Local particle-ghost symmetry

arXiv:1505.05242v1 [hep-th] 20 May 2015

Local particle-ghost symmetry Yoshiharu K AWAMURA∗ Department of Physics, Shinshu University, Matsumoto 390-8621, Japan May 20, 2015

Abstract 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 symmetry is a kind of fermionic symmetry, different from the space-time supersymmetry and the BRST symmetry. Subsidiary conditions on states guarantee the unitarity of systems.

1 Introduction Graded Lie algebras or Lie superalgebras have been frequently used to formulate theories and construct models in particle physics. Typical examples are supersymmetry (SUSY) [1, 2, 3, 4] and BRST symmetry [5, 6, 7]. The space-time SUSY [8, 9] is a symmetry between ordinary particles with integer spin and those with half-integer spin, and the generators called supercharges are spacetime spinors that obey the anti-commutation relations [10, 11]. The BRST symmetry is a symmetry concerning unphysical modes in gauge fields and abnormal fields called Faddeev-Popov ghost fields [12]. Though both gauge fields and abnormal fields contain negative norm states, theories become unitary on the physical subspace, thanks to the BRST invariance [13, 14]. The BRST and anti-BRST charges are anti-commuting space-time scalars. 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 [15, 16, 17, 18]. Models have fermionic symmetries different from the space-time SUSY and the BRST symmetry. We refer to this type of novel symmetries as “particle-ghost symmetries”. 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 ∗

E-mail: [email protected]

1

such as a wormhole [19]. Then, it is expected that an fundamental theory possesses local symmetries, and global continuous symmetries can appear as accidental ones in lowerenergy scale. In the system with global particle-ghost symmetries, the unitarity holds by imposing subsidiary conditions on states by hand. In contrast, there is a possibility that the conditions are realized as remnants of local symmetries in a specific situation. 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. Subsidiary conditions on states guarantee the unitarity of systems. 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 Sect. II, and carry out the quantization of the system containing scalar and gauge fields in Sect. III. Section IV is devoted to conclusions and discussions. In appendix A, we study the system that gauge fields are auxiliary ones.

2 Systems with local fermionic symmetries 2.1 Scalar fields with local fermionic symmetries Recently, the system described by the following Lagrangian density has been studied [15, 16, 17, 18], L ϕ,cϕ = ∂µ ϕ† ∂µ ϕ − m 2 ϕ† ϕ + ∂µ cϕ† ∂µ cϕ − m 2 cϕ† cϕ ,

(1)

where ϕ is an ordinary complex scalar field and cϕ is the fermionic counterpart obeying the anti-commutation relations. The system has a global OSp(2|2) symmetry that consists of U (1) and fermionic symmetries. The unitarity holds by imposing suitable subsidiary conditions relating the conserved charges on states. Starting from (1), the model with the local OSp(2|2) symmetry is constructed by introducing gauge fields. The resultant Lagrangian density is given by L = LM + LG , © ª© ª L M = (∂µ − i g A µ − i g B µ )ϕ† − g C µ−cϕ† (∂µ + i g A µ + i g B µ )ϕ + g C +µ cϕ © ª© ª + (∂µ − i g A µ + i g B µ )cϕ† − g C µ+ ϕ† (∂µ + i g A µ − i g B µ )cϕ − g C −µϕ

− m 2 ϕ† ϕ − m 2 cϕ† cϕ , © ª© ª L G = − ∂µ A ν − ∂ν A µ + i g (C µ+C ν− −C ν+C µ− ) ∂µ B ν − ∂ν B µ ª 1© − ∂µC ν+ − ∂νC µ+ + 2i g (B µC ν+ − B νC µ+ ) 2 © ª · ∂µC −ν − ∂νC −µ − 2i g (B µC −ν − B νC −µ ) ,

(2)

(3)

where A µ and B µ are the gauge fields relating the (diagonal) U (1) symmetries, C µ+ and C µ− are gauge fields relating the fermionic symmetries, and g is the gauge coupling constant. The quantized fields of C µ± obey the anti-commutation relations.

2

The L is invariant under the local U (1) transformations, δ A ϕ = −i ǫϕ, δ A ϕ† = i ǫϕ† , δ A cϕ = −i ǫcϕ , δ A cϕ† = i ǫcϕ† , 1 δ A A µ = ∂µ ǫ, δ A B µ = 0, δ A C µ+ = 0, δ A C µ− = 0, g

(4)

δB ϕ = −i ξϕ, δB ϕ† = i ξϕ† , δB cϕ = i ξcϕ , δB cϕ† = −i ξcϕ† , 1 δB A µ = 0, δB B µ = ∂µ ξ, δB C µ+ = −2i ξC µ+ , δB C µ− = 2i ξC µ− g

(5)

and the local fermionic transformations, δF ϕ = −ζcϕ , δF ϕ† = 0, δF cϕ = 0, δF cϕ† = ζϕ† , 1 δF A µ = −i ζC µ− , δF B µ = 0, δFC µ+ = 2i ζB µ + ∂µ ζ, δFC µ− = 0, g

(6)

δ†F ϕ = 0, δ†F ϕ† = ζ† cϕ† , δ†F cϕ = ζ† ϕ, δ†F cϕ† = 0,

1 δ†F A µ = −i ζ†C µ+ , δ†F B µ = 0, δ†FC µ+ = 0, δ†FC µ− = −2i ζ† B µ + ∂µ ζ† , g

(7)

where ǫ and ξ are infinitesimal real functions of x, and ζ and ζ† are Grassmann-valued functions of x. The L M and L G are simply written as 1 L M = (D µ Φ)† (D µ Φ) − m 2 Φ† Φ and L G = − Str(Fµν F µν ), 4

