Research
Gravitation, Mathematical Physics and Field Theory
Conjugate spinor field equation for massless spin- 3/2 field in de
Sitter ambient space
1Department of Physics, Kermanshah Branch,
Islamic Azad University, Kermanshah, Iran. e-mail:
s.falahi@iauksh.ac.ir
2Department of Physics, Kermanshah Branch,
Islamic Azad University, Kermanshah, Iran. e-mail:
sajadparsamehr@iauksh.ac.ir
Abstract
The quantum field theory in de Sitter ambient space provides us with a
comprehensive description of massless gravitational fields. Using the
gauge-covariant derivative in the de Sitter ambient space, the gauge invariant
Lagrangian density has been found. In this paper, the equation of the conjugate
spinor for massless spin-3/2 field is obtained using the Euler-Lagrange
equation. Then the field equation is written in terms of the Casimir operator of
the de Sitter group. Finally, the gauge invariant field equation is
presented.
Keywords: De Sitter space-time; conjugate spinor; field equation; massless spin-3/2
PACS: 04.62.+v; 11.15.-q; 11.10.-z
1.Introduction
The experimental data and cosmological observations show that the universe is
expanding with a constant positive acceleration, i.e., the
space-time can be non-flat 1-6. Since the simplest curved space-time that corresponds to
these observations is de Sitter space-time and this space-time has maximal symmetry,
the group of ten parametric SO (1,4) is the kinematic group of the de Sitter space.
Therefore, quantum field theory, gauge theory and quantum cosmology are investigated
in this space-time 7-16.
Supersymmetry has been introduced as one of the fundamental principles of all the
efforts to achieve the grand unified theory. It acts in such a way that the
relationship between the bosons (integer-valued spin) and the fermions (half-integer
spin) is established. In the supersymmetry, all the particles have a partner. For
example, for a graviton ( gravity-carrier particle with a spin-2), a partner with a
spin-32 called gravitino can be considered. With the development of quantum
field theory, gauge theories and their very successful results, everyone began to
think about quantization and gauge everything. As far as gauge theories are
concerned, it should be considered that the structure of a particular symmetry
called gauge symmetry, should remain invariant. Quantum gravity theories that use
supersymmetry are called supergravity and seek to unify the gravitational
interaction with other fundamental interactions. It should be noted that
supergravity is a local supersymmetry theory 17. Therefore, gauge theory can be extended to gravity
18.
To understand physical systems, we must obtain the equation of motion of physical
quantities or, equivalently, the system’s Lagrangian. In Lagrangian mechanics,
changes in a physical system are described through solving the Euler-Lagrange
equation for that system’s behavior. The spin-32 field equation was first introduced by William Rarita and Julian
Schwinger in 1941 using the Euler Lagrange method 19.
The field quantization and the gauge theory are reformulated in the ambient space
formalism 12. In the de Sitter
ambient space formalism the spinor fields can be written in terms of the de Sitter
plane wave. The dS plane wave cannot be defined properly since the
plane wave solution has the singularity in the limit x→∞ (x is the ambient space coordinate) 14. Therefore, in ambient space
formalism only the massless fields with spin 0,12,1,32,2 can propagate. We assume that the interactions between the elementary
systems in the universe are governed by the gauge principle and formulated through
the gauge-covariant derivative which is defined as a quantity that preserves the
gauge invariant transformation of the Lagrangian. In quantum field theory, the
Gupta-Bleuler quantization method is used to eliminate the infrared divergence of
the two-point function 20-22. According to this method, the quantum field affects the
space including all states with positive and negative norm. In this formalism, of
all the degrees of freedom, only two are physical 12,13.
In the previous work, we introduced Lagrangian for massless spin-32 field in de Sitter space-time. Due to the complex and difficult
calculations, we intend to present more details of the calculations in this article.
The notation of ambient space is briefly reviewed in Sec. 2. In Sec. 3, through this
notation, we calculate the conjugate spinor field equation also the equation of this
field is invariant. A brief conclusion is presented in Sec. 4. Finally, details of
mathematical calculations are given in two appendices.
