The Dirac Conjecture and the Non-uniqueness of Lagrangian

Wang Yong-Long

Department of Physics,School of Science,Linyi University

The First Sino-Americas Workshop and School on the Bound-State Problem in Continuum QCD Oct. 22-26, 2013, USTC, Hefei

Department of Physics,Nanjing University

Introductions

Non-uniqueness of Lagrangian

Cawley’s Example

Outline

. arXiv:1306.3580

“Counterexample”

Conclusions

IntroductionsIntroductions

Dynamical Systems

NewtonFormalism

LagrangeFormalism

HamiltonFormalism

SingularLagrangianSystems

ConstrainedHamiltonianSystems

GaugeTheories

QuantizationOf GaugeSystems

Symmetries

The Dirac Conjecture

Quantization of Gauge Systems, edited by M. Henneaux, C. Teitelboim, Princeton University, 1991

Gauge Fields Introduction to Quantum Theory, edited by L. D. Faddeev and A. A. Slavnov, The Benjamin, 1980.

Classical and Quantum Constrained Systems and Their Symmetries. Zi-Ping Li, Press of Beijing University of Technology, 1993 (in Chinese) Constrained Hamiltonian Systems and Their Symmetries. Zi-Ping Li. Press of Beijing University of Technology, 1999 (in Chinese)

Symmetries in Constrained Canonical Systems, Li Zi-Ping, Science Press, 2002

Quantum Symmetries in Constrained Systems. Zi-Ping Li, Ai-Min Li. Press of Beijing University of Technology, 2011 (in Chinese)

Symmetries in Constrained Hamiltonian Systems and Applications. Yong-Long Wang, De-Yu Zhao, Shandong People’s Publishing House, 2012 (in Chinese)

①①

Regular systems Canonical Systems

Quantization of Gauge Systems

IntroductionsIntroductions

CanonicalQuantization

Path IntegralQuantization

Dirac’sFormalism

Faddeev-Jackiw’sFormalism

BRST Batalin-Fradkin-Vilkovsky

Faddeev-Popov

Faddeev-Senjanovic

BRST Batalin-Fradkin-Vilkovsky

The Dirac Conjecture

②②

IntroductionsIntroductions③③

Lagrange Formalism Hamilton Formalism

),,(),( 1 NqqqqqL ),(),( qqLqppqHC

),,1( Niq

Lp

ii

.isofrankThe2

Rqq

L

ji

),,1(0),( Mmpqm mmCT HH

MRN

[ , ] 0m m TH

0),( pqkThe higher-stage constraints

The primary constraints

0

0),(

0),( ,equality" strong" denotes ""

0),( ,equality"weak " denotes ""

p

f

q

f

pqf

pqf

pqf

mm

mmmmCT

C

m

H

UHH

qqLqpH

pq

~~

),(

0),(

first class constraints :

[ , ] 0 ( 1, , )( )

second class constraints :

( 1, , )

m

m m

mm

m MM M M

m M

1

[ , ] [ , ] [ , ] [ , ] 0

[ , ] [ , ]

m m T m C n m n m m m

n mn m mn m n

H H U

U C H C

According to the consistency of the , we can obtain the Lagrange multipliers with respect to primary second-class constraints

m

IntroductionsIntroductions④④

0],[ mm

0

0

( ) {[ , ] [ , ]}

( ) {[ , ] [ , ]}

[ , ] [ , ]

m m

m m

m m

g t g t g H g

g g t g t g H g

t g H t g

In terms of the total Hamiltonian, for a general dynamical variable g depending only on the q’s and the p’s, with initial value g0, its value at time ist

IntroductionsIntroductions⑤⑤

small arbitrary

ations transformgauge ofgenerator ism

tions.transforma

gauge ofgenerator are],[ of bracketsPoisson The nm

IntroductionsIntroductions⑥⑥

get we, functions generating with

onansformaticontact tr second aapply and, functions

generatingon with ansformaticontact tr afirst Apply )1(

m

nn

m

0 [ , ] [ [ , ], ]n n m n n mg g g g g

(2) We apply the two contact transformations in succession

in reverse order, we obtain finally

0 [ , ] [ [ , ], ]m m n m m ng g g g g

[[ , ], ]

[[ , ], ]

[[ , ], ] [[ , ], ] [ , ], ] 0

[ ,[ , ]]

m n

n m

m n m n

g

g

A B C B C A C A B

g g g g

IntroductionsIntroductions⑦⑦

It is arbitrary

[ , ] is generator of gauge transformations.m n

P. C. Dirac, Can. J. Math. 2, 147(1950); Lectures on Quantum Mechanics

1

1 1

[ , ] [ , ] [ , ] 0

1, ,

m m m T m m m mH H

m M

IntroductionsIntroductions⑧⑧

The secondary constraints can be deduced by the consistency of the primary constraints as

(1)The original Lagrangian equations of motion are inconsistent.

