Transition amplitude for free massless particle
The propagator for massless particle of arbitrary spin is represented as BFV-BRST path integral in index spinor formalism. The classical formulation of the theory is investigated and it is carried out its Hamiltonization procedure. The structure functions are obtained. The BRST-charge of the model i...
Saved in:
| Published in: | Вопросы атомной науки и техники |
|---|---|
| Date: | 2001 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2001
|
| Subjects: | |
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/79423 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Transition amplitude for free massless particle / V.G. Zima, S.A. Fedoruk // Вопросы атомной науки и техники. — 2001. — № 6. — С. 53-59. — Бібліогр.: 8 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859940307547717632 |
|---|---|
| author | Zima, V.G. Fedoruk, S.A. |
| author_facet | Zima, V.G. Fedoruk, S.A. |
| citation_txt | Transition amplitude for free massless particle / V.G. Zima, S.A. Fedoruk // Вопросы атомной науки и техники. — 2001. — № 6. — С. 53-59. — Бібліогр.: 8 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | The propagator for massless particle of arbitrary spin is represented as BFV-BRST path integral in index spinor formalism. The classical formulation of the theory is investigated and it is carried out its Hamiltonization procedure. The structure functions are obtained. The BRST-charge of the model is calculated and it is shown, that it has the first rank. The expression for transition amplitude is transformed to the form of amplitude for a system with only the first class constraints. It is shown, that complexification of some phase variable results in the Gupta-Bleuler formalism. In these frameworks it is considered quantization procedure.
|
| first_indexed | 2025-12-07T16:10:30Z |
| format | Article |
| fulltext |
TRANSITION AMPLITUDE FOR FREE MASSLESS PARTICLE
V.G. Zima1, S.A. Fedoruk2
1Kharkov National University, Kharkov, Ukraine
e-mail: zima@postmaster.co.uk
2Ukrainian Engineering Pedagogical Academy, Kharkov, Ukraine
e-mail: fed@postmaster.co.uk
The propagator for massless particle of arbitrary spin is represented as BFV-BRST path integral in index spinor
formalism. The classical formulation of the theory is investigated and it is carried out its Hamiltonization procedure.
The structure functions are obtained. The BRST-charge of the model is calculated and it is shown, that it has the
first rank. The expression for transition amplitude is transformed to the form of amplitude for a system with only the
first class constraints. It is shown, that complexification of some phase variable results in the Gupta-Bleuler
formalism. In these frameworks it is considered quantization procedure.
PACS: 0460G, 0370
1. INTRODUCTION
Calculation of the propagator for a particle is an
important part of the quantization problem. The most
powerful modern method for solving this problem, as
well as the problem of quantization in general, is the
Batalin-Fradkin-Vilkovisky Becchi-Rouet-Stora-Tyutin
(BFV-BRST) approach [1]. Massive particle with spin
has been in details considered in the paper [2]. However
till now calculation of transition amplitude for massless
particle with spin in the BFV-BRST approach is not
brought to the desirable level of transparency both
computations in definite approach to description of spin
and for their connection in various approaches
(pseudoclassical mechanics, twistorial formalism, index
spinor etc.) [3-5]. It justifies further efforts on finding
the propagator of particle with spin by the BFV-BRST
method with using of various sorts of spinning
variables.
In this paper the propagator of massless particle with
arbitrary spin in the usual space-time dimension D=4 is
represented as the BFV-BRST path integral. For
description of spin it is used the index spinor formalism
[6], which is seemed sufficiently general and convenient
and having clear physical sense.
Section 2 presents Hamiltonian analysis of the
massless particle with spin in index spinor formulation.
Covariant and irreducible separation of the constraints
in the classes is fulfilled and we prove the first rank of
the BRST charge. It is analyzed the gauge
transformations of the phase variables and
corresponding variation of the action. In section 3 we
construct the transition amplitude for the massless
spinning particle as integral in BFV-BRST approach.
Quantum theory is formulated in relativistic gauges with
derivatives for the Lagrange multipliers. The presence
of the second class constraints in the theory leads to
modification of the integral measure and to some
complication of the formulation. We transform the
expression for transition amplitude to the form of
amplitude for a system with only first class constraints.
The half of the initial second class constraints plays the
role of gauge fixing conditions for the other second
class constraints. Section 4 presents the
complexification procedure for the ghost variables and
the corresponding constraints in the path integral for
transition amplitude. As result we obtain path integral in
the Gupta-Bleuler approach. Section 5 describes general
features of the BFV-BRST quantization with using
Gupta-Bleuler procedure. Section 6 contains some
concluding comments.