(8)

respectively. In L M , D µ and Φ are the covariant derivative and the doublet of fermionic transformation defined by Dµ ≡

µ

∂µ + i g A µ + i g B µ g C µ+ −g C µ− ∂µ + i g A µ − i g B µ

¶

and Φ ≡

µ

ϕ cϕ

¶

,

(9)

respectively. In L G , Str is the supertrace defined by StrM = a − d where M is the 2 × 2 matrix given by µ ¶ a b M= . (10) c d The Fµν is defined by 1 Fµν ≡ [D µ , D ν ] = ig

µ

A µν + B µν − iC µν

+ −iC µν A µν − B µν

¶

,

(11)

A µν = ∂µ A ν − ∂ν A µ + i g (C µ+C ν− −C ν+C µ− ),

(12)

+ C µν = ∂µC ν+ − ∂νC µ+ + 2i g (B µC ν+ − B νC µ+),

(14)

+ − where A µν , B µν , C µν and C µν are the field strengths given by

B µν = ∂µ B ν − ∂ν B µ ,

3

(13)

− C µν = ∂µC ν− − ∂νC µ− − 2i g (B µC ν− − B νC µ−).

(15)

Under the transformations (4) – (7), the field strengths are transformed as + − δ A A µν = 0, δ A B µν = 0, δ A C µν = 0, δ A C µν = 0, + δB C µν

+ − − δB A µν = 0, δB B µν = 0, = −2i ξC µν , δB C µν = 2i ξC µν , − + − = 0, δF A µν = −i ζC µν , δF B µν = 0, δFC µν = 2i ζB µν , δFC µν − + + = −2i ζ† B µν . = 0, δ†FC µν , δ†F B µν = 0, δ†FC µν δ†F A µν = −i ζ†C µν

(16) (17) (18) (19)

Using the global fermionic transformations,

δ˜F ϕ = −cϕ , δ˜F ϕ† = 0, δ˜F cϕ = 0, δ˜F cϕ† = ϕ† ,

δ˜F A µ = −iC µ− , δ˜F B µ = 0, δ˜FC µ+ = 2i B µ , δ˜FC µ− = 0,

(20)

δ˜F† A µ = −iC µ+ , δ˜F† B µ = 0, δ˜F†C µ+ = 0, δ˜F†C µ− = −2i B µ ,

(21)

δ˜F† ϕ = 0, δ˜F† ϕ† = cϕ† , δ˜F† cϕ = ϕ, δ˜F† cϕ† = 0,

L is rewritten as L = δ˜F δ˜F† L ϕ,A = −δ˜F† δ˜F L ϕ,A ,

(22)

where L ϕ,A is given by © ª© ª L ϕ,A = (∂µ − i g A µ − i g B µ )ϕ† − g C µ− cϕ† (∂µ + i g A µ + i g B µ )ϕ + g C +µ cϕ 1 (23) − m 2 ϕ† ϕ − A µν A µν . 4

2.2 Spinor fields with local fermionic symmetries For spinor fields, we consider the Lagrangian density, L ψ,cψ = i ψγµ ∂µ ψ − mψψ + i c ψ γµ ∂µ cψ − mc ψ cψ ,

(24)

where ψ is an ordinary spinor field and cψ is its bosonic counterpart obeying commutation relations. This system also has global U (1) and fermionic symmetries, and the unitarity holds by imposing suitable subsidiary conditions on states. Starting from (24), the Lagrangian density with local symmetries is constructed as sp

L sp = L M + L G , © ª sp L M = i ψγµ (∂µ + i g A µ + i g B µ )ψ + g C µ+ cψ − mψψ © ª + i c ψ γµ (∂µ + i g A µ − i g B µ )cψ − g C µ−ψ − mc ψ cψ ,

(25)

† 0 γ and γµ are the γ matrices satisfying where L G is given by (3), ψ ≡ ψ† γ0 , c ψ ≡ cψ sp {γµ , γν } = 2ηµν . The L M is rewritten as sp

L M = i ΨΓµ D µ Ψ − mΨΨ, 4

(26)

where Γµ and Ψ are the extension of γ-matrices and the doublet of fermionic transformation defined by ¶ µ ¶ µ µ ψ γ 0 µ and Ψ ≡ , (27) Γ ≡ cψ 0 γµ respectively. The L sp is invariant under the local U (1) transformations, † † δ A ψ = −i ǫψ, δ A ψ† = i ǫψ† , δ A cψ = −i ǫcψ , δ A cψ = i ǫcψ , 1 δ A A µ = ∂µ ǫ, δ A B µ = 0, δ A C µ+ = 0, δ A C µ− = 0, g † † δB ψ = −i ξψ, δB ψ† = i ξψ† , δB cψ = i ξcψ , δB cψ = −i ξcψ , 1 δB A µ = 0, δB B µ = ∂µ ξ, δB C µ+ = −2i ξC µ+ , δB C µ− = 2i ξC µ− g

(28)

(29)

and the local fermionic transformations, † δF ψ = −ζcψ , δF ψ† = 0, δF cψ = 0, δF cψ = −ζψ† , 1 δF A µ = −i ζC µ− , δF B µ = 0, δFC µ+ = 2i ζB µ + ∂µ ζ, δFC µ− = 0, g

(30)

† † δ†F ψ = 0, δ†F ψ† = −ζ† cψ , δ†F cψ = ζ† ψ, δ†F cψ = 0,

1 δ†F A µ = −i ζ†C µ+ , δ†F B µ = 0, δ†FC µ+ = 0, δ†FC µ− = −2i ζ† B µ + ∂µ ζ† , g

