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arXiv:1204.1361v2 [hep-th] 3 Aug 2012

PREPRINT-V2-08-2012

Converting Classical Theories to Quantum Theories by

Solutions of the Hamilton-Jacobi Equation

Zhi-Qiang Guo and Iv´ an Schmidt

Departamento de F´ ısica y Centro Cient´ ıfico Tecnol´ ogico de Valpara´ ıso,

Universidad T´ ecnica Federico Santa Mar´ ıa,

Casilla 110-V, Valpara´ ıso, Chile

E-mail: zhiqiang.guo@usm.cl, ivan.schmidt@usm.cl

Abstract: By employing special solutions of the Hamilton-Jacobi equation and tools from lattice theories, we

suggest an approach to convert classical theories to quantum theories for mechanics and field theories. Some

nontrivial results are obtained for a gauge field and a fermion field. For a topologically massive gauge theory,

we can obtain a first order Lagrangian with mass term. For the fermion field, in order to make our approach

feasible, we supplement the conventional Lagrangian with a surface term. This surface term can also produce

the massive term for the fermion.

Keywords: De Donder-Weyl Theory, Topologically Massive Gauge Theory, Mass Generation for Fermion

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Contents

1 Introduction

2

2 Examples of Mechanics

2.1Linear Theories: Harmonic Oscillator

2.1.1 Path Integral Quantization for the Harmonic Oscillator

2.1.2Hamilton-Jacobi Equation for the Harmonic Oscillator

2.1.3The Pauli Short-Time Kernel

2.1.4 Deriving Discretized Lagrangians from Special Solutions of the Hamilton-Jacobi Equation

2.2Nonlinear Theories: Double-Well Potential

2.2.1Tools for Solving the Nonlinear Hamilton-Jacobi Equation

2.2.2Derivation of the Lattice Lagrangian

2

2

2

3

4

5

6

6

8

3Examples of Scalar Field Theories

3.1Linear Theories: Massive Scalar Theories

3.1.1Covariant Hamilton-Jacobi Equations

3.1.2Solutions of Covariant Hamilton-Jacobi Equations

3.1.3Derivation of the Discretized Lagrangian

3.2Nonlinear Theories: Scalar Theories with λφ4Potential

3.2.1De Donder-Weyl Equation and Its Solution

3.2.2Derivation of the Discretized Lagrangian

9

9

9

10

11

13

13

15

4 Topologically Massive Gauge Theory

4.1De Donder-Weyl Equation for Gauge Theories

4.2Derivation of the Discretized Lagrangian

15

16

18

5Mass Generating Mechanism for a Fermion Field

5.1De Donder-Weyl Theory for a Fermion Field

5.2Derivation of the Lattice Lagrangian for a Fermion field

20

20

23

6SU(2) Yang-Mills Gauge Theory

6.1De Donder-Weyl Equation of the Yang-Mills Gauge Theory

6.2 Derivation of the Lattice Lagrangian

23

23

26

7Further Discussions

7.1Discussions on Solution Dependence

7.2comparisons with conventional lattice gauge theories

27

27

28

8Conclusions28

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1Introduction

Quantum theories have achieved tremendous success in the passed century. There are two conventional ap-

proaches to convert classical theories to quantum theories: canonical quantization and path integral quan-

tization.These two approaches employ two basic objects in classical theories: the Hamiltonian and the

Lagrangian, which are two equivalent tools to formulate classical theories. However, besides these, we know

there is a third equivalent way to formulate classical theories: the Hamilton-Jacobi equation. In this paper,

we show that it is possible to derive quantum theories from classical theories by employing special solutions of

the Hamilton-Jacobi equation, together with some tools from lattice theories. This third approach turns out

to be consistent with the path integral approach for most of the cases. However, we also can obtain several

new sectors for gauge fields and fermion fields. Specifically, we can obtain a massive Lagrangian of the first

order for the topological massive gauge theory introduced in [1, 2]; While for a fermion field, we find that the

mass of a fermion can be produced by a surface term, which is a mass generating mechanism similar to the

topological massive gauge theory.