2. HAMILTONIAN ANALYSIS OF
MASSLESS PARTICLE IN INDEX SPINOR
FORMULATION
A spinning particle of arbitrary mass can be
described in terms of commuting variables
),,,( ζζxz = where µx is a four-vector of the
space-time coordinate and αζ is a Weyl index spinor
[6]. In the first-order formalism the Lagrangian for such
a particle has the form
)ˆ()(
2
22 jpmpepL −−+−= ζζλω , (1)
where
ζσζζσζτωω dididxd +−==
is bosonic ‘superform’, which is invariant with respect
to the transformations of the N=1 ‘bosonic
supersymmetry’
εζδεζδεζ σζε σδ ==−= ;;iix
with commuting Weyl parameter αε . Massive ( 0>m )
particle has been considered in details in [2]. Here we
deal with the massless ( 0=m ) case.
On the constraint surface for massless particle the
classical Pauli-Lyubanskii vector
)()ˆ( 2 ζσζζζ pppw −=
is proportional on shell to particle momentum, jpw = ,
thus the constant j has a sense of ‘classical helicity’.
The primary constraints of the model (1) are
0ˆ ≈−≡ ζζζ pipd , 0ˆ ≈−−= ppid ζζζ ; (2)
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 53-59. 53
0≈− ppx , 0≈pp ; (3)
0≈ep , 0≈λp . (4)
The mass condition for massless particle
02/2 ≈≡ pT (5)
and the spin constraint
0ˆ ≈−≡ jph ζζ , (6)
which explicitly enter the Lagrangian (1), appear as the
secondary constraints for preservation of constraints (4)
in the Dirac procedure [7].
The constraints (3) are a pair of the self-conjugate
second-class constraints. By introducing of Dirac
brackets [7] for them one can identifies xp and p in all
other expressions and excludes p and pp from the
consideration. For all remaining canonical variables
these Dirac brackets (DB) are coincided with initial
Poisson brackets (PB). So the index indicating type of
brackets can be omitted.
On the surface of spinor constraints (2) the spin
constraint (6) is equivalent to the constraint
( ) 0
2
≈−−≡ jppiS ζζ ζζ (7)
which will be called the spin constraint also.
Nonzero PB of the constraint algebra are
,ˆ2},{ pidd =ζζ
,2
},{ ζζ didS −= .2
},{ ζζ didS = .
Thus, the constraint T and S belong to the first class.
Spinor constraints ζd and ζd are mixes of constraints
of the first and second classes because of singularity of
a matrix ,p̂ 02ˆdet ≈−= Tp .
On a constraint surface at 0≠j spinors ζ and ζp̂
form a basis in the space of spinors with nondotted
indexes,
.)ˆ()ˆ()ˆ( β
αα
ββ
α ζζζζδζζ ppp −=
Covariant and irreducible separation of the spinor
constraints (4) in the classes is achieved by projection
on these spinors and conjugated ones. For the
projections of spinor constraints,
,ˆζφ ζ pd≡ ,ζζχ d≡ ,ˆ ζζφ dp= ,ζχ ζd≡
we have algebra
( )jSiT +−= 4},{ φφ , ),(2},{ jhi +=χχ
,},{},{},{ φχφφχφ iS === .0},{ =χS
Hence, the projections φ and φ are independent
constraints of the first class. The projections χ and χ
are the second-class constraints. The PBs of the first-
class constraints ).,,( φφSTFa = are the same as the
Dirac brackets
}],}{,{
},}{,[{)ˆ2(},{},{ 1
AB
BApiBABA D
χχ
χχζζ
−
−+= −
introducing for the second-class constraints χ , χ .
In the algebra
c
c
abba FUFF 2},{ =
of the first class constraints
aF , ;,,, φφSTa =
,,,, φφ φφ ≡≡≡≡ FFSFTF ST
the first rank structural functions c
abU are differ from
zero only. Already the second rank structural functions
21
321
aa
bbbU , which are determined in general with
ambiguity by the equality
,2
2
21
321
1
321 ][ a
aa
bbb
a
bbb FUD =
where
,},{ 1
3213
1
21
1
321
a
kb
k
bbb
a
bb
a
bbb UUFUD +=
one can take equal to zero.
Due to reparametrization invariance of the action the
Hamiltonian is a linear combination of the first class
constraints
.a
aFSeTH λφκκ φλ ≡+++=
As the generating functions of gauge
transformations the first class constraints aF generate
the following transformations for the coordinates
},{ a
aFzz ξδ = , the momenta },{ a
azz Fpp ξδ = and
the Lagrange multipliers ca
bc
ba U λξξδ λ += with local
parameters )(τξ a . The last equality is a necessary
condition for invariance of the Hamiltonian action with
respect to the gauge transformations. For considered
model of massless particle we have
ζσκσζκεδ ζζ
~~ pipipx −+= , 0=pδ ,
pii ~
2
ζκζϕζδ += , ζζζ κϕδ ppipip ˆ
2
−−= ,
))((4 φφ λκλκεδ −+−= jSie , ϕδ λ = ,
)( κ λϕ λκδ λ φφ −+= i . (8)
The corresponding variation of the Hamiltonian
action
∫ −= f
i
a
aA
zA FzpdA
τ
τ
λτ )(
is equal to
f
i
jpA
τ
τ
φκκ φϕεδ
+++=
2
2
and vanishes (outside of the constraint surface) only if
( ) ( ) 0== fi τετε , ( ) ( )fi τϕτϕ = ,
0)()( == fi τκτκ , 0)()( == fi τκτκ .