(31)

where ǫ and ξ are infinitesimal real functions of x, and ζ and ζ† are Grassmann-valued functions of x. Using the global fermionic transformations, † δ˜F ψ = −cψ , δ˜F ψ† = 0, δ˜F cψ = 0, δ˜F cψ = −ψ† ,

δ˜F A µ = −iC µ− , δ˜F B µ = 0, δ˜FC µ+ = 2i B µ , δ˜FC µ− = 0,

(32)

δ˜F† A µ = −iC µ+ , δ˜F† B µ = 0, δ˜F†C µ+ = 0, δ˜F†C µ− = −2i B µ ,

(33)

† † δ˜F† ψ = 0, δ˜F† ψ† = −cψ , δ˜F† cψ = ψ, δ˜F† cψ = 0,

L sp is rewritten as L = δ˜F δ˜F† L ψ,A = −δ˜F† δ˜F L ψ,A ,

(34)

where L ψ,A is given by ª © 1 L ψ,A = i ψγµ (∂µ + i g q A µ + i g B µ )ψ + g C µ+ cψ − mψψ − A µν A µν . 4

(35)

3 Quantization We carry out the quantization of the system with scalar and gauge fields described by L = LM + LG . 5

3.1 Canonical quantization 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 [16], the conjugate momenta are given by ¶ µ ∂L = (∂0 − i g A 0 − i g B 0 )ϕ† − g C 0− cϕ† , (36) π≡ ∂ϕ˙ R µ ¶ ∂L † π ≡ = (∂0 + i g A 0 + i g B 0 )ϕ + g C 0+ cϕ , (37) † ∂ϕ˙ L ¶ µ ∂L = (∂0 − i g A 0 + i g B 0 )cϕ† − g C 0+ ϕ† , (38) πc ϕ ≡ ∂c˙ϕ R Ã ! ∂L † πc ϕ ≡ = (∂0 + i g A 0 − i g B 0 )cϕ − g C 0− ϕ, (39) † ∂c˙ϕ L Ã ! ! Ã ∂L ∂L µ µ µ0 = 2B , ΠB ≡ = 2A µ0 , (40) ΠA ≡ ˙ ˙ ∂ Aµ L ∂B µ R Ã Ã ! ! ∂L ∂L +µ −µ −µ0 ΠC ≡ =C , ΠC ≡ = C +µ0 , (41) + − ˙ ˙ ∂C µ ∂C µ L

R

where O˙ = ∂O /∂t , and R and L stand for the right-differentiation and the left-differentiation, respectively. From (40) and (41), we obtain the primary constraints, Π0A = 0, Π0B = 0, ΠC+0 = 0, ΠC−0 = 0.

(42)

Using the Legendre transformation, the Hamiltonian density is obtained as −µ +µ µ µ ˙ + ϕ˙ † π† + πcϕ c˙ϕ + c˙ϕ† π†cϕ + A˙ µ Π A + ΠB B˙µ + C˙µ+ ΠC + ΠC C˙µ− − L H = πϕ

+ λ A Π0A + Π0B λB + λC+ ΠC+0 + ΠC−0 λC−

= ππ† + πcϕ π†cϕ + (D i Φ)† (D i Φ) + m 2 Φ† Φ − i g A 0 (πϕ − ϕ† π† + πcϕ cϕ − cϕ† π†cϕ )

− i g B 0 (πϕ − ϕ† π† − πcϕ cϕ + cϕ† π†cϕ + 2C i+ΠC+i − 2ΠC−i C i− )

− g C 0+(πcϕ − ϕ† π†cϕ + iC i− ΠiA − 2i B i ΠC+i )

− g (cϕ† π† − πcϕ ϕ − iC i+ ΠiA + 2i B i ΠC−i )C 0−

+ Π Ai ΠiB + A i j B i j + ∂i A 0 ΠiA + ΠiB ∂i B 0 1 + ΠC+i ΠC−i + C i j C i j + ∂i C 0+ ΠC+i + ΠC−i ∂i C 0− 2 + λ A Π0A + Π0B λB + λC+ ΠC+0 + ΠC−0 λC− ,

(43)

where Roman indices i and j denote the spatial components and run from 1 to 3, λ A , λB , λC+ and λC− are Lagrange multipliers, and A˙ 0 + λ A , B˙0 + λB , C˙0+ + λC+ and C˙0− + λC− are rewritten as λ A , λB , λC+ and λC− in the final expression. 6

Secondary constraints are obtained as follows, d Π0A dt d Π0B

³ ´ © 0 ª † † † † = Π A , H PB = i g πϕ − ϕ π + πcϕ cϕ − cϕ πcϕ + ∂i ΠiA = 0,

³ © ª = Π0B , H PB = i g πϕ − ϕ† π† − πcϕ cϕ + cϕ† π†cϕ dt ´ +2C i+ ΠC+i − 2ΠC−i C i− + ∂i ΠiB = 0,

d ΠC+0 dt d ΠC−0 dt

´ ³ © ª = ΠC+0 , H PB = g πcϕ − ϕ† π†cϕ + iC i− ΠiA − 2i B i ΠC+i + ∂i ΠC+i = 0, ´ ³ © ª = ΠC−0 , H PB = g cϕ† π† − πcϕ ϕ − iC i+ ΠiA + 2i B i ΠC−i + ∂i ΠC−i = 0,

(44)

(45) (46) (47)

R where H is the Hamiltonian H = H d 3 x and {A, B}PB is the Poisson bracket. The Poisson bracket for the system with canonical variables (Q k , P k ) and (Q k† , P k† ) is defined by [16] ©

·µ

µ ¶ ¶ µ ¶ ∂g ∂g |Q k | ∂ f f , g PB ≡ − (−) ∂P k R ∂Q k L R ∂P k L k à ! à ! à ! à ! # ∂ f ∂g ∂ f ∂g +(−)|Q k | − , ∂Q k† R ∂P k† L ∂P k† R ∂Q k† L ª

X

∂f ∂Q k

¶ µ

(48)

where |Q k | is the number representing the Grassmann parity of Q k , i.e., |Q k | = 1 for the Grassmann odd Q k and |Q k | = 0 for the Grassmann even Q k . There appear no other constraints, and all constraints are first class ones and generate local transformations. We take the gauge fixing conditions, A 0 = 0, B 0 = 0, C +0 = 0, C −0 = 0, ∂i A i = 0, ∂i B i = 0, ∂i C +i = 0, ∂i C −i = 0.