(2)One kind of equations reduces as 0=0.(3)To determine the arbitrary function of t

he Lagrangian multiplier. (for second-class constraints)

(4)Turn up new constraints.

1[ , ] [ , ] 0m m m m mH

IntroductionsIntroductions⑨⑨

1m generators1m generators

? left by Dirac

class-first

class-second

1

1~

m

m

m

Dirac conjecture: All first-class constraints are generators of gauge transformations, not only primary first-class ones.

Non-uniqueness of LagrangianNon-uniqueness of Lagrangian①①

1 ( )( , ) ( , ) ( , ) m md

L q q L q q L q qdt

1 ( )( , ) ( , ) ( , ) m m

T T m m

dH q p H q p H q p

dt

( , ) ( , ) ( , )C m mH q p H q p U q p

m

The prime Hamiltonian consists of the canonical Hamiltonian and all primary second-class constraints, the number can be determined by the rank of the matrix .[ , ]m n M M

denotes all first-class primary constraints.

Classical Mechanics, H. Goldstein, 1980.Classical and Quantum Constrained Systems and Their Symmetries. Zi-Ping Li, 1993 (in Chinese)

②②Non-uniqueness of LagrangianNon-uniqueness of Lagrangian

0

10

0

( )

[ , ]

{[ , ] [ , ] [ ,[ , ]]}T

m m m m m m T

dgg t g t

dt

g g H t

g g H g g H t

time.of

functionsarbitrary

are and

because arbitrary,

is

mm

mmm

]],[,[]],[,[ mmmmmm gHg

as at time ,

on time depending explicitly without variablephysicalarbitrary an

of valueobtain thecan we, valueinitial with the, theUsing 01

tgt

gHT

③③[ , ] : is arbitrary.

is generator of gauge transformations.

[ ,[ , ]] : is arbitrary when

[ , ] is first-class.

[ , ] is generator of gauge transformatio

m m m

m

m m m

m

m

g

g H

H

H

ns.

[ ,[ , ]] : is arbitrary when

[ , ] is first-class.

[ , ] is generator of gauge transformations.

m m m m m m

m m

m m

g

Non-uniqueness of LagrangianNon-uniqueness of Lagrangian

④④Non-uniqueness of LagrangianNon-uniqueness of Lagrangian

1

1

1: is generator of gauge transformations.

m

m mm

1 1

1

2

( , ) ( , )

( )( )( , ) ( , ) m mm m

L q q L q q

ddL q q L q q

dt dt

1 1

1

2

( , ) ( , )

( )( )( , ) ( , )

T T

m mm mT m m

H q p H q p

ddH q p H q p

dt dt

A new annulation

Terminate: No new constraint.

⑤⑤Non-uniqueness of LagrangianNon-uniqueness of Lagrangian

0th-stage

( , )L q q q

qqLpi

),(

m

T m mH H

1 ( , ) ( )m m

dL L q p

dt

1st-stage

],[1 Tmm H

1 1

11T m mH H

⑥⑥Non-uniqueness of LagrangianNon-uniqueness of Lagrangian

)( 11

1

im

ii

idt

dLL

ith-Stage

],[ 1

1

iTmm H

ii

111 ii mmi

iT HH

1

11( )

S

S Sm S

dL L

dt

Sth-Stage1

1[ , ]S S

Sm m TH

1 11 S S

ST S m mH H

No new constraints.

End!

⑦⑦Non-uniqueness of LagrangianNon-uniqueness of Lagrangian

(2) The total time derivatives of constraints to Lagrangian may turn up new constraints. In terms of the stage total Hamiltonian, the consistencies of constraints can generate all constraints implied in the constrained system.

The Dirac Conjecture is valid.