This circumstance makes directly admissible only
‘relativistic gauges’ [1], i.e. the gauges with derivatives,
54
which impose restrictions on e , φλ , φλ , expressing
they in terms of the other phase space variables. It
should be stressed that the last conditions on ε and κ ’s
are not necessary on the constraint surface in contrast
with the rest parameter .ϕ
In some relations it will be convenient to pass from
the complex second class constraints χχ , to real
constraints 0≈h and
,0)(
2
1 ≈+≡ ζζ ζζ ppg
which are equivalent to complex ones at the account of
spin constraint (7). Let's note the identities
.; ShigShig +−−≡+−≡ χχ
The brackets of the constraints h and g with all first
class constraints are equal to zero, and with each other it
is
jhpgh +== ζζ ˆ},{ .
If the column of constraints h and g is multiplied
on an arbitrary matrix from the group ),2( ℜSL , the
real constraints forming a new column, will be
equivalent initial and have the same brackets with each
other and with the first class constraints as initial ones.
In particular such transformation allows changing by
places h and g , having replaced them, for example, on
g− and h respectively. Below, if not opposite is told,
the constraints h and g are determined up to an
arbitrary ),2( ℜSL -transformation.
Let's note, that the constraints h and g differ from
the constraints 2/)( S−χ and 2/)( Si −χ by a
complex unimodular transformation.
3. TRANSITION AMPLITUDE AS THE BFV-
BRST PATH INTEGRAL
The most profound method for calculation of
transition amplitude for constrained systems is the
BFV--BRST formalism [1]. In this approach, for each
first-class constraint aF the set of coordinates of the
initial phase space is supplemented by ‘dynamical’
Lagrange multipliers aλ with the same Grassmannian
parity, their canonically conjugate momenta aπ ,
a
bb
a δπλ =},{ , and the ghost variables of the opposite
parity. The ghost sector contains Grassmannian odd
ghosts aC , antighosts aC~ and their canonically
conjugate quantities aP~ and aP ,
}~,{}~,{ b
aa
bb
a CPPC == δ . The ghost numbers of the
ghost variables are
1)~()~()()( =−=−== CghPghPghCgh .
The variables λ , π , C , P are real, whereas P~ , C~ are
pure imaginary.
The variables of original phase space are subjected
to the second-class constraints, but the algebra of the
first-class constraints aF remains the same even after
introducing the DBs. Thus, the BRST charge has a rank
one and is a linear combination of the first-class
constraints, aF and aπ , of the extended phase space
;~
2
1 bca
cba
a
a
a
a CCUPPCF ++=Ω π
{ } { } 0,, =ΩΩ=ΩΩ DBPB .
The BRST charge is real, ,Ω=Ω Grassmannian odd,
( ) 1=Ωε , and has the ghost number one, ( ) 1=Ωgh .
The path integral for the transition amplitude,
( )
),(exp|},{det|2
],[
2
1
,
effji
i
iZ
AiGG
GpZD
∏
∏∫
×
×=Ζ Ψ
τ
τ
π
δ
(9)
includes the usual Liouville measure [ ]ZpZD , in the
phase space of BFV-BRST approach parameterized by
the coordinates ),,( PCzZ = and canonically
conjugate variables )~,~,( CPpp zZ = . This means that
in the standard finite-dimensional approximations of the
path integral, the product of differentials of each pair of
the canonically conjugate real bosonic variables in the
measure is divided by π2 . The differential of each
variable that remains without its pair, in accordance
with the boundary conditions under consideration, is
also divided by π2 . Similar multipliers are absent for
the Grassmannian quantities. In the Hamiltonian
approach, the multipliers corresponding to the
realification Jacobian of the using complex variables do
not appear in the measure.
Fulfillment of the second-class constraints
),()( ghGi = , which commute with the first class
constraints, is provided by the functional δ -functions in
expression (9). The multipliers corresponding to the
realification Jacobian do not arise in the product
∏
τ
δ
,
)(
i
iG of δ -functions of the complex second-class
constraints. The measure is normalized by the
determinant of Poisson brackets matrix for the second-
class constraints, ,)ˆ(},det{ 2ζζ pGG ji = which is
equal 2j on the surface of the second-class constraints.