(49)

The system is quantized by regarding variables as operators and imposing the following relations on the canonical pairs, [ϕ(x, t ), π(y , t )] = i δ3 (x − y ), [ϕ† (x, t ), π† (y , t )] = i δ3 (x − y ), 3

{cϕ† (x, t ), π†cϕ (y , t )} = −i δ3 (x − y ),

{cϕ (x, t ), πcϕ (y , t )} = i δ (x − y ), ¶ µ ∂i ∂ j 3 j j δ (x − y ), [A i (x, t ), Π A (y , t )] = i δi − ∆ µ ¶ ∂i ∂ j 3 j j [B i (x, t ), ΠB (y , t )] = i δi − δ (x − y ), ∆ µ ¶ ∂i ∂ j 3 j +j + {C i (x, t ), ΠC (y , t )} = −i δi − δ (x − y ), ∆ µ ¶ ∂i ∂ j 3 j −j − {C i (x, t ), ΠC (y , t )} = i δi − δ (x − y ), ∆

(50) (51) (52) (53) (54) (55)

where [O 1 , O 2 ] ≡ O 1 O 2 − O 2 O 1 , {O 1 , O 2 } ≡ O 1 O 2 + O 2 O 1 , and only the non-vanishing ones are denoted. Here, we define the Dirac bracket using the first class constraints and 7

the gauge fixing conditions, and replace the bracket with the commutator or the anticommutator. On the reduced phase space, the conserved U (1) charges N A and NB and the conserved fermionic charges Q F and Q F† are constructed as Z

³ ´ d 3 x πϕ − ϕ† π† + πcϕ cϕ − cϕ† π†cϕ , Z ´ ³ NB = −i d 3 x πϕ − ϕ† π† − πcϕ cϕ + cϕ† π†cϕ + 2C i+ΠC+i − 2ΠC−i C i− , Z ´ ³ Q F = − d 3 x πcϕ − ϕ† π†cϕ + iC i− ΠiA − 2i B i ΠC+i , Z ´ ³ † Q F = − d 3 x cϕ† π† − πcϕ ϕ − iC i+ ΠiA + 2i B i ΠC−i . N A = −i

(56) (57) (58) (59)

The following algebraic relations hold: 2

Q F 2 = 0, Q F† = 0, {Q F ,Q F† } = N A , [N A ,Q F ] = 0, [N A ,Q F† ] = 0, [NB ,Q F ] = −2Q F , [NB ,Q F† ] = 2Q F† , [N A , NB ] = 0.

(60)

The above charges are generators of global U (1) and fermionic transformations such that δ˜ A O = i [ǫ0 N A , O ], δ˜B O = i [ξ0 NB , O ], δ˜F O = i [ζ0Q F , O ], δ˜†F O = i [Q F† ζ†0 , O ],

(61)

where ǫ0 and ξ0 are real parameters, and ζ0 and ζ†0 are Grassmann parameters. Note that δ˜F and δ˜F† in (20) and (21) are related to δ˜F and δ˜†F as δ˜F = ζ0 δ˜F , δ˜†F = ζ†0 δ˜F† . The system contains negative norm states originated from cϕ , cϕ† and C i± . In the presence of negative norm states, the probability interpretation cannot be endured. To formulate our model in a consistent manner, we use a feature that conserved charges can be, in general, set to be zero as subsidiary conditions. We impose the following subsidiary conditions on states by hand, N A |phys〉 = 0, NB |phys〉 = 0, Q F |phys〉 = 0, Q F† |phys〉 = 0.

(62)

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

3.2 Unitarity Let us study the unitarity of physical S matrix in our system, using the Lagrangian density of free fields, L 0 = ∂µ ϕ† ∂µ ϕ − m 2 ϕ† ϕ + ∂µ cϕ† ∂µ cϕ − m 2 cϕ† cϕ − 2∂µ A i ∂µ B i − ∂µC i+ ∂µC −i ,

(63)

where the gauge fixing conditions (49) are imposed on. The L 0 describes the behavior of asymptotic fields of Heisenberg operators in L = L M + L G .

