Generating functional in classical Hamiltonian mechanics
We give a brief survey of a path-integral formulation of classical Hamiltonian dynamics that means a functional-integral representation of classical transition probabilities. This functional exhibits a hidden BRST and anti-BRST invariance. Therefore a simple expression, in terms of superfields, is r...
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| Дата: | 2001 |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2001
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| Назва видання: | Вопросы атомной науки и техники |
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| Цитувати: | Generating functional in classical Hamiltonian mechanics / L.G. Zazunov // Вопросы атомной науки и техники. — 2001. — № 6. — С. 383-385. — Бібліогр.: 10 назв. — англ. |
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irk-123456789-800552015-04-10T03:03:09Z Generating functional in classical Hamiltonian mechanics Zazunov, L.G. Quantum fluids We give a brief survey of a path-integral formulation of classical Hamiltonian dynamics that means a functional-integral representation of classical transition probabilities. This functional exhibits a hidden BRST and anti-BRST invariance. Therefore a simple expression, in terms of superfields, is received for the generating functional. We extend the results for discrete classical systems to continuum mechanics. 2001 Article Generating functional in classical Hamiltonian mechanics / L.G. Zazunov // Вопросы атомной науки и техники. — 2001. — № 6. — С. 383-385. — Бібліогр.: 10 назв. — англ. 1562-6016 PASC: 03.40.-i, 03.65.Db http://dspace.nbuv.gov.ua/handle/123456789/80055 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Quantum fluids Quantum fluids |
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Quantum fluids Quantum fluids Zazunov, L.G. Generating functional in classical Hamiltonian mechanics Вопросы атомной науки и техники |
| description |
We give a brief survey of a path-integral formulation of classical Hamiltonian dynamics that means a functional-integral representation of classical transition probabilities. This functional exhibits a hidden BRST and anti-BRST invariance. Therefore a simple expression, in terms of superfields, is received for the generating functional. We extend the results for discrete classical systems to continuum mechanics. |
| format |
Article |
| author |
Zazunov, L.G. |
| author_facet |
Zazunov, L.G. |
| author_sort |
Zazunov, L.G. |
| title |
Generating functional in classical Hamiltonian mechanics |
| title_short |
Generating functional in classical Hamiltonian mechanics |
| title_full |
Generating functional in classical Hamiltonian mechanics |
| title_fullStr |
Generating functional in classical Hamiltonian mechanics |
| title_full_unstemmed |
Generating functional in classical Hamiltonian mechanics |
| title_sort |
generating functional in classical hamiltonian mechanics |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2001 |
| topic_facet |
Quantum fluids |
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http://dspace.nbuv.gov.ua/handle/123456789/80055 |
| citation_txt |
Generating functional in classical Hamiltonian mechanics / L.G. Zazunov // Вопросы атомной науки и техники. — 2001. — № 6. — С. 383-385. — Бібліогр.: 10 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT zazunovlg generatingfunctionalinclassicalhamiltonianmechanics |
| first_indexed |
2025-07-06T03:59:26Z |
| last_indexed |
2025-07-06T03:59:26Z |
| _version_ |
1836868558396588032 |
| fulltext |
GENERATING FUNCTIONAL
IN CLASSICAL HAMILTONIAN MECHANICS
L.G. Zazunov
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
e-mail: zazun@online.kharkiv.com
We give a brief survey of a path-integral formulation of classical Hamiltonian dynamics that means a functional-
integral representation of classical transition probabilities. This functional exhibits a hidden BRST and anti-BRST
invariance. Therefore a simple expression, in terms of superfields, is received for the generating functional. We
extend the results for discrete classical systems to continuum mechanics.
PASC: 03.40.-i, 03.65.Db
1. INTRODUCTION
The functional-integral approach initially developed
for quantum mechanics by R. Feynman [1], is now
widely used in classical statistical physics and statistical
hydrodynamics. But recently this method is widely
spread in classical Hamiltonian mechanics as well. A
first attempt to provide a path-integral formulation of
classical mechanics was made in [2-4]. There the
evolution of dynamical system is considered as a
functional-integral representation of classical transition
probabilities with such measure as it gives a weight
“one” to the classical paths and a weight “zero” to all
other, just as functional δ-function for the classical
equations of motion. With the help of anticommuting
“ghosts” this measure can be rewritten as the exponents
of a certain action S . We provide an interpretation for
the “ghosts” fields as being the well-known Jacobi
fields (or “geodesic deviations”), which describe
behavior of dynamical systems. The remarkable point
about the action S is that it exhibits an unexpected
BRST and anti-BRST type invariant. This invariance
allows for simple expressions, in terms of superfields, of
this generating functional.