In addition, for every ‘moment of time’ τ the factor
( ) 12 −π should be introduced into the measure on each
pair of the real bosonic second-class constraints.
The effective Hamiltonian action is
..)( tbZeff AHZpdA
f
i
+−= ∫ Ψ
τ
τ
τ .
This expression can contain, and in our case it indeed
contains, uncertainty, which should be eliminated
during the calculation of amplitude. The question is in
the ordering constants connected with the possible
presence of products of canonically conjugate
(noncommuting) variables in the BRST Hamiltonian
ΨH .
55
For the theory with reparametrization invariance, the
BRST Hamiltonian ΨH is the ‘BRST derivative’ of the
gauge fermion Ψ : { }ΨΩ=Ψ ,H . In the amplitude (9),
one can use on equal footing both Poisson and Dirac
brackets because, in our case, the Poisson brackets of
the first class constraints (entering into Ω ) and the
arbitrary function of phase space variables differ from
the Dirac brackets by addends which are proportional to
the second class constraints only. Thus these terms
vanish on the second-class constraint surface. The gauge
fermion Ψ is Grassmannian odd, 1)( =Ψε , pure
imaginary, Ψ−=Ψ , and has a negative ghost number,
1)( −=Ψgh . The relativistic gauge with derivatives
for the Lagrange multipliers ( 0=λ ) corresponds to
a
aP λ~=Ψ , then
a
a
a
a PPFH ~+=Ψ λ .
As it is known [1], the transition amplitude does not
depend on a choice of the gauge fermion if the path
integral is taken over the paths, which belong to the one
class of equivalence with respect to the BRST
transformation. Such class is extracted by choosing the
appropriate gauge and boundary conditions.
For standard formulated theory with the first class
constraints the canonical gauges, connecting coordinates
and momenta, and ‘relativistic’ gauges, fixing
derivatives of Lagrange multipliers in term of the phase
variables, are physically equivalent. The proof of the
equivalence involves a permutation of some limit
transition and path integration.
The boundary conditions for the ghosts and
antighosts as well as for the momenta of Lagrange
multipliers consist in vanishing of these quantities in
initial and final moments. For real basic coordinates the
boundary conditions fix both initial and final values. For
complex basic coordinates, in general, due to
noncommutativity of mutually conjugate quantities, the
boundary conditions fix eigenvalues of corresponding
operators either in bra or in ket vectors, i.e. either in
initial or in final states only. Therefore, unification of
conditions on considered states and canonical conjugacy
of complex conjugate quantities lead to situation when
boundary conditions fix initial values of a half of
complex coordinates and final values of the rest of
complex conjugate for them coordinates. In considered
model, because of presence of the second-class
constraints and occurrence of some gauge and physical
variables in uniform object (index spinor), the choice of
correct boundary conditions is not quite trivial.
Let us transform the expression (9) to standard form
of transition amplitude for systems without second-class
constraints. For this purpose we introduce auxiliary
variables hλ , hπ , hP , hC , hP~ , hC~ with natural
Grassmannian parities and use obvious equalities
;1)~(exp]~,[ =− ∫∫ h
hh
h PPdiPPD
f
i
τ
τ
τ
;)()(2
))((exp],[
∏
∫∫
=
=+−
τ
τ
τ
δπ δ
πλτπλ
gh
ghdiD h
h
h
h
f
i
{ }
{ } ( ) ,},{det,
)~,(exp]~,[
2
1
∏∏
∫∫
=
=−
ττ
τ
τ
τ
ji
h
h
h
h
GGgh
СghСdiССD
f
i
where the integrand variables do not satisfied any
boundary conditions. Then, after permutation of the
limit transition and path integration, we have
( +′=Ζ ∫∫ ′→Ψ ZpdipZD ZZ
f
i
[exp],[lim
0
τ
τ
ε
τ
−−+++ ΨHPCCP h
h
h
h
h
h
~~2 εελπε
{ } )..]~,~
tbh
h
h
hh
h AiCghCghPP +−−−− πλ (10)
where ).~,~,,(),,,,( hhhZZ
hhh CPppPCZZ πλ ≡≡′ ′ After
change of variables hh CC →ε , hh CС ~~ →ε ,
hh ππε →2 with unit super-Jacobian we obtain
{ },}],{[exp
],[lim
..1
0
tbZ
Z
AiZpdi
pZD
f
i
+ΨΩ ′−′×
×′=Ζ
′
′→Ψ
∫
∫
τ
τ
ε
τ
(11)
where
h
h
h PCg π++Ω=Ω ′ ,
gCP h
h
h
~1~
21 ε
λ ++Ψ=Ψ .