2.Notations
It has been discovered today that the universe is accelerating, with a small, but
non-zero and positive cosmological constant, Therefore, it can be concluded that the
shape and geometry of the universe is curved. So, in the first-order approximation,
we can use the de Sitter space-time to explain the curved universe. This space-time
is a 4-dimensional hyperboloid that can be embedded in a Minkowski 5-dimensional
space-time 14,15:
XH={x∈\R5|x⋅x=ηαβxα×xβ=-H-2},α,β=0,1,2,3,4,
(1)
where ηαβ=diag(1,-1,-1,-1,-1) and H is the Hubble parameter. The metric is defined as
follows:
ds2=ηαβdxαdxβ|x2=-H-2=gμνdSdXμdXν, μ=0,1,2,3,
(2)
where Xμ are the components of the coordinate four-vector in a system of
intrinsic coordinates on a hyperboloid and xα is the five dimensional Minkowski space-time (de Sitter ambient space).
Two Casimir operators of the group include:
Q(1)=-12LαβLαβ, α,β=0,1,2,3,4,
(3)
Q(2)=-WαWα , Wα=18ϵαβγδηLβγLδη,
(4)
where ϵαβγδη is an anti-symmetric tensor, and Lαβ are ten infinitismal generators in de Sitter space. They can be written
as a linear combination: Lαβ=Mαβ+Sαβ. Where Mαβ is the orbital part and Sαβ is the spinoral part. In this formalism, the space Mαβ is represented as
Mαβ=-i(xα∂β-xβ∂α)=-i(xα∂β⊤-xβ∂α⊤),
(5)
where ∂β⊤=θβ α∂α is the transverse derivative (x.∂⊤=0) and θαβ=ηαβ+H2xαxβ is the projection tensor on de sitter hyperboloid. For half-integer spin
fields s=l+1/2, the spinoral part is defined as:
Sαβ(s)=Sαβ(l)+Sαβ(12),
(6)
where Sαβ for spin (1/2) field is:
Sαβ=-i4[γα,γβ],
(7)
and the γ-matrices satisfy following relation:
{γα,γβ}=2ηαβI.
(8)
A proper display for them is :
γ0=I00-I,γ4=0I-I0, γ1=0iσ1iσ10,γ2=0-iσ2-iσ20, γ3=(0iσ3iσ30),
(9)
γα†=γ0γαγ0(γ4)2=-1(γ0)2=1,
(10)
where I is unit 2 x 2 matrix and σi are the Pauli matrices. It should be noted that for massless spin-(3/2)
field Q(3/2)(1) is :
Q32(1)Ψα=Q0(1)Ψα+⧸x⧸∂⊤Ψα+2xα∂⊤⋅Ψ-112Ψα+γα⧸Ψ,
(11)
Q0(1)=-∂α⊤∂α⊤ is the “scalar” Casimir operator.
3.Conjugate spinor field equation for massless spin-3/2 field
It is believed that the gauge theory is the basis of fundamental particle
interactions. The Lagrangian for massless spin-(3/2) field is peresented, in the
linear approximation, by using the gauge theory and defining the gauge covariant
derivative. In the ambient space notation the gauge covariant derivative can be
defined as DβΨ=∇β⊤+i(ΨβA)†γ0QA, with A = 1,…N13. The vector-spinor field equation Ψα(x) is obtained from the usual Euler-Lagrange equations. This Lagrangian is
invariant under the gauge transformation: Ψα→Ψαg=Ψα+∇α⊤ψ, and Ψ̃α→Ψ̃αg=Ψ̃α+∂α⊤ψ̃ (see more details 23):
L=(∇~α⊤Ψ~β-∇~β⊤Ψ~α)(∇⊤αΨβ-∇⊤βΨα),
(12)
∇α⊤ is a transverse-covariant derivative which is defined to obtain an
invariant Lagrangian according to the following equation:
∇β⊤Ψα1....αl≡(∂β⊤+γβ⊤x̸)Ψα1....αl-∑n=1lxαnΨα1..αn-1βαn+1..αl,
(13)
also for the conjugate spinor Ψ̃α:
∇~β⊤Ψ~α1....αl≡∂β⊤Ψ~α1....αl-∑n=1lxαnΨ~α1..αn-1βαn+1..αl,
(14)
where x̸=γαxα and γα⊤=θαβγβ. The above equations are specifically designed for our calculations:
∇α⊤Ψβ=∂α⊤Ψβ+γα⊤x̸Ψβ-xβΨα,
(15)
∇~β⊤Ψ~α=∂β⊤Ψ~α-xαΨ~β.