8

From (63), free field equations for ϕ, ϕ† , cϕ , cϕ† , A i , B i and C i± are derived. By solving the Klein-Gordon equations, we obtain the solutions Z ³ ´ d 3k a(k)e −i kx + b † (k)e i kx , (64) ϕ(x) = p (2π)3 2k 0 Z ³ ´ d 3k † ϕ (x) = p a † (k)e i kx + b(k)e −i kx , (65) (2π)3 2k 0 s Z ´ k0 ³ † i kx −i kx a ( k )e − b( k )e , (66) π(x) = i d 3 k 2(2π)3 s Z ´ k0 ³ −i kx † i kx a( k )e − b ( k )e π† (x) = −i d 3 k , (67) 2(2π)3 Z ³ ´ d 3k c(k)e −i kx + d † (k)e i kx , (68) cϕ (x) = p (2π)3 2k 0 Z ³ ´ d 3k † † i kx −i kx cϕ (x) = p c (k)e + d (k)e , (69) (2π)3 2k 0 s Z ´ k0 ³ † i kx −i kx c ( k )e − d ( k )e , (70) πcϕ (x) = i d 3 k 2(2π)3 s Z ´ k0 ³ −i kx † i kx π†cϕ (x) = −i d 3 k c( k )e − d ( k )e , (71) 2(2π)3 p where k 0 = k2 + m 2 and kx = k µ xµ . In the same way, by solving the free Maxwell equations, we obtain the solutions, Z ³ ´ d 3k −i kx ∗α † i kx A i (x) = p εα a ( k )e + ε a ( k )e , (72) α α i i (2π)3 2k 0 Z ³ ´ d 3k α −i kx ∗α † i kx B i (x) = p εi b α (k)e + εi b α (k)e , (73) (2π)3 2k 0 Z ³ ´ d 3k −i kx ∗α † i kx εα c ( k )e + ε d ( k )e , (74) C i+(x) = p α i α i (2π)3 2k 0 Z ³ ´ d 3k − † i kx α −i kx C i (x) = p ε∗α c ( k )e + ε d ( k )e , (75) α i i α (2π)3 2k 0 s Z ´ k 0 ³ ∗α † i kx α −i kx ε b ( k )e − ε b ( k )e , (76) ΠiA (x) = 2i d 3 k α i 2(2π)3 i α s Z ´ k 0 ³ ∗α † i kx α −i kx ΠiB (x) = 2i d 3 k ε a ( k )e − ε a ( k )e , (77) i α 2(2π)3 i α s Z ´ k 0 ³ ∗α † i kx α −i kx ε c ( k )e − ε d ( k )e ΠC+i (x) = i d 3 k , (78) α i 2(2π)3 i α s Z ´ k0 ³ α −i kx ∗α † i kx ε c ( k )e − ε d ( k )e , (79) ΠC−i (x) = −i d 3 k α α i 2(2π)3 i 9

where k 0 = |k| and εα are polarization vectors satisfying the relations, i ′

α ∗i α k i εα = δαα′ , i = 0, εi ε

X α

j

∗jα εα = δi − i ε

ki k j

k2

.

(80)

The index α represents the helicity of gauge fields. By imposing the same type of relations as (50) – (55), we have the relations, [a(k), a † (l)] = δ3 (k − l), [b(k), b † (l)] = δ3 (k − l), †

3

†

3

{c(k), c (l)} = δ (k − l), {d (k), d (l)} = −δ (k − l), 1 1 † † 3 3 [aα (k), b α ′ (l)] = − δαα′ δ (k − l), [b α (k ), a α′ (l)] = − δαα′ δ (k − l), 2 2 {cα (k), cα† ′ (l)} = δαα′ δ3 (k − l), {d α (k), d α† (l)} = −δαα′ δ3 (k − l),

(81) (82) (83) (84)

and others are zero. The states in the Fock space are constructed by acting the creation operators a † (k), † † b † (k), c † (k), d † (k), aα (k), b α (k), cα† (k) and d α† (k) on the vacuum state |0〉, where |0〉 is defined by the conditions a(k)|0〉 = 0, b(k)|0〉 = 0, c(k)|0〉 = 0, d (k)|0〉 = 0, aα (k)|0〉 = 0, b α (k)|0〉 = 0, cα (k)|0〉 = 0 and d α (k)|0〉 = 0. We impose the following subsidiary conditions on states to select physical states, N A |phys〉 = 0, NB |phys〉 = 0, Q F |phys〉 = 0, Q F† |phys〉 = 0.

(85)

Note that Q F† |phys〉 = 0 means 〈phys|Q F = 0. We find that all states, except for the vacuum state, are unphysical because they do not satisfy (85). This feature is understood as a counterpart of the quartet mechanism [13, 14]. The projection operator P (n) on the states with n particles is given by P (n) =

1 n † (n−1) a P a + b † P (n−1) b + c † P (n−1) c − d † P (n−1) d n ´¾ X³ † (n−1) † (n−1) † (n−1) † (n−1) + −2aα P b α − 2b α P aα + cα P cα − dα P dα ,

(86)

α

where n ≥ 1 and we omit k, for simplicity. Using the transformation properties,

δ˜F a = −c, δ˜F a † = 0, δ˜F b = 0, δ˜F b † = −d † , δ˜F c = 0, δ˜F c † = a † , δ˜F d = b, δ˜F d † = 0, δ˜F aα = −i d α , δ˜F aα† = −i cα† , δ˜F bα = 0, δ˜F bα† = 0, δ˜F cα = 2i bα , δ˜F cα† = 0, δ˜F d α = 0, δ˜F d α† = 2i bα† ,

(87)

P (n) is written in a simple form as © ª P (n) = i Q F , R (n) ,

where R (n) is given by ½ ´¾ X ³ † (n−1) 1 † (n−1) † (n−1) † (n−1) (n) c P a +b P d +i aα P cα + dα P aα . R = n α 10

(88)

(89)

From (88), we find that any state with n ≥ 1 is unphysical from 〈phys|P (n) |phys〉 = 0. Then, we understand that every field becomes unphysical, and only |0〉 remains as the physical state. This is also regarded as a field theoretical version of the Parisi-Sourlas mechanism [20]. The system is also formulated using hermitian fermionic charges defined by Q 1 ≡ Q F + Q F† and Q 2 ≡ i (Q F − Q F† ). They satisfy the relations Q 1Q 2 + Q 2Q 1 = 0, Q 1 2 = N A and Q 2 2 = N A . Though Q 1 , Q 2 and N A form elements of the N = 2 (quantum mechanical) SUSY algebra [21], our system does not possess the space-time SUSY because N A is not our Hamiltonian but the U (1) charge N A . Only the vacuum state is selected as the physical states by imposing the following subsidiary conditions on states, in place of (85), N A |phys〉 = 0, NB |phys〉 = 0, Q 1 |phys〉 = 0, Q 2 |phys〉 = 0.