It is easy to verify 0},{ =Ω ′Ω ′ . Since Ω ′ is real,
odd and has ghost number one, so that it may be
interpreted as BRST charge for only first-class
constraints T , S , φ , φ , h . Simultaneously 1Ψ is
interpreted as gauge fermion, since it is odd and have
negative ghost number. Underline that ‘new’ BRST-
charge Ω ′ includes one second-class constraint h along
with first-class constraints. The second-class constraint
g of initial model enters into gauge fermion 1Ψ and
plays a role of ‘nonrelativistic’ canonical gauge
condition.
Expression (11) depends on the parameter ε
through the gauge fermion 1Ψ only and therefore this
path integral does not depend on ε . It makes possible in
(11) at first to omit passage to the limit 0→ε and then
to do passage to the limit ∞→ε . Then in limit ∞→ε
(permutations of the limit transitions with the path
integration are made within the framework of the usual
assumptions of properties last) we obtain
}}],{
[{exp],[
..tb
ZZ
Ai
ZpdipZD
f
i
′+Ψ ′Ω ′−
−′′=Ζ ′′Ψ ∫∫
τ
τ
τ
56
where A
AP λ ′′=Ψ ′ ~ , ),( haA λλλ =′ , )~,~(~
haA PPP =′ is
gauge fermion for system with BFV-BRST phase space
Z ′ , Zp ′ extended by auxiliary variables. This gauge
fermion corresponds ‘relativistic’ gauge with usual
boundary conditions for initial and auxiliary variables.
Origin of boundary conditions for auxiliary variables
hh
h CC π,~, is a result of the mentioned limit
transitions. The expression (11) has standard form for
the transition amplitude in BFV-BRST approach for
reparametric-invariant system without second-class
constraints. Thus the calculation of propagator for initial
system with first- and second-class constraints is
reduced to the calculation of propagator for a system
with only first-class constraints.
We carry out the calculation of transition amplitude
in the coordinate representation for the variables Az
and in the mixed representation for the ghosts, i.e. we
choose the boundary conditions
µµ τ ii xx =)( , µµ τ ff xx =)( ;
αα ζτζ 11 )( = , αα ζτζ
22 )( = ;
0)()( == iafa τπτπ ;
0)()( == f
a
i
a CC ττ , 0)(~)(~ == faia CC ττ ,
where the marks )2,1( of spinors must be understood as
),( if for the holomorphic choice and as ),( fi for the
antiholomorphic one. The boundary values are not fixed
for the rest of variables. The boundary conditions
imposed are BRST-invariant and ensure vanishing of
the BRST charge on the boundaries. This provides the
form-invariance of amplitude.
The choice of boundary conditions for the index
spinor is covariant. Such choice is not unique. Using
combinations of the index spinor and its conjugate
momentum with other variables of the phase space, one
can propose a variety of covariant boundary conditions
on index variables. All they are in essence equivalent
and reflect a concrete choice of the quantum description
of a spin (i.e., realization of the Hilbert space of
quantum states). We restrict ourselves to the
consideration of two basic variants. As the simplest
ones, they are described in the literature now.
The boundary conditions imposed are BRST
invariant and ensure vanishing of the BRST charge on
the boundaries (for that it is sufficiently the conditions
on π and C ). One can understand the vanishing of the
boundary values of the BRST charge as a classical
manifestation of the quantum condition 0ˆ =ΨΩ phys .
With boundary conditions, the correctness of the
variational principle, i.e. independence of any variation
of the action from the boundary values of the variation
for variables, which are not fixed at the boundary, needs
introducing the boundary term
)( 1122.. ζζε ζζζ ppA tb −= .
Here 1+=ζε corresponds to the holomorphic choice
of the boundary conditions and 1−=ζε corresponds to
the antiholomorphic one.
4. COMPLEXIFICATION OF PHASE
VARIABLES
Complexification of some phase variables leads to
the Gupta-Bleuler formalism [6] as to the result of
calculation of the path integral by the saddle-point
method. The Gupta-Bleuler formalism simplifies
physical interpretation and mathematical calculations
however it ‘violates’ simple formulations of some
fundamental physical principles. In operator
quantization this obstacle appears as necessity to use an
indefinite metric in the state space and so demands
certain freedom of formulations and exceptional caution
in their application outside of the formally justified
region. Of course, in the BFV-BRST approach, where
indefiniteness of the metric is an element of basic
formulation, it does not form any obstacle but attempts
of ‘direct’ application of the Gupta-Bleuler procedure
here collide with the problem of reality (Hermiticity) for
BRST-charge, gauge fermion and Hamiltonian, i.e.
ultimately with the unitary problem. In general case
these difficulties are not overcome now [8]. Here we do
not pursue purpose of solution for general problem
restricting consideration with particular model.
Therefore our consideration is justified in the
framework of usual assumption on the properties of path
integral.