(16)
Now we want to obtain the field equation for the conjugate spinor (Ψ̃α=Ψα†γ0) by using the Euler-Lagrange equation in the linear approximation. The
Euler-Lagrange equation is:
δLδΨm-∂l⊤δLδ(∂l⊤Ψm)=0.
(17)
First, we extend each terms of the (11):
A=(∇⊤αΨβ-∇⊤βΨα)=∂⊤αΨβ+γα⧸xΨβ-∂⊤βΨα-γβ⧸xΨα,
(18)
B=(∇~α⊤Ψ~β-∇~β⊤Ψ~α)=(∂α⊤Ψ~β-xβΨ~α-∂β⊤Ψ~α+xαΨ~β).
(19)
By use the Euler-Lagrange equation, we consider the following terms:
δLδΨm=(γα⧸xδmβ-γβ⧸xδmα)(∇~α⊤Ψ~β-∇~β⊤Ψ~α),
(20)
and
δLδ(∂l⊤Ψm)=(δlαδmβ-δlβδmα)(∇~α⊤Ψ~β-∇~β⊤Ψ~α),
(21)
if β=m, then, one obtains:
δLδΨm=γα⧸x(∇~α⊤Ψ~β-∇~β⊤Ψ~α),
(22)
δLδ(∂l⊤Ψm)=δlα(∇~α⊤Ψ~β-∇~β⊤Ψ~α).
(23)
Now, by placing the above expressions in the Euler-Lagrange equation, the field
equation is obtained as follows:
γα⧸x(∇~α⊤Ψ~β-∇~β⊤Ψ~α)-∂⊤α(∇~α⊤Ψ~β-∇~β⊤Ψ~α)=0,
(24)
this equation can be written in the summarized form:
(∂⊤α-γαx̸)(∇~α⊤Ψ~β-∇~β⊤Ψ~α)=0.
(25)
In the Appendix A, the field equations for the
conjugate spinor is obtained by using the second order Casimir operator:
(Q32(1)+52)Ψ~α+∂α⊤(x̸Ψ̸~+∂⊤⋅Ψ~)-2(γαΨ̸~+x̸∂̸⊤Ψ~α-Ψ~α)=0.
(26)
In this here we present the gauge invariant field equation for the conjugate spinor.
Given the definition of Q32(1), we rewrite (25) as follows:
-Q0Ψ~β+Ψ~β-2xβ(∂⊤⋅Ψ~)-∂β⊤(∂⊤⋅Ψ~)\na+x̸∂̸⊤Ψ~β-xβx̸⧸Ψ~-x̸∂β⊤⧸Ψ~=0.
(27)
We show that the above equation is invariant under the following gauge
transformation:
Ψ̃α→Ψ̃αg=Ψ̃α+∂α⊤ψ̃,
(28)
with Ψ as an arbitrary spinor field. Therefore, (26) comes as follows:
-Q0(Ψ~β+∂β⊤ψ~)+Ψ~β+∂β⊤ψ~-2xβ∂α⊤(Ψ~α+∂α⊤ψ~)-∂β⊤∂α⊤(Ψ~α+∂α⊤ψ~)+x̸∂̸⊤(Ψ~β+∂β⊤ψ~)-xβx̸(Ψ~+\slashed∂⊤ψ~)-x̸∂β⊤(Ψ~+\slashed∂⊤ψ~)=0.
(29)
Consequently, we can derive:
⟹=-Q0Ψ~β+Ψ~β-2xβ(∂⊤⋅Ψ~)-∂β⊤(∂⊤⋅Ψ~)+x̸∂̸⊤Ψ~β-xβx̸⧸Ψ~-x̸∂β⊤⧸Ψ~-Q0∂β⊤ψ~+∂β⊤ψ~-2xβ∂α⊤∂α⊤ψ-∂β⊤∂α⊤∂α⊤ψ+\slashedx\slashed∂⊤∂β⊤ψ~-xβx̸⧸\slashed∂⊤ψ~-\slashedx∂β⊤\slashed∂⊤ψ~⏟=0=0.