(90)

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. sp The system with spinor and gauge fields described by L sp = L M + L G is also quantized, in a similar way. We find that the theory becomes harmless but empty leaving the vacuum state alone as the physical state, after imposing subsidiary conditions corresponding to (62).

3.3 BRST symmetry 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 L T = L M + L G + L gf + L FP ,

L gf = −∂µ b A A µ − ∂µ b B B µ +C +µ ∂µ φc + ∂µ φ†c C −µ 1 + α(b 2A + b B2 + 2φ†c φc ), 2 L FP = −i ∂µ c A (∂µ c A − i g φC −µ + i g φ†C +µ ) − i ∂µ c B ∂µ cB

(91)

+ i (∂µ φ − 2i g cB C +µ + 2i g φB µ )∂µ φ †

− i ∂µ φ (∂µ φ† − 2i g cB C −µ − 2i g φ† B µ ),

(92)

†

where c A , cB , φ and φ† are FP ghosts, c A , c B , φ and φ are FP anti-ghosts, b A , b B , φc and φ†c are NL fields, and α is a gauge parameter. These fields are scalar fields. c A , cB , c A and †

c B are fermionic, and b A and b B are bosonic. In contrast, φ, φ† , φ and φ are bosonic, and φc and φ†c are fermionic because the relevant symmetries are fermionic. 11

The L T is invariant under the BRST transformation,

δBRSTϕ = −i g c A ϕ − i g cB ϕ − g φcϕ , δBRST ϕ† = i g c A ϕ† + i g cB ϕ† − g φ† cϕ† ,

δBRSTcϕ = −i g c A cϕ + i g cB cϕ − g φ† ϕ, δBRST cϕ† = i g c A cϕ† − i g cB cϕ† + g φϕ† ,

δBRSTc A = −i g φ† φ, δBRST cB = 0, δBRSTφ = −2i g cB φ, δBRST φ† = 2i g cB φ† , δBRST A µ = ∂µ c A − i g φC µ− + i g φ†C µ+, δBRSTB µ = ∂µ cB ,

δBRSTC µ+ = −2i g cB C µ+ + 2i g φB µ + ∂µ φ, δBC µ− = 2i g cB C µ− + 2i g φ† B µ − ∂µ φ† , †

δBRSTc A = i b A , δBRSTc B = i bB , δBRSTφ = i φc , δBRSTφ = −i φ†c , δBRSTb A = 0, δBRST bB = 0, δBRSTφc = 0, δBRSTφ†c = 0,

(93)

where the transformations for ϕ, ϕ† , cϕ , cϕ† , A µ , B µ , C µ+ and C µ− are obtained by regarding the sum of transformations δ A +δB +δF +δ†F as δBRST and replacing ǫ, ξ, ζ and ζ† with g c A , g cB , g φ and −g φ† , and those for c A , cB , φ and φ† are determined by the requirement that δBRST has a nilpotency property, i.e., δBRST2 O = 0. The sum of the gauge fixing terms and FP ghost terms is simply written as © † L gf + L FP = i δBRST ∂µ c A A µ + ∂µ c B B µ +C +µ ∂µ φ + ∂µ φ C −µ ª 1 † − α(c A b A + c B b B − φ†c φ − φ φc ) . 2

(94)

µ

According to the Noether procedure, the BRST current J BRST and the BRST charge Q BRST are obtained as µ

J BRST = b A (∂µ c A − i g φC −µ + i g φ†C +µ ) − c A ∂µ b A + b B ∂µ cB − cB ∂µ b B − φc (∂µ φ − 2i g cB C +µ + 2i g φB µ ) + φ∂µ φc

− φ†c (∂µ φ† − 2i g cB C −µ − 2i g φ† B µ ) + φ† ∂µ φ†c †

− 2g cB φ∂µ φ − 2g cB φ† ∂µ φ − g φ† φ∂µ c A

− 2∂ν (c A B µν ) − 2∂ν (cB A µν ) − ∂ν (φC −µν ) − ∂ν (φ†C +µν )

(95)

and Q BRST ≡

Z

3

0

d x J BRST =

Z

© d 3 x b A (∂0 c A − i g φC −0 + i g φ†C +0 ) − c A ∂0 b A

+ b B ∂0 cB − ∂0 b B cB − φc (∂0 φ − 2i g cB C +0 + 2i g φB 0 ) + φ∂0 φc

− φ†c (∂0 φ† − 2i g cB C −0 − 2i g φ† B 0 ) + φ† ∂0 φ†c ª † − 2g cB φ∂0 φ − 2g cB φ† ∂0 φ − g φ† φ∂0 c A ,

(96)

respectively. Here we use the field equations. The BRST charge is a conserved charge (dQ BRST /d t = 0), and it has the nilpotency property such as Q BRST2 = 0. By imposing the following subsidiary condition on states, Q BRST|phys〉 = 0, 12

(97)

it is shown that any negative 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 cϕ , cϕ† and C µ± , and it is necessary to impose additional conditions corresponding to (62) on states in order to project out such harmful states.