Complex linear unimodular transformation of the
second-class constraints and corresponding auxiliary
variables
=
h
g
A
d
d
,
=
=
−
h
h
T
d
d
d
d A
π
λ
λ
λ
π
π 1 ,
,~~ 1
=
−
h
h
T
d
d
C
C
A
C
C
=
h
h
d
d
P
P
A
P
P
~~ ,
where
−−
−
−
=
1
1
2
1
i
i
i
A ,
−
−
=−
iii
A
11
2
11 ,
allows writing the equalities (9) in terms of complex
auxiliary variables. Then instead of (10) we obtain
{
} .]~},{1
)(
2
)(~
[exp],[
..tb
d
d
dd
d
cZzc
iACghC
ighiighPP
HZpdipZD
c
f
i
c
+−
−−−+−−
−−′=Ζ Ψ′Ψ ∫∫
ε
π
ε
λ
τ
τ
τ
As result of such replacement a complex Lagrange
multiplier dλ and its canonically conjugated
momentum dπ appear. Here one should take into
account corresponding modification of boundary
conditions and boundary term. Return to prelimit form
57
(9) of the amplitude is necessary because of the
constraints h and g still enter in it on equal footing.
As has been shown above the propagator ΨZ does
not depend on ε and so in the limit ∞→ε the
amplitude ΨZ takes the form
{
}.)](~~~
[exp],[
..tb
dd
d
d
d
d
d
d
dZZc
AihigPPHPCCP
ZpdipZD
f
i
c
++−−−++
++=Ζ
Ψ
Ψ ∫∫
λ
λπτ
τ
τ
Equality
}~,)({
)(~
d
d
d
d
d
dd
d
PPChig
higPPH
λπ
λ
+Ψ+++Ω=
=++=Ψ
allows us to write the amplitude in the form
{
}..}],{
[exp],[
tbcc
cZZc
Ai
ZpdipZD
c
f
i
c
+ΨΩ−
−=Ζ ∫∫Ψ
τ
τ
τ
where ( ) d
d
d
c PChig π+++Ω=Ω is the new “BRST
charge” and d
dc P λ~+Ψ=Ψ is “gauge fermion”.
The new “BRST charge" cΩ satisfies to the
conventional nilpotency condition 0},{ =ΩΩ cc but
includes, along with the first class constraints, the
complex second-class constraint higd +≡ and is not
real. The “gauge fermion” cΨ is complex as well due
to nonreality of the Lagrange multiplier dλ and
antighost dP~ . All that is not unessential for developing
consideration because of can be regard as a formal
method.
Relativistic gauge with derivatives for Lagrange
multipliers corresponds to the choice a
ac P λ~=Ψ
then
the BRST Hamiltonian },{ ccc
H ΨΩ=Ψ is equal to
( ) ( ) ,~4~4
~~
~~~
φφφφ
φ
φ
φ
φ
φ
φ
φ
φ
λλ
λλ
λλλ
CPJSiCPJSi
CPiCPi
CPiCPiPPFH
TT
SS
SSa
a
a
ac
+−++
+−+
++−+=Ψ
where .,,,, dSTa φφ=
5. THE SECOND CLASS CONSTRAINTS
AND COMPLEX STRUCTURE ON THE
CONSTRAINED PHASE SPACE
It should be instructive to give a short review of the
main ideas, which are concern to the splitting of the
second-class constraints into holomorphic, and
antiholomorphic sets. We start from a phase manifold
M with a symplectic form
Mω , which is restricted by
a set of real constraints Aϕ , kA ,...,1= . The pullback
of Mω on the constraint surface N via its identical
imbedding into the initial phase space M defines a
presymplectic form Nω there. The Poisson brackets of
the constraints do not vanish on the constraint surface in
general, i. e.
{ } ABC
C
ABPBBA ZC += ϕϕϕ ,
where the structure functions should satisfy the Jacobi
identity. The Hamiltonian vector field space falls into a
subset being tangent to the constraint surface and a
subset, which is skew orthogonal to it. The tangent
fields correspond to the first class constraints aϕ ,
la ,...,1= , and another ones αϕ , lk −= ,...,1α , to the
second-class constraints.
In quantization procedure, if any anomalies are
absent, to the first class constraints one can apply the
Dirac prescription, i. e. impose such constraints on the
physical states .0ˆ =Ψaϕ The BFV-BRST
quantization is far-reaching generalization of this
prescription.
However, if the second-class constraints are present
the BFV-BRST quantization, with its excellent
covariance properties, has very limited applicability.
The majority of the developed methods need in
covariant separation of constraints in classes and either
transform the second class constraints into the first class
one due to extending of the phase space and at the price
of essential losing of transparent physical picture or turn
to the Gupta-Bleuler quantization. It should be noted
that covariant anomaly-free solution of the second-class
constraints with complete exclusion of corresponding
variables is impossible in the nontrivial cases by
definition. The Gupta-Bleuler procedure inserts minimal
deformation of the physics, however in general also
needs in covariant separation of constraints and
existence of complex structure which leads to involutive
(anti)holomorphic set of constraints.