(30)
For a gauge invariant, the last terms must be zero. In order to prove this, we use
the following auxiliary relationships:
[∂α⊤,∂̸⊤]=x̸∂α⊤-xα∂̸⊤,
(31)
[∂α⊤,Q0]=-6∂α⊤-2(Q0+4)xα.
(32)
Thus, the last terms of equation (29) are
-Q0∂β⊤ψ~+∂β⊤ψ~-2xβ∂α⊤∂α⊤ψ-∂β⊤∂α⊤∂α⊤ψ+x∂⊤∂β⊤ψ~-xβx̸⧸∂⊤ψ~-x∂β⊤∂⊤ψ~=2xβQ0ψ~+∂β⊤ψ~-xβx̸⧸\∂⊤ψ~-6∂β⊤ψ~-2Q0xβψ~-8xβψ~+xβx̸⧸\∂⊤ψ~+∂β⊤ψ~.
(33)
After some simplification, the equation (32) becomes:
⇒2xβQ0ψ̃-4∂β⊤ψ̃-2Q0xβψ̃-8xβψ̃=2(xβQ0-Q0xβ)ψ̃-4∂β⊤ψ̃-8xβψ̃.
(34)
Finally, using the auxiliary relationship [xα,Q0]=2∂α⊤+4xα, we obtain:
=2(2∂β⊤+4xβ)ψ̃-4∂β⊤ψ̃-8xβψ̃=0,
(35)
proving that the field equation is invariant.
4.Conclusions
In order to better understand the evolution of the universe, it is necessary to
extend the theory of quantum fields, field interactions, or gauge theory,
supersymmetry and supergravity in the de Sitter space-time. We studied the conjugate
spinor field equation for massless gravitational field by the Euler-Lagrange
equation in the de Sitter ambient space formalism. The field equation in terms of
the Casimir operator is obtained. We have shown that the field equation of the
conjugate spinor for massless spin-(3/2) field is gauge invariant. Studies of this
kind are of particular interest given the recent observations of gravitational waves
(LIGO Collaboration); the graviton is a particle that is believed to carry the force
of gravity, which would be accompanied by the gravitino in a supersymmetric theory
in curved space.
Acknowlegements
We would like to express our heartfelt thank and sincere gratitude to Professor M.V.
Takook and E. Yusofi for their helpful discussions. This work has been supported by
the Islamic Azad University, Kermanshah Branch, Kermanshah, Iran.
References
1. S. Perlmutter et al., Measurement of and from
42 highredshift supernovae, Astrophys. J. 517 (1999) 565.
https://doi.org/10.1086/307221
[ Links ]
2. A.G. Riess et al., Observational evidence from
supernovae for an accelerating universe and a cosmological constant,
Astron. J. 116 (1998) 1009.
https://doi.org/10.1086/300499
[ Links ]
3. J. P. Henry, U. G. Briel, and H. Bohringer, The evolution of
galaxy clusters, Scientific American 279 (1998) 52.
https://doi.org/10.1038/scientificamerican1298-52
[ Links ]
4. J. P. Henry, Measuring cosmological parameters from the evolution
of cluster X-ray temperatures, Astrophys. J. 534 (2000) 565.
https://doi.org/10.1086/308783
[ Links ]
5. P. de Bernardis, et al., A flat universe from high-resolution
maps of the cosmic microwave background radiation, Nature 404
(2000) 955. https://doi.org/10.1038/35010035
[ Links ]
6. P. A. R. Ade, et al., Detection of B-Mode polarization at degree
angular scales by BICEP2, Phys. Rev. Lett. 112, (2014) 241101.
https://doi.org/10.1103/PhysRevLett.112.241101
[ Links ]
7. T.S. Bunch, P.C.W. Davies, Quantum field theory in de sitter
space: renormalization by point-splitting, Proc. R. Soc. Lond.