4 Conclusions and discussions 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, but 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 one 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 [17]. After the appearance of physical fields, Q F -singlets and Q F -doublets coexist with exact fermionic symmetries. The Lagrangian density is, in general, written in the form as L Total = L S + L D + L mix = L S + δ˜F δ˜F† (∆L ). Here, L S , L D and L mix stand for the Lagrangian density for Q F -singlets, Q F -doublets and interactions between Q F -singlets and Q F -doublets. Under the subsidiary conditions N A |phys〉 = 0, NB |phys〉 = 0, Q F |phys〉 = 0 and Q F† |phys〉 = 0 on states, all Q F -doublets become unphysical and would not give any physical effects on Q F singlets. Because Q F singlets would not receive any radiative corrections from Q F doublets, the theory is free from the gauge hierarchy problem if all heavy fields form Q F doublets [15]. The system seems to be same as that described by L S alone, and to be impossible to show the existence of Q 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 [15, 22]. 13

In most cases, our ghost fields require non-local interactions [15] and the change of degrees of freedom can occur in systems with infinite numbers of fields [17]. Then, they might suggest that fundamental objects are not point particles but extended objects such as strings and membranes. Hence, it would be interesting to explore systems with particle-ghost symmetries and their applications in the framework of string theories.1

Acknowledgments This work was supported in part by scientific grants from the Ministry of Education, Culture, Sports, Science and Technology under Grant No. 22540272.

A System with auxiliary gauge fields Let us study the system without L G described by © ª ª© L M = (∂µ − i g A µ − i g B µ )ϕ† − g C µ−cϕ† (∂µ + i g A µ + i g B µ )ϕ + g C +µ cϕ © ª© ª + (∂µ − i g A µ + i g B µ )cϕ† − g C µ+ ϕ† (∂µ + i g A µ − i g B µ )cϕ − g C −µϕ − m 2 ϕ† ϕ − m 2 cϕ† cϕ .

(98)

In this case, gauge fields do not have any dynamical degrees of freedom, and are regarded as auxiliary fields. The conjugate momenta of ϕ, ϕ† , cϕ and cϕ† are same as those obtained in (36) – (39). The conjugate momenta of A µ , B µ , C µ+ and C µ− become constraints, µ

µ

−µ

+µ

Π A = 0, ΠB = 0, ΠC = 0, ΠC = 0.

(99)

Using the Legendre transformation, the Hamiltonian density is obtained as −µ +µ µ µ H M = πϕ˙ + ϕ˙ † π† + πcϕ c˙ϕ + c˙ϕ† π†cϕ + A˙ µ Π A + ΠB B˙µ + C˙µ+ ΠC + ΠC C˙µ− − L µ

µ

−µ

+µ

+ λ Aµ Π A + ΠB λBµ + λC+µ ΠC + ΠC λC−µ

= ππ† + πcϕ π†cϕ + (D i Φ)† (D i Φ) + m 2 Φ† Φ

− i g A 0 (πϕ − ϕ† π† + πcϕ cϕ − cϕ† π†cϕ ) − i g B 0 (πϕ − ϕ† π† − πcϕ cϕ + cϕ† π†cϕ )

− g C 0+(πcϕ − ϕ† π†cϕ ) − g (cϕ† π† − πcϕ ϕ)C 0− µ

µ

−µ

+µ

+ λ Aµ Π A + ΠB λBµ + λC+µ ΠC + ΠC λC−µ ,

(100)

where λ Aµ , λBµ , λC+µ and λC−µ are Lagrange multipliers. Secondary constraints are obtained as µ

d ΠA dt 1

© µ ª µ = Π A , HM PB = g j A = 0,

(101)

Objects called ghost D-branes have been introduced as an extension of D-brane and their properties have been studied [23, 24].

14

µ

d ΠB

© µ ª µ = ΠB , HM PB = g j B = 0,

dt +µ d ΠC dt −µ d ΠC dt

o n +µ = ΠC , HM

+µ

(102)

= g jC = 0,

(103)

© −µ ª −µ = ΠC , HM PB = g jC = 0,

(104)

PB

R −µ +µ µ µ where HM is the Hamiltonian HM = H M d 3 x, and j A , j B , jC and jC are the currents of U (1) and fermionic symmetries given by ³ ´ 0 † † † † j A = i πϕ − ϕ π + πcϕ cϕ − cϕ πcϕ , £© ª © ª j Ai = i (∂i − i g A i − i g B i )ϕ† − g C −i cϕ† ϕ − ϕ† (∂i + i g A i + i g B i )ϕ + g C +i cϕ © ª © ª¤ + (∂i + i g A i − i g B i )cϕ − g C −i ϕ cϕ − cϕ† (∂i + i g A i − i g B i )cϕ − g C −i ϕ , (105) ³ ´ 0 † † † † j B = i πϕ − ϕ π − πcϕ cϕ + cϕ πcϕ , £© ª © ª j Bi = i (∂i − i g A i − i g B i )ϕ† − g C −i cϕ† ϕ − ϕ† (∂i + i g A i + i g B i )ϕ + g C +i cϕ © ª © ª¤ − (∂i + i g A i − i g B i )cϕ − g C −i ϕ cϕ + cϕ† (∂i + i g A i − i g B i )cϕ − g C −i ϕ , (106) jC+0 = πcϕ − ϕ† π†cϕ , © ª © ª jC+i = (∂i − i g A i − i g B i )ϕ† − g C −i cϕ† cϕ − ϕ† (∂i + i g A i − i g B i )cϕ − g C −i ϕ , (107)

jC−0 = cϕ† π† − πcϕ ϕ, © ª © ª jC−i = cϕ† (∂i + i g A i + i g B i )ϕ + g C +i cϕ − (∂i − i g A i + i g B i )cϕ† − g C +i ϕ† ϕ.

