are obtained as

where

In the same way, tertiary constraints are obtained as

from the invariance under the time evolution of

the conditions

thermore, new constraints do not appear from the conditions

The constraints are classified into the first class ones

and the second class ones

The determinant of Poisson bracket between second class ones does not vanish on constraints.

Using

The same algebraic relations hold as those in (60).

The above charges are conserved and generators of global U(1) and fermionic transformations for scalar fields. They satisfy the relations,

where

After taking the following gauge fixing conditions for the first class ones (113),

the system is quantized by regarding variables as operators and imposing the same type of relations (50) and (51) on the canonical pairs. From (120), it is reasonable to impose the following subsidiary conditions on states,

Then, they guarantee the unitarity of our system, though it contains negative norm states originated from