A., 360 (1978) 117.
https://doi.org/10.1098/rspa.1978.0060
[ Links ]
8. B. Allen, Vacuum states in de Sitter space, Phys. Rev.
D 32 (1985) 3136.
https://doi.org/10.1103/PhysRevD.32.3136
[ Links ]
9. J. Bros, U. Moschella, Two-point functions and quantum field in
the de Sitter Universe, Rev. Math. Phys. 8 (1996) 327.
https://doi.org/10.1142/S0129055X96000123
[ Links ]
10. N.A. Chernikov, E. A. Tagirov, Quantum theory of scalar field in
de Sitter space-time, Ann. Inst. Henri Poincar´e IX, A9 (1968)
109. http://www.numdam.org/item/AIHPA-1968-9-2-109-0/
[ Links ]
11. M.V. Takook, Entropy of quantum fields in de sitter space-time,
Annals of Physics 367 (2016) 6.
https://doi.org/10.1016/j.aop.2015.11.013
[ Links ]
12. M.V. Takook, Quantum field theory in de Sitter universe:
ambient space formalism (2016) http://arxiv.org/abs/1403.1204v2
arXiv:1403.1204v2.
[ Links ]
13. S. Parsamehr, M. Enayati, M.V. Takook, Super-gauge field in de
Sitter universe, Eur. Phys. J. C 76 (2016) 260.
https://doi.org/10.1140/epjc/s10052-016-4027-7
[ Links ]
14. M.V. Takook, Théorie quantique des champs pour des
systèmes élémentaires “massifs” et de “masse nulle” sur l’espace- temps de
de Sitter, Thèse de l’université Paris VI (1997).
[ Links ]
15. P. Bartesaghi, J.P. Gazeau, U. Moschella, M.V. Takook, Dirac
fields and thermal effects in the de Sitter universe, Class. Quant.
Grav. 18 (2001) 4373.
https://doi.org/10.1088/0264-9381/18/21/302
[ Links ]
16. E. Yusofi, M. Mohsenzadeh, Scale-dependent power spectrum from
initial excited-de Sitter modes, JHEP, 09 (2014) 020.
https://doi.org/10.1007/JHEP09(2014)020
[ Links ]
17. P. Nath and R. Arnowitt, Generalized Supergauge Symmetry as a
New Framework for Unified Gauge Theories, Phys. Lett. B 56
(1975) 177. https://doi.org/10.1016/0370-2693(75)90297-X
[ Links ]
18. D. Z. Freedman, P. van Nieuwenhuizen, and S. Ferrara, Progress
toward a theory of supergravity, Phys. Rev. D 13 (1976)
3214-3218. https://doi.org/10.1103/PhysRevD.13.3214
[ Links ]
19. W. Rarita, J. Schwinger, On a Theory of Particles with
Half-Integral Spin Phys. Rev. 60(1941) 61.
https://doi.org/10.1103/PhysRev.60.61
[ Links ]
20. B. Binegar, C. Fronsdal, W. Heidenreich, Conformal QED,
J. Math. Phys. 24 (1983) 2828.
https://doi.org/10.1063/1.525664
[ Links ]
21. F. Strocchi, Spontaneous symmetry breaking in local gauge
quantum field theory; the Higgs mechanism, Commun. Math. Phys.
56 (1977) 57. https://doi.org/10.1007/BF01611117
[ Links ]
22. T. Garidi, J.P. Gazeau, S. Rouhani, M.V. Takook, Massless vector
field in de Sitter universe, J. Math. Phys. 49 (2008) 032501.
https://doi.org/10.1063/1.2841327
[ Links ]
23. S. Parsamehr, Lagrangian for massless gravitational field in de
Sitter space, Eur. Phys. J. Plus 132 (2017) 291.
https://doi.org/10.1140/epjp/i2017-11565-x
[ Links ]
Appendix
A. The field equation in terms of Casimir operator
Here we want to show how the field equation is written in terms of the Casimir
operator:
(∂⊤α-γαx̸)(∂α⊤Ψ~β-xβΨ~α-∂β⊤Ψ~α+xαΨ~β)=0,
(A.1)
∂⊤α(∂α⊤Ψ~β-xβΨ~α-∂β⊤Ψ~α+xαΨ~β)⏟1-γαx̸(∂α⊤Ψ~β-xβΨ~α-∂β⊤Ψ~α+xαΨ~β)⏟2=0.