In this article we give a brief survey of applicating
the method of generating functional in classical
mechanics. The results for Hamiltonian systems with
finite degrees of freedom extend to continuum
mechanics. We illustrate how the method of the
generating functional, so much useful in the quantum
theory and statistical physics, is found also in
determination of some problems in Hamiltonian
mechanics.
2. THE CONSTRUCTION OF AN
EFFECTIVE ACTION
Consider the continuous medium and let its
mechanic properties are described by the set of
functions ( ),, txαϕ where α =1,2,…n. The equations of
classical evolution are presented by a set of functional-
form equations:
( )[ ] ,0, =ϕα txF
(1)
where α is corresponding to different field components.
Now accordingly to [2,3] we introduce a classical
generating functional Z(J) in the form of a functional
integral:
[ ] ( ) ( ) ( )[ ]
( ) ( )[ ] ,,,exp
,,,
∫
∏ ∫
αα
αα
α
α
ϕ
×ϕ−ϕδϕ=
txtxJxdtd
txtxtxDJZ cl
(2)
where ( )txcl ,αϕ is the solution of Eq. (1), ),( txJ
α are
the external sources. Then the δ – functional may be
written such as
( ) ( )[ ] ( )( )[ ] .det,,,
δ ϕ
δ
ϕδ=ϕ−ϕδ α
ααα
F
txFtxtx cl
The calculation of the determinant in last expression is
very a difficult problem and depends essentially from
the structure of the functional ( )( )., txF ϕα But we
leave these difficulties, reexpressing the determinant
over two anticommuting “ghosts” ( ) ).,(,, txctxc
αα
Then, representing the delta-functional as a
functional-integral over auxiliary field ),( txλ one
obtains a field theory representation of the generating
functional Z(J):
( )
],),(),(
),,,(exp[
dtxdtxtxJ
cciScDDcDDJZ eff
αα
αα
α
αα
ϕ
+λϕλϕ=
∫
∏
(3)
where
( ) ( ) ( )[ ]
( ) ( ) ( ).,,
,,,,,
tycFtxctdtdydxdi
dtxdtxFtxccSeff
′
δ ϕ
ϕδ′
−ϕλ=λϕ
β
β
α
α
αα
∫
∫
(4)
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 383-385. 127
So, we obtain a way to apply the field theory methods to
problems of classical mechanics after expressing the
generating functional (2) through the effective action
(3), or extending the structure of physical field variables
with the auxiliary field ),( txλ and two Grassman
anticommuting fields, ),( txc
and ),( txc
, called
“ghosts”.
3. HIDDEN BRST SYMMETRY
Turn our attention on the wonderful property of the
symmetry of effective action (3) that exhibits BRST-
like [5,6] invariance of the form:
( ) ( ),,, txctx
αα ε=δ ϕ ( ) ,0, =δ α txc
( ) ( ),,, txitxc
αα λε=δ ( ) .0, =δ λ α tx (5)
where ε and ε are the anticommuting number
parameters. It is possible to prove that the invariance of
the effective action (3) under transformations (5) does
not depend on the shape and structure original
functional. Therefore arises a question about a character
of symmetry (3) under anti-BRST transformations:
( ) ( ),,, txitxc
αα ε λ−=δ
( ) ,0, =δ α txc
( ) ,0, =δ λ α tx
).,(),( txctx
αα ε−=δ ϕ (6)
It is possible to prove [7] that after transformation (6)
we have the following expression for the variation of
effective action:
( ) ( ) ( ) ( )
.,,
δ ϕ
ϕδ
−
δ ϕ
ϕδ′λ′−=δ
α
β
β
α
βα∫
FFtyctxtdtdydxdSeff
In general it is nonzero. But this variance of the
effective action vanisches only if the functional F[
),( txϕ ] satisfies the condition of “potentiality”, so if it
may be present as:
( )[ ] ( )
( ) .
,
,
tx
StxF
α
α δ ϕ
ϕδ=ϕ
It is evident, that the functional S(φ) means a classical
action, and the Eq. (1) describes a condition of it
minimum. So, the classical dynamics equations obtained
from the minimum action principle can be described by
means of generating functional (3) with the effective
action
( ) ( ) ( )
( ) ( ) ( ) ( ) ,,
,,
,
,
,,,,
2
tdtdydxdtyc
tytx
Stxci
dtxd
tx
StxccSeff
′′
δ ϕδ ϕ
δ−
δ ϕ
δλ=λϕ
β
βα
α
α
α
∫
∫
(7)
which has property of invariance under the BRST and
anti-BRST transformations.