For the case of even variables the existence of such
complex structure was proved in neighborhood of
constraint surface with using of Darboux’s theorem [8].
In our case such structure is inseparable property of the
model and does not need in any proof.
A (pseudo) Hilbert-space, which is constructed on
the phase space M enlarged for first-class constraints
by ghost and antighost coordinates in the BFV-BRST
quantization procedure and does not have a positive
definite metric in general, is too large. It contains the
physical Hilbert space as a subspace. The physical states
are extracted as BRST-closed states which are
annihilated by a Hermitian nilpotent BRST operator
,Ω̂ .0ˆ =ΨΩ phys The physical states are defined up
to BRST-exact states, i. e. states physΨ and
,ˆ ΛΩ+Ψ phys with arbitrary ,Λ are physically
equivalent. The space of physical states is defined by
the cohomologies of the BRST operator, i. e. by the
elements of the coset space .ˆIm/ˆKer ΩΩ
If the second-class constraints are present, it seems
natural to regard a part of the splitted second-class
constraints as the first-class ones and the other part as
gauge-fixing condition. Actually it is BFV-BRST
modification Gupta-Bleuler prescription. The problems
58
in such approach first of all lie in covariant separation of
the constraints in the classes and in the Hermiticity
conditions. Covariant separation is not a problem for
our model of particle with spin as one can see from the
consideration above.
A standard way to ensure uniform conditions on ket
and bra vectors in the Gupta-Bleuler approach consists
in modification of scalar product. This modification
takes transparent form if one accounting its aim realizes
the imaginary unit i (in the mixed real-complex
subspace of the phase space for the complex second-
class constraints) as a matrix
−
=
01
10
i and
introduces a metric operator
−
=
10
01
G so that
iGGi −= . Now redefining the scalar product as
( )ΨΦ≡ΨΦ G,, we have
ΨΦΩ=ΨΩΦ ,ˆˆ,
due to the identity GG ∗Ω=Ω ˆˆ . So we introduce an
indefinite metric in the state space.
One should realize the metric operator in terms of
the variables of the model. Such a problem has well
known solution in holomorphic representation for
harmonic oscillator.
We imply an evident equality
aaaa e
da
dae
∗∗ −
∗
− −=
where the operator a
da
d ˆ=− ∗ is, by definition,
Hermitian conjugated to the creation operator +a in the
holomorphic representation with diagonal creation
operator ∗+ = aâ . Such a problem does not appear for
the first class constraints written in terms of complex
variables because of the complex structure is fictive
because it introduced for convenient and does not
change the basic principle of the theory. In our
formulation a role of a plays that or other of the
complex second-class constraints. Let’s notice, that one
can add linear combination of the first class constraints
to a understood in this way in the exponent, as
working on physical states they should give zero. It
allows appreciably simplify expression in the exponent
of our model. In a result we come to expression for
scalar product already used in work [6].
6. CONCLUSION
In this paper the propagator of free massless particle
with arbitrary spin is represented as BFV-BRST path
integral within the index spinor formalism. Classical
formulation of the theory is given. Its Hamiltonization is
carried out. The constraints are investigated and the
structure functions are obtained. The BRST-charge is
found and it is shown that its rank is 1. Expression for
transition amplitude as path integral is transformed to
the form of transition amplitude for system with the first
class constraints only. Complexification of some phase
variables is carried out that allows one simplifies
calculations and, that is more important, physical
interpretation.
This work was supported in part by INTAS Grant
INTAS-2000-254 and by the Ukrainian National Found
of Fundamental Researches under the Project
№ 02.07/383. We would like to thank I.A. Bandos,
A. Frydryszak, E.A. Ivanov, S.O. Krivonos, J. Lukier-
ski, A.J. Nurmagambetov and D.P. Sorokin for interest
to the work and for many useful discussions.
REFERENCES
1. E.S. Fradkin and G.A. Vilkovisky.
Quantization of relativistic systems with
constraints // Phys. Lett. 1975, v. B55, p. 224-226;
I.A. Batalin and G.A. Vilkovisky. Relativistic S-
matrix of dynamical system with boson and
fermion constraints // Phys. Lett. 1977, v. B69,
p. 309-312; E.S. Fradkin and T.E. Fradkina.
Quantization of relativistic systems with boson and
fermion first- and second-class constraints //
Phys. Lett. 1978, v. B72, p. 343-347; I.A. Batalin
and E.S. Fradkin. Quantization of the Bose-Fermi-
systems subject to the first and second class
constraints. Proc. Int. Seminar “Group Theory
Methods in Physics”, v. 2, Zvenigorod, 1979,
p. 247-269.