(108)

In the same way, tertiary constraints are obtained as d j A0 dt d j B0

© ª = j A0 , HM PB = −∂i j Ai = 0,

ª © = j B0 , HM PB = −∂i j Bi = 0,

dt d jC+0

dt d jC−0 dt

© ª = jC+0 , HM PB = −∂i jC+i = 0, © ª = jC−0 , HM PB = −∂i jC−i = 0,

(109) (110) (111) (112)

from the invariance under the time evolution of j A0 = 0, j B0 = 0, jC+0 = 0 and jC−0 = 0. © ª © ª On the other hand, the conditions d j Ai /d t = j Ai , HM PB = 0, d j Bi /d t = j Bi , HM PB = © © ª ª 0, d jC+i /d t = jC+i , HM PB = 0 and d jC−i /d t = jC−i , HM PB = 0 are not new constraints but the relations to determine λ Ai , λBi , λC+i and λC−i . Furthermore, new constraints do not appear from the conditions d (∂i j Ai )/d t = 0, d (∂i j Bi )/d t = 0, d (∂i jC+i )/d t = 0 and d (∂i jC−i )/d t = 0. The constraints are classified into the first class ones Π0A = 0, Π0B = 0, ΠC+0 = 0, ΠC−0 = 0 15

(113)

and the second class ones ΠiA = 0, ΠiB = 0, ΠC+i = 0, ΠC−i = 0, j Ai = 0, j Bi = 0, jC+i = 0, jC−i = 0,

j A0 = 0, j B0 = 0, jC+0 = 0, jC−0 = 0,

∂i j Ai = 0, ∂i j Bi = 0, ∂i jC+i = 0, ∂i jC−i = 0.

(114)

a The determinant of Poisson bracket between second class ones {φ2nd } does not vanish on constraints. Using j A0 , j B0 , jC+0 and jC−0 , the conserved U (1) and fermionic charges are constructed as Z Z ³ ´ 3 0 3 † † † † N A ≡ −i d x j A = −i d x πϕ − ϕ π + πcϕ cϕ − cϕ πcϕ , (115) Z Z ³ ´ NB ≡ −i d 3 x j B0 = −i d 3 x πϕ − ϕ† π† − πcϕ cϕ + cϕ† π†cϕ , (116) Z Z ³ ´ Q F ≡ − d 3 x jC+0 = − d 3 x πcϕ − ϕ† π†cϕ , (117) Z Z ³ ´ Q F† ≡ − d 3 x jC−0 = − d 3 x cϕ† π† − πcϕ ϕ . (118)

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, n n n n o o o o N A , φaˆ = 0, NB , φaˆ = 0, Q F , φaˆ = 0, Q F† , φaˆ = 0, (119) PB

PB

PB

PB

where φaˆ are first class constraints (113) and the Hamiltonian HM . From (101) – (104) and (119), following relations can be considered as first class constraints, N A = 0, NB = 0, Q F = 0, Q F† = 0.

(120)

After taking the following gauge fixing conditions for the first class ones (113), A 0 = 0, B 0 = 0, C +0 = 0, C −0 = 0,

(121)

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, N A |phys〉 = 0, NB |phys〉 = 0, Q F |phys〉 = 0, Q F† |phys〉 = 0.

(122)

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

16

References [1] A. Neveu and J. H. Schwarz, Nucl. Phys. B31, 86 (1971). [2] P. Ramond, Phys. Rev. D3, 2415 (1971). [3] Y. Aharonov, A. Casher and L. Susskind, Phys. Lett. B35, 512 (1971) [4] J. L. Gervais and B. Sakita, Nucl. Phys. B34, 632 (1971). [5] C. Becchi, A. Rouet and R. Stora, Comm. Math. Phys. 42, 127 (1975). [6] C. Becchi, A. Rouet and R. Stora, Ann. Phys. 98, 287 (1976). [7] V. Tyutin, Lebedev Physical Institute Report No. 39, (1975), arXiv:0812.0580 [hepth]. [8] J. Wess and B. Zumino, Nucl. Phys. B70, 39 (1974). [9] J. Wess and J. Bagger, Supersymmetry and Supergravity, second edition (Princeton, 1991). [10] A. Salam and J. Strathdee, Phys. Rev. D11, 1521 (1975). [11] R. Haag, J. T. Lopuszanski ´ and M. Sohnius, Nucl. Phys. B88, 257 (1975). [12] L. D. Faddeev and N. Popov, Phys. Lett. B25, 29 (1967). [13] T. Kugo and I. Ojima, Phys. Lett. B73, 459 (1978). [14] T. Kugo and I. Ojima, Prog. Theor. Phys. Supplement 66, 1 (1979). [15] Y. Kawamura, arXiv:1311.2365 [hep-ph]. [16] Y. Kawamura, arXiv:1406.6155 [hep-th]. [17] Y. Kawamura, Int. J. Mod. Phys. A30, 1550056 (2015), arXiv:1409.0276 [hep-th]. [18] Y. Kawamura, arXiv:1502.00751 [hep-th]. [19] S. Giddings and A. Strominger, Nucl. Phys. B306, 890 (1988). [20] G. Parisi and N. Sourlas, Phys. Rev. Lett. 43, 744 (1979). [21] E. Witten, J. Diff. Geom. 17, 661 (1982). [22] Y. Kawamura, to appear in Int. J. Mod. Phys. A, arXiv:1503.03960 [hep-ph]. [23] T. Okuda and T. Takayanagi, JHEP 03, 062 (2006). [24] S. Terashima, JHEP 05, 067 (2006).

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