PRD32,405(1985); PRD42,2726(1990)

(1) The Poisson brackets [ , ] are generators of

gauge transformations that can be deduced from the

non-uniqueness of Lagrangian. The Dirac conjecture

is implied in the non-uniqueness of Lagrangian.

m H

Cawley’s ExampleCawley’s Example①①

N

nnnnn zyzxzyxzyxL

1

2 )2

1(),,,,,(

R. Cawley, PRL, 42,413(1979); PRD21, 2988(1980)

.,0,

are and, respect to with Momenta

nzynx

nnn

xppzp

zyx

nnn

.,at vanishing ionsfor variat )( principle

al variation theof extremals" on the evaluated" means "" where

,0,02

1,0

are equations Lagrangian-Euler The

0

2

0f

it

t

nnnznynx

tttqdtLSS

zyxLzLzL

f

nnn

L. Lusanna, Phys. Rep. 185,1(1990); Riv. Nuovo Cimento 14(3), 1(1991)

Cawley’s ExampleCawley’s Example②②

2

Under the Noether transformations , one gets

the Noether identities as

10, 0.

2n

n

y n

δy ε(t)

p z

2

There are 2 constraints

10, 0.

2ny n

N

p z

1

The "tertiary constraints" given by Cawley are genuine

first-order equations of motion

[ , ] 0, .n n n n

NF F

x n n T T x z n yn

p z z H H p p p

L. Lusanna, Riv. Nuovo Cimento 14(3), 1-75(1991)

Cawley’s ExampleCawley’s Example③③0th-stage

N

nnnnn zyzxzyxzyxL

1

2 )2

1(),,,,,(

0nn yp

2

1 1

1( )

2n n n

N N

T x z n n n yn n

H p p y z p

1th-stage 1

1

( , , , , , ) ( )N

x y z n nn

dL x y z p p p L

dt

1 21[ , ] 0

2n n n T nH z

1 21

1

1

2

N

T n nn

H H z

2th-stage2 1 1( , , , , , ) ( )x y z n n

dL x y z p p p L

dt

2 1 1

1[ , ] 0nn n n T n xH z p

2 1

1n

N

T T n n xn

H H z p

Cawley’s ExampleCawley’s Example④④

3th-stage3 2 2( , , , , , ) ( )x y z n n

dL x y z p p p L

dt

2

3 2 2 2 2[ , ] 0n n n T xH p

3 2 2

1n

N

T T xn

H H p

2 2 2

1

2

The extended Hamiltonian is

1 1( ),

2 2

which generates canonical equations are

2 , ,

10, ,

2

n n n n n

n n n

n n n n

N

E x z n n n y n n n n x n xn

n z n n n x n n n x

x y n z n n n n n x

H p p y z p z z p p

x p z p y z p

p p z p y z z p

Cawley’s ExampleCawley’s Example⑤⑤

2

2

010, 0

2 00, 0 0

n

n

n n n

y

y n

n

n x x x

pp z

zz p p p

0

0

0

2

2

n

n

x

xn

n

p

pz

z

Cawley’s ExampleCawley’s Example⑥⑥

2

2

0

1( )

2( ) 0

(0) , ( ) contradiction(0) ( )

suppose (0) ( )

n n n n

nn n T

n nn n

S dt x z y zz t

r r r T rz z T

z z T

2

2

0

1( )

2( ) 0

(0) , ( )(0) ( )

suppose (0) ( )

n n n n

nn n T

n nn n

S dt x z y zz t

r r r T rz z T

z z T

In the Cawley example, we must consider the secondary constraints.

A. A. Deriglazov, J. Phys. A40, 11083(2007); J. Math. Phys. 50,012907(2009)

““Counterexample”Counterexample”①①

Lagrangian: ( , , , , ) exL x z x y z xz y

Canonical equations:

, , ,

, 0, 0

xz x y

xx y z

x p y z p p e

p z e p p x

The total Hamiltonian: [ ]xT x z yH p p p e

Euler-Lagrange equations:

0, 0, 0.x xx e x z ye

Momenta: , ,xx y zp z p e p x

Primary constraint: 0xyp e

The Secondary constraint: 0.xe x

““Counterexample”Counterexample”②②

1

The extended Hamiltonian:

( ) ,x xT x z y zH p p p e e p

Canonical equations:

, , ,

, 0, 0 ,

, 0

xz x

x xx z y z

x xy

x p y z p e

p e e p p p

p e e x

ConclusionsConclusions①①

(1)The Dirac conjecture is valid to a system with singular Lagrangian.

(2) The extended Hamiltonian shows symmetries more obviously than the total Hamiltonian in a constrained system.

Thanks!