This paper is organized as follows. In section 2, starting with classical mechanics, we introduce the

basic tools and methodology used to derive quantum mechanics from special solutions of the Hamilton-Jacobi

equation. We also introduce tools to find solutions of the Hamilton-Jacobi equation for nonlinear theories in

section 2. We turn to scalar field theories in section 3. In this section, we introduce the covariant Hamilton-

Jacobi equation for field theories, that is, the De Donder-Weyl approach [3–5] for field theories. Along with

the discussions regarding mechanics, we introduce tools for solving the De Donder-Weyl equation for nonlinear

field theories, and stress the differences between mechanics and field theories. In section 4 and section 5, we

discuss the topological massive gauge theories and fermion fields separately. Section 6 is devoted to SU(2)

Yang-Mills theories. We only obtain restricted solutions for Yang-Mills theories, and no firm conclusions can

be drawn from these restricted solutions. We provide further discussions and conclusions in sections 7 and 8.

2 Examples of Mechanics

We discuss mechanics in this section. Take the harmonic oscillator, for example. The logic structure of these

discussions is as follows: First we present the path integral quantization for the harmonic oscillator and its

lattice definition; Then we display its Hamilton-Jacobi equation and find several solutions for the Hamilton-

Jacobi equation. Based on the discussions above, we can find a close relation between the lattice definition

of path integral quantization and a special solution of the Hamilton-Jacobi equation. This relation will make

it feasible to convert classical theories to quantum theories by the special solution of the Hamilton-Jacobi

equation.

2.1Linear Theories: Harmonic Oscillator

2.1.1 Path Integral Quantization for the Harmonic Oscillator

For a massive harmonic oscillator in one dimension, its Lagrangian is given by

L(x, ˙ x) =1

2m˙ x2−1

2mω2x2,

(2.1)

where ˙ x =

defined by the path integral in configuration space

dx

dtis defined. For a time evolution from ta to tb, the Green function or the Feynman kernel is

K(xb,tb;xa,ta) =

?x(tb)=xb

x(ta)=xa

Dx(t)NexpiS(tb,ta),S(tb,ta) =

?tb

ta

L(x, ˙ x)dt,

where N is a normalization factor. Here and hereafter we use natural units, so the Planck constant ? = 1 is

assumed. Discretizing the interval [ta, tb] into N equivalent smaller intervals with length ǫ =tb−ta

N

, that is,

– 2 –

Page 4

let ta= t0, t1= t0+ ǫ, ··· , tk= t0+ kǫ, ··· , tb= tN, then the path integral (2.2) can be regarded as the

limit of multi-integrals and can be expressed as the following lattice version

K(xb,tb;xa,ta) = lim

N→∞

ǫ→0

N−1

?

j=1

??

dxj

?m

2πiǫ

?1

2exp

?

iǫ

N−1

?

j=1

m

2

??xj+1− xj

ǫ

?2

− ω2?xj+1+ xj

2

?2???

.

(2.2)

For infinitesimal time evolution ǫ, we have the short-time Feynman kernel

K(xj+1,tj+ ǫ;xj,tj) =

?m

2πiǫ

?1

2exp

?

iǫm

2

??xj+1− xj

ǫ

?2

− ω2?xj+1+ xj

2

?2??

.

(2.3)

So the Feynman kernel (2.2) of finite time evolution also can be regarded as multi-convolutions of short-

time Feynman kernels (2.3)

K(xb,tb;xa,ta) = lim

N→∞

ǫ→0

? ?

···

?

dxN−1···dx2dx1

K(xb,tN−1+ ǫ;xN−1,tN−1)···K(x2,t1+ ǫ;x1,t1)K(x1,ta+ ǫ;xa,ta).