2. V.G. Zima and S. Fedoruk. Weinberg
propagator of a free massive particle with an
arbitrary spin from the BFV-BRST path integral //
Class. Quantum Grav. 1999, v. 16, p. 3653-3671.
3. S. Monaghan. BRST Hamiltonian
quantization of the supersymmetric particle in
relativistic gauges // Phys. Lett. 1989, v. B181,
p. 101-105; C. Batlle, J. Gomis and J. Roca.
BRST-invariant path integral for a spinning
relativistic particle // Phys. Rev. v. D40, p. 1950-
1955.
4. G. Papadopoulos. The path integral
quantization of the spinning particles with
extended supersymmetries // Class. Quantum
Grav. 1989, v. 6, p. 1745-1757; M. Pieri and
V.O. Rivelles. BRST quantization of spinning
relativistic particles with extended
supersymmetries // Phys. Lett. 1990, v. B251,
p. 421-426; M. Pieri and V.O. Rivelles. Comments
on “BRST quantization of the extended
supersymmetric spinning particle” // Phys. Rev.
1990, v. D43, p. 2054-2055; Q. Liu and G. Ni.
BRST quantization of the extended
supersymmetric spinning particle // Phys. Rev.
1990, v. D41, p. 1307-1311.
5. I, Bandos, A. Maznytsia, I. Rudychev and
D. Sorokin. On the BRST quantization of the
massless bosonic particle in twistor-like
formulation // Int. Journ. of Mod. Phys. 1997, v.
12, p. 3259-3273.
6. V.G. Zima and S. Fedoruk. Spinning (super-)
particle with a commuting index spinor // JETP
Letters. 1995, т. 61, p. 251-256.
59
7. P.A.M. Dirac. Lectures on Quantum
Mechanics, New York: Yeshiva University, 1964.
8. W. Kalau. On Gupta-Bleuler quantization of
systems with second-class constraints // Int. Journ.
of Mod. Phys. 1993, v. A8, p. 391-406.
60
TRANSITION AMPLITUDE FOR FREE MASSLESS PARTICLE
V.G. Zima1, S.A. Fedoruk2
1Kharkov National University, Kharkov, Ukraine
e-mail: zima@postmaster.co.uk
2Ukrainian Engineering Pedagogical Academy, Kharkov, Ukraine
REFERENCES
|
| id | nasplib_isofts_kiev_ua-123456789-79423 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T16:10:30Z |
| publishDate | 2001 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Zima, V.G. Fedoruk, S.A. 2015-04-01T19:27:44Z 2015-04-01T19:27:44Z 2001 Transition amplitude for free massless particle / V.G. Zima, S.A. Fedoruk // Вопросы атомной науки и техники. — 2001. — № 6. — С. 53-59. — Бібліогр.: 8 назв. — англ. 1562-6016 PACS: 0460G, 0370 https://nasplib.isofts.kiev.ua/handle/123456789/79423 The propagator for massless particle of arbitrary spin is represented as BFV-BRST path integral in index spinor formalism. The classical formulation of the theory is investigated and it is carried out its Hamiltonization procedure. The structure functions are obtained. The BRST-charge of the model is calculated and it is shown, that it has the first rank. The expression for transition amplitude is transformed to the form of amplitude for a system with only the first class constraints. It is shown, that complexification of some phase variable results in the Gupta-Bleuler formalism. In these frameworks it is considered quantization procedure. This work was supported in part by INTAS Grant INTAS-2000-254 and by the Ukrainian National Found of Fundamental Researches under the Project № 02.07/383. We would like to thank I.A. Bandos, A. Frydryszak, E.A. Ivanov, S.O. Krivonos, J. Lukierski, A.J. Nurmagambetov and D.P. Sorokin for interest to the work and for many useful discussions. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Quantum field theory Transition amplitude for free massless particle Амплитуда перехода свободной безмассовой частицы Article published earlier |
| spellingShingle | Transition amplitude for free massless particle Zima, V.G. Fedoruk, S.A. Quantum field theory |
| title | Transition amplitude for free massless particle |
| title_alt | Амплитуда перехода свободной безмассовой частицы |
| title_full | Transition amplitude for free massless particle |
| title_fullStr | Transition amplitude for free massless particle |
| title_full_unstemmed | Transition amplitude for free massless particle |
| title_short | Transition amplitude for free massless particle |
| title_sort | transition amplitude for free massless particle |
| topic | Quantum field theory |
| topic_facet | Quantum field theory |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/79423 |
| work_keys_str_mv | AT zimavg transitionamplitudeforfreemasslessparticle AT fedoruksa transitionamplitudeforfreemasslessparticle AT zimavg amplitudaperehodasvobodnoibezmassovoičasticy AT fedoruksa amplitudaperehodasvobodnoibezmassovoičasticy |