4. HAMILTONIAN DYNAMICS,
CONSTRUCTION OF
“SUPERHAMILTONIAN”
So far, our study of classical motion was founded by
the functional of action without using Hamilton’s
properties of its equations. Now, let us consider the
consequence, if our system is described by Hamilton
equations. The equations of motion for the continuum
system with Hamiltonian H(φ) has the form:
( ) ( ) ( )
( ) ,
,
,
tx
Hxtx abb
αδ ϕ
ϕδω=ϕ (8)
where )(xω is the local symplectic two-form and the
corresponding symplectic matrix has the property:
( ) ( ) .a
cbc
ab xx δ=ωω
Then we construct the corresponding generating
functional for equations (8) using the above-mentioned
method. Now from (4) and (7) we obtain the expression
for the generating functional of Hamilton mechanics:
( ) [ ] ×λϕ=ψψη ∏ ∫ ∫α
α
α
α
a
dtLcDDcDDJZ exp,,,
,)],(),(),(),(
),(),(),(),([exp
dtxdtxtxctxctx
txtxtxtxJ
α
α
α
α
α
αα
α
ψ+ψ
+λη+ϕ∫ (9)
where the Lagrange’s function (“super-Lagrangian”)
has the form:
.)],()]()[,(
)])(),()[,([),,,(
xdtxcHtxci
xtxtxccL
b
bc
aca
bta
b
aba
a
ϕδδω+δ∂
+ϕω−ϕλ=λϕ ∫ (10)
Here we introduce the new external sources, ),( txη
and ),( txψ corresponding to the additional fields
),( txλ and ),( txc
respectively. From the effective
Lagrangian (10) we obtain as usually the general
Hamilton’s function (“super-Hamiltonian”):
)].,()(),(
)(),([),,,(
txcHtxci
HtxdtxdccH
b
bc
ac
b
ab
ϕδδω
+ϕδωλ=λϕ
α
α∫ (11)
Here an essential moment is that expression (11) can be
interpreted as a Hamiltonian’s function to be correct, we
have to make sure that the fields ),( txλ and ),( txϕ ,
as well as ),( txc
and ),( txc
are pairs of canonically
conjugate variables with the following (graded) Poisson
structure:
( ) ( ){ } ( ),,,, xxtxtx abba ′−δδ=′λϕ
( ) ( ){ } ( ).,,, xxitxctxc abab ′−δδ−=
The formulation of classical mechanics on the basis
of introducing the generating functional with an
effective Lagrangian (10) gives a possibility to describe
from the unified point of view such problems as
obtaining the equations of dynamics, problems of
ergodicity and stochastization of Hamilton system. We
128
).,(),( txctx
αα ε−=δ ϕ
note here that after field's variation in generating
functional (9) with the Lagrange’s function (10) and by
using the minimum action principle we obtain the
following equations:
( ) ( ) ,0, =ϕδω−ϕ Htx b
aba (12.a)
They, obviously, form the initial Hamilton’s equations.
The equations of “motion” of “ghosts” ),( txc
and
),( txc
are defined by the form of the second variation
of the effective action (7) and we find:
( )[ ] ( ) ,0, =ϕδδω−δ∂ txcH b
bc
aca
bt
(12.b)
( )[ ] .0),( =ϕδδω+δ∂ txcH abc
aca
bt
(12.c)
It was shown in [4] that the “ghosts” ),( txc
and
),( txc
can be interpreted as Jacobi fields ),( txδ ϕ ,
i.e. the infinitesimal displacement between two classical
trajectories (or “geodesic deviations”) [9]. Such
correspondence between the “ghosts” and fields of
Jacobi explains that there is a possibility to apply them
in such problems as beginning of a dynamical chaos in
the Hamilton’s system with the exponential running of
trajectories.
5. SUPERFIELDS IN CLASSIC MECHANICS
Now note the importance meaning of the invariance
of effective action (7) under simultaneous BRST and
anti-BRST transformations. Just this extended
symmetry of the action gives a superfield description of
classical mechanics. Let us define the following
superfield ),,( θφ tx on the space of initial fields as
their combination:
( ) ( ) ( )
( ) ( ),,,
,,,,,
txitxc
txctxtx
b
ab
b
ab
aaa
λθ ωθ+ωθ
+θ+ϕ=θθφ
(13)
where θ are the Grassmann variables. In terms of
superfield Lagrangian (10) and also Hamiltonian (11)
assume a very simple form:
( ) ( ) ,
2
1, ∫
φ−φωφθθ=φφ ab
ab
a HddiL (14)
( ) ( ).∫ φθθ=φ HddiH (15)
Note, that the functional dependence of “super-
Hamiltonian” (15) from the superfield has such
structure as an initial Hamiltonian in dynamics
equations (8). The generating functional (9) in terms of
superfield obtaines a form:
∏ ∫ ∫ ∫α αα
αα θθφΓ
+θθφ
φ=Γ
].