(2.4)

The Gaussian integrals in (2.2) can be performed in sequence and we can get a closed formulation for the

Feynman kernel

K(xb,tb;xa,ta) =

?

mω

2πisinω(tb− ta)

?1

2exp

?

imω

2sinω(tb− ta)[cosω(tb− ta)(x2

b+ x2

a) − 2xbxa]

?

.

(2.5)

2.1.2Hamilton-Jacobi Equation for the Harmonic Oscillator

The Hamilton-Jacobi equation was independently introduced by Hamilton and Jacobi from different ap-

proaches.In this section, we only give a pedagogical introduction based on the independent integral of

Hilbert [6], which has been applied to field theories by De Donder [3] and Weyl [4, 5]. We caution that our

introductions only work well for regular Lagrangians, which we always work with in this paper. For non-regular

Lagrangians, we refer the reader to the more rigorous discussions in [4, 5, 7–9].

For a system of mechanics in one dimension, its classical aspects can be formulated by a Lagrangian

L

?

mation p =∂L

∂q, we can get the corresponding Hamiltonian H(p,q,t) = ˙ qp−L. The Hamilton-Jacobi equation

can be derived as follows. Suppose there is a function S(q,t) that depends only the coordinate but not its time

derivative ˙ q(t), then the Lagrangian can be regarded as an independent integral of Hilbert with the following

meaning

q(t), ˙ q(t),t

?

, where q(t) is the coordinate and ˙ q =

dq

dtis defined. After performing the Legendre transfor-

L =dS

dt=∂S

∂t+dq

dt

∂S

∂q,

(2.6)

which also can be expressed as

∂S

∂t+dq

dt

∂S

∂q− L = 0.

(2.7)

Designating p =∂S

∂q, we can get the Hamilton-Jacobi equation straightforwardly

∂S

∂t+dq

dt

∂S

∂q− L =∂S

∂t+dq

dtp − L =∂S

∂t+ H

?

q,∂S

∂q,t

?

= 0.

(2.8)

For the harmonic oscillator (2.1), its Hamiltonian is given by

H =

1

2mp2+1

2m2ω2x2.

(2.9)

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Designating p =∂S

∂x, we derive its Hamilton-Jacobi from eq. (2.8) as

∂S

∂t+

1

2m

?∂S

∂x

?2

+1

2mω2x2= 0.

(2.10)

We can find solutions of eq. (2.10) following two different approaches as follows:

Type-(I): Assuming the solution is a polynomial of x, we have

S =1

2f(t)x2+ h(t)x + g(t).

(2.11)

Substituting this assumption into eq. (2.10), and letting the coefficients of x to be zeros, we can get equations

1

2

df(t)

dt

+

1

2mf(t)2+1

dh(t)

dt

dg(t)

dt

2mω2= 0,

(2.12)

+1

mf(t)h(t) = 0,

1

2mh(t)h(t) = 0.

(2.13)

+ (2.14)

Solving these ordinary differential equations (ODEs), we can get the solution for S to be

(Ia) :

S =m

2

ω

sinω(t − t0)[cosω(t − t0)(x2+ x2

ω

cosω(t − t0)[−sinω(t − t0)(x2+ x2

0) − 2xx0],

(2.15)

(Ib) : S1=m

2

0) − 2xx0],

(2.16)

where x0, t0are integration constants.

Type-(II): Assuming that

S = −E(t − t0) + W(x),

(2.17)

we can get a solution which we are familiar with in classical mechanics

S = −E(t − t0) +E

marctan

?

mωx

√2mE − m2ω2x2

?

+1

2x

?

2mE − m2ω2x2,

(2.18)

where E, t0are constants of integral.

2.1.3The Pauli Short-Time Kernel

Based on the discussions above, we should have noticed the resemblance between the Feynman kernel (2.5)

and the solution (2.15) of the Hamilton-Jacobi equation. Of course, this resemblance is not so surprising. The

reasons are as follows: For a Lagrangianof quadratic interaction, its Feynman kernel can be given exactly by the

Wentzel-Kramers-Brillouin semiclassical approximation; While the Wentzel-Kramers-Brillouin approximation

involves the classical action of the system. Therefore, the resemblance between (2.5) and (2.15) only holds

for quadratic interaction or for linear theories. For nonlinear theories, there would be no such resemblance.