(A.2)
We first consider the expression 1:
∂⊤α∂α⊤Ψ̃β-xβΨ̃α-∂β⊤Ψ̃α+xαΨ̃β=∂⊤α(∂α⊤Ψ̃β)-∂⊤α(xβΨ̃α)-∂⊤α(∂β⊤Ψ̃α)+∂⊤α(xαΨ̃β),
(A.3)
=-Q0Ψ̃β-(∂⊤αxβ)Ψ̃α-xβ(∂⊤αΨ̃α)-∂⊤α(∂β⊤Ψ̃α)+4Ψ̃β,
(A.4)
=-Q0Ψ̃β-(δβα+xβxα)Ψ̃α-xβ(∂⊤⋅Ψ̃)-∂⊤α(∂β⊤Ψ̃α)+4Ψ̃β,
(A.5)
=-Q0Ψ̃β-Ψ̃β-xβ(∂⊤⋅Ψ̃)-(∂β⊤∂⊤αΨ̃α+xβ∂⊤αΨ̃α-xα∂β⊤Ψ̃α)+4Ψ̃β,
(A.6)
=-Q0Ψ̃β+3Ψ̃β-2xβ(∂⊤⋅Ψ̃)-∂β⊤(∂⊤⋅Ψ̃)+xα∂β⊤Ψ̃α,
(A.7)
=-Q0Ψ̃β+2Ψ̃β-2xβ(∂⊤⋅Ψ̃)-∂β⊤(∂⊤⋅Ψ̃).
(A.8)
We first consider the expression 2:
-γαx̸(∂α⊤Ψ~β-xβΨ~α-∂β⊤Ψ~α+xαΨ~β),
(A.9)
=-(2xα-⧸xγα)(∂α⊤Ψ~β-xβΨ~α-∂β⊤Ψ~α+xαΨ~β),
(A.10)
=2xα∂β⊤Ψ~α-2xαxαΨ~β+x̸∂̸⊤Ψ~β-x̸γαxβΨ~α-x̸γα∂β⊤Ψ~α+x̸xαΨ~α,
(A.11)
=x̸∂̸⊤Ψ~β-xβx̸⧸Ψ~-x̸∂β⊤⧸Ψ~-Ψ~β,
(A.12)
in the above calculations, we have used the terms x⋅Ψ=0 and x⋅∂⊤=0. According to 1 and 2 we can write the equation of motion as
follow:
⟹=-Q0Ψ~β+Ψ~β-2xβ(∂⊤⋅Ψ~)-∂β⊤(∂⊤⋅Ψ~)+x̸∂̸⊤Ψ~β-xβx̸⧸Ψ~-x̸∂β⊤⧸Ψ~=0.
(A.13)
Finally, given Q32(1) definition, we have:
(Q32(1)+52)Ψ~α+∂α⊤(x̸Ψ̸~+∂⊤⋅Ψ~)-2(γαΨ̸~+x̸∂̸⊤Ψ~α-Ψ~α)=0.
(A.14)
B. The auxiliary relationships
Here are some of the auxiliary relationships used in this article:
∂α⊤,∂β⊤=xβ∂α⊤-xα∂β⊤,[∂α⊤,xβ]=θαβ,xα,∂̸⊤=-γα⊤,[γα⊤,∂α⊤]=-4x̸,Q0,x̸=-4x̸-2∂̸⊤,[xα,Q0]=2∂α⊤+4xα,[x̸,∂̸⊤]=4-2∂̸⊤x̸,γα⊤=γα+xαx⋅γ,x̸,∂α⊤=-γα⊤,[x̸,γα⊤]=2xα-2γαx̸,∂α⊤,∂̸⊤=x̸∂α⊤-xα∂̸⊤,[γα⊤,∂α⊤]=-4x̸,∂α⊤,Q0=-6∂α⊤-2Q0+4xα,[∂̸⊤,γα⊤]=-2γα⊤∂̸⊤+2∂α⊤+γα⊤x̸+4xα,[Q0,γα⊤]=-8xαx̸-2x̸∂α⊤-2γα⊤-2xα∂̸⊤.
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