)(
exp[)(
dtdd
dtddL
iDZ
Now we see the similarities between this functional
integral and the quantum mechanical one. Obviously,
the analysis of phase space structure of classical
dynamics reduces to the study of the superfield theory.
6. CONCLUSION
In this paper we have reviewed the general
framework of a path-integral approach to classical
Hamiltonian dynamics. We confirm that our path-
integral method is the correct counterpart of the
operatorial approach to classical mechanics. The crucial
elements of the operatorial formalism mentioned above
are the “classical commutation relations” which follow
from the classical path- integral.
In this approach we had a lot of extra variables
(auxiliary fields) besides those labelling the phase-space
of the original mechanical system. It was discovered
that these extra variables had a beautiful geometric
meaning and therefore unexpected universal symmetries
generated by the functional technique were received.
We call the reader’s attention to a very important
point. The invariance that we have found does not rely
on any particular form of the Lagrangian. It is valid for
any action, so its consequences must be very general
independently on the initial construction of the phase
space of the classical system.
ACKNOWLEDGMENTS
The author is obliged to A.S. Bakai for constant
interest to this work and also to Ju.P. Stepanovsky for
useful remarks.
REFERENCES
1. R.P. Feynman and A.R. Hibbs. Quantum
Mechanics and Path-Integrals, Moscow: ”Mir”,
1968, 382 p. (in Russian).
2. E. Gozzi. Hidden BRS invariance in classical
mechanics // Phys. Lett. B, 1988, v. 201, p. 525-
528.
3. E. Gozzi, M. Reuter, W.D. Thaker. Hidden
BRS invariance in classical mechanics. II // Phys.
Rev. D. 1989, v. 40, p. 3363-3375.
4. E. Gozzi, M. Reuter. Algebraic
characterization of ergodicity // Phys Lett. B, 1989,
v. 233, p. 283-287.
5. T.-P. Cheng and L.-F. Li. Gauge Theory of
Elementary Particle Physics, Moscow: “Mir”,
1987, 624 p. (in Russian).
6. D.M. Gitman and I.V. Tjutin. Canonic
quantization of fields with constraints, Moscow:
“Nauka”, 1986, 216 p. (in Russian).
7. R.A. Kraenkel. Anti-BRS invariance and
Lagrangianity in classical mechanics // Europhys.
Lett. 1988, v. 6, p. 381-385 p.
8. E. Gozzi, M. Reuter. Lyapunov exponents,
path-integrals and forms // Chaos, Solitons &
Fractals. 1994, v 4, p. 1117-1139.
9. C. De Witt-Morette et al. Path integration in
non-relativistic quantum mechanics // Phys. Rep.
1979, v. 50, p. 255-372.
10. J. Wess and J. Bagger. Supersymmetry and
Supergravity. Moscow: “Mir”, 1986, 180 p. (in
Russian).
129
IN CLASSICAL HAMILTONIAN MECHANICS
L.G. Zazunov
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
e-mail: zazun@online.kharkiv.com
We give a brief survey of a path-integral formulation of classical Hamiltonian dynamics that means a functional-integral representation of classical transition probabilities. This functional exhibits a hidden BRST and anti-BRST invariance. Therefore a simple expression, in terms of superfields, is received for the generating functional. We extend the results for discrete classical systems to continuum mechanics.
1. INTRODUCTION
The functional-integral approach initially developed for quantum mechanics by R. Feynman [1], is now widely used in classical statistical physics and statistical hydrodynamics. But recently this method is widely spread in classical Hamiltonian mechanics as well. A first attempt to provide a path-integral formulation of classical mechanics was made in [2-4]. There the evolution of dynamical system is considered as a functional-integral representation of classical transition probabilities with such measure as it gives a weight “one” to the classical paths and a weight “zero” to all other, just as functional δ-function for the classical equations of motion. With the help of anticommuting “ghosts” this measure can be rewritten as the exponents of a certain action . We provide an interpretation for the “ghosts” fields as being the well-known Jacobi fields (or “geodesic deviations”), which describe behavior of dynamical systems. The remarkable point about the action is that it exhibits an unexpected BRST and anti-BRST type invariant. This invariance allows for simple expressions, in terms of superfields, of this generating functional.
ACKNOWLEDGMENTS
REFERENCES
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