However, we will show that a connection between the discretized Lagrangian and special solutions of the

Hamilton-Jacobi equation can still be constructed, not only for linear theories but also for nonlinear theories.

In quantum mechanics, the link connecting them is Pauli’s formula or Pauli short-time kernel [10, 11].

For infinitesimal time evolution from t to t + ǫ, the Pauli short-time kernel is defined by

KP(q,t + ǫ;q′,t) =

?1

2πi

?1

2?

−∂2S(q,q′;ǫ)

∂q∂q′

?1

2

expiS(q,q′;ǫ).

(2.19)

Pauli proved that this kernel will satisfy the same equation as the Feynman short-time kernel (2.3) if the

function S(q,q′;ǫ) satisfies the Hamilton-Jacobi equation

∂S(q,q′;ǫ)

∂ǫ

+ H(q,∂S

∂q) = 0.

(2.20)

– 4 –

Page 6

For the harmonic oscillator, the solutions of its Hamilton-Jacobi equation are given by eqs. (2.15), (2.16) and

(2.18). Obviously the Type-(Ia) solution eq. (2.15) is suitable to formulate the Pauli short-time kernel while

the solutions Type-(Ib) (2.16) and Type-(II) (2.18) are not. The Pauli short-time kernel for the harmonic

oscillator is therefore given by

KP(xj+1,tj+1;xj,tj) =

?

mω

2πisinωǫ

?1

2exp

?

im

2

ω

sinωǫ

?

cosωǫ(x2

j+1+ x2

j) − 2xj+1xj

??

.

(2.21)

Here we use the subscript P to differentiate the Pauli short-time kernel from the Feynman short-time kernel.

The convolution of the Pauli short-time kernel (2.21) has the semi-group property, that is,

KP(xj+2,tj+ 2ǫ;xj,tj) =

?

dxj+1KP(xj+2,tj+1+ ǫ;xj+1,tj+1)KP(xj+1,tj+ ǫ;xj,tj).

(2.22)

So its multi-convolutions will give the same results as the Feynman kernel (2.5). The calculations of the multi-

convolutions of the Pauli short-time kernel (2.21) are simple and straightforward; While the multi-convolutions

of the Feynman short-time kernel (2.3) are very complicated.

2.1.4Deriving Discretized Lagrangians from Special Solutions of the Hamilton-Jacobi Equation

The discussions in section 2.1.3 suggest us an approach to convert classical theories to quantum theories. The

logic is as follows: Employing the multi-convolutions of the Pauli short-time kernel (2.21), we can get the

same results as that we get with the Feynman short-time kernel (2.3); While the Pauli short-time kernel (2.21)

can be determined by the special Type-(Ia) solution eq. (2.15) of the Hamilton-Jacobi equation. The logical

deductions above can be reversed: we can begin with the Type-(Ia) solution eq. (2.15), then we can get a

discretized version of the path integral via the multi-convolutions of Pauli’s formula (2.19).

The equivalence of the results of these two approaches can be understood in another way. Expanding the

Pauli short-time kernel (2.21) with the small parameters ǫ and keeping the terms of leading order, we get

2?cosωǫ(x2

j+1+ x2

j) − 2xj+1xj

?

j+1+ x2

?= (1 + cosωǫ)(xj+1− xj)2− (1 − cosωǫ)(xj+1+ xj)2

mω

2πisinωǫ

2πiǫ

??

(2.23)

?1

ǫ→0

− − − → exp

2

ǫ→0

− − − →

?m

?

?1

2

(2.24)

exp

?

im

2

ω

sinωǫ

?

cosωǫ(x2

j) − 2xj+1xj

iǫ

?m

2

?xj+1− xj

ǫ

?2

−1

2mω2?xj+1+ xj

2

?2??

.

(2.25)

What we should realize from this approximation is the discretized Lagrangian in the lattice definition (2.2) of

the path integral is recovered from the Type-(Ia) solution eq. (2.15) of the Hamilton-Jacobi equation in the

approximation of small ǫ.

The derivation of the discretized Lagrangian can be generally summarized in the following procedure.

For a solution of the Hamilton-Jacobi equation S(x,t;x0,t0) with some constants (x0,t0) which need to be

designated by the initial conditions and boundary conditions, its corresponding discretized Lagrangian can be

defined by

˜Llattice=S(xb,tb;x0,t0) − S(xa,ta;x0,t0)

tb− ta

.

(2.26)

Here [ta,tb] is the lattice interval. The constants (x0,t0) need to be designated appropriately. Substituting

the Type-(Ia) solution eq. (2.15) into eq. (2.26), we get

˜Llattice =

1

tb− ta

−m

2

?m

ω

2

ω

sinω(tb− t0)[cosω(tb− t0)(x2

sinω(ta− t0)[cosω(ta− t0)(x2

b+ x2

0) − 2xbx0]

?

a+ x2

0) − 2xax0]

.

(2.27)

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Page 7

Taking the limit x0→ xa, then the limit t0→ ta, we get

Llattice= lim

t0→ta

lim

x0→xa

˜Llattice=

1

tb− ta

?m

2

ω

sinω(tb− ta)[cosω(tb− ta)(x2

b+ x2

a) − 2xbxa]

?

.

For infinitesimal lattice spacing tb−ta= ǫ, following the procedure in eq. (2.25), eq. (2.28) can be approximated

as

Llattice=

1

tb− ta

?m

?xb− xa

2

ω

sinω(tb− ta)[cosω(tb− ta)(x2

?2

b+ x2

a) − 2xbxa]

?

tb−ta→0

− − − − − − →m

2

tb− ta

−1

2mω2?xb+ xa

2

?2

.

(2.28)

The lattice Lagrangian is recovered again.

2.2Nonlinear Theories: Double-Well Potential

In this section, taking the double-well potential for example, we deal with nonlinear theories. For nonlinear

theories, the Hamilton-Jacobi equation is generally difficult to solve. We introduce the tools to handle the

nonlinear theories and show that the discussions in section (2.1) can also apply to nonlinear theories.

2.2.1Tools for Solving the Nonlinear Hamilton-Jacobi Equation

For a particle in the double-well potential, its Lagrangian is given by

L(x, ˙ x,t) =1

2m˙ x2−1

8g2?x2− v2?2,

∂ ˙ x, we get its Hamiltonian

(2.29)

where v is constant. After Legendre transformation p =∂L

H(x,p,t) =

1

2mp2+1

8g2?x2− v2?2.

(2.30)

The canonical Hamiltonian equations of motion are

˙ x =∂H

∂p

=p

m,

(2.31)

˙ p = −∂H

∂x= −1

2g2x?x2− v2?.

(2.32)

While the corresponding Euler-Lagrange equation of motion is

m¨ x +1

2g2x?x2− v2?= 0.

(2.33)

Designating p =∂S

∂x, its Hamilton-Jacobi equation is

∂S

∂t+

1

2m

?∂S

∂x

?2

+1

8g2?x2− v2?2= 0.

(2.34)

This nonlinear equation is difficult to solve. However, employing the “embedding method” introduced in [12],

we can find a series solution for eq. (2.34); While this series solution is enough to satisfy our purpose in this

paper.

The “embedding method” is as follows. Suppose that we seek a series solution of the type

S(x,t) = S∗(t) + A(t)[x − f(t)] + R(t)[x − f(t)]2

+ K(t)[x − f(t)]3+ M(t)[x − f(t)]4+ N(t)[x − f(t)]5+ ··· ,

(2.35)

– 6 –

Page 8

where f(t) is a function which will be given later. So it seems like we expand S(x,t) in a series around

a function f(t) in eq. (2.35). The potential function V (x) =

expanded by the following identities

1

8g2?x2− v2?2is a polynomial, which can be

(x − f(t) + f(t))2− v2?2

(x − f(t) + f(t))2= [x − f(t)]2+ 2f(t)[x − f(t)] + f(t)2.

V (x) =1

8g2(x2− v2)2=1

8g2?

,

(2.36)

Notice that we expand these functionals around a function f(t) but not 0, so the combination [x−f(t)] always

remains. Substituting the expansions (2.35) and (2.36) into the Hamilton-Jacobi equation (2.34), collecting

the terms of [x − f(t)] of the same power, and letting the coefficients of this series to be zeros, we get

?0

?

?

?

?

where we only display terms of power not higher than 4 while those of higher power are omitted for convenience.

These ODEs are generally difficult to solve. The key ingredient of the “embedding method” is we can use

some assumptions to simplify these ODEs. We suppose

?

x − f(t):

dS∗(t)

dt

dA(t)

dt

+

1

2mA2+1

?df(t)

?df(t)

dK(t)

dt

?df(t)

8g2?f(t)2− v2?2− Adf(t)

−A

m

?

?df(t)

−A

m

2m

dt

= 0,

(2.37)

x − f(t)

?

?2

?3

?4

:

− 2R

dt

?

+1

2g2f(t)?f(t)2− v2?= 0,

mR2+1

−A

m

1

?9K2+ 16MR?+1

(2.38)

x − f(t)

:

dR(t)

dt

− 3H

dt

−A

m

+2

4g2?3f(t)2− v2?= 0,

+6

2g2f(t) = 0,

(2.39)

x − f(t)

:

− 4M

dt

?

?

mKR +1 (2.40)

x − f(t)

:

dM(t)

dt

− 5N

dt

+

8g2= 0,

(2.41)

df(t)

dt

−A

m= 0,

(2.42)

dA(t)

dt

+1

2g2f(t)?f(t)2− v2?= 0,

(2.43)

then eq. (2.38) is satisfied. We noticed that equations (2.42) and (2.43) coincide with the canonical Hamiltonian

equations (2.31) and (2.32) if we make the replacements x → f(t) and p → A. So f(t) satisfies the Euler-

Lagrange equation automatically

m¨f +1

2g2f?f2− v2?= 0.

(2.44)

Eq. (2.37) can be transformed as

dS∗(t)

dt

= Adf(t)

dt

−

1

2mA2−1

8g2?f(t)2− v2?2= L∗(f,˙f,t).

(2.45)

We see under the assumptions (2.42) and (2.43), the righthand side of (2.45) coincides with the Lagrangian

evaluated on the configuration f(t). While eqs. (2.39), (2.40) and (2.41) are simplified to be

dR(t)

dt

+2

mR2+1

dK(t)

dt

1

2m

4g2?3f(t)2− v2?

+6

= 0,

(2.46)

mKR +1

2g2f(t) = 0,

(2.47)

dM(t)

dt

+

?9K2+ 16MR?+1

8g2= 0.

(2.48)

We see that eq. (2.46) only includes coefficients of power less than 3, eq. (2.47) only includes coefficients of

power less than 4, and eq. (2.48) only includes coefficients of power less than 5. So coefficients of higher power

– 7 –

Page 9

terms decouple from that of lower power terms. This is a great advantage of the “embedding method”. From

the discussions above, we learned an important point in the “embedding method” , which is we need to know

a solution for the original Hamiltonian canonical equations or the Euler-Lagrange equation. This solution

of the equation of motion will help us find a series solution for the Hamilton-Jacobi equation. Obviously,

this “embedding method” is useless if our purpose is to find the solution of Hamiltonian canonical equations.

However, here the “embedding method” just satisfies our purpose because we try to find a solution of the

Hamilton-Jacobi equation.

A solution of eq. (2.44) can be expressed by the Jacobi’s function

f(t) = x0JacobiDN

?

λ(t − t0),k

?

, λ =

gx0

2√m, k2= 2

?

1 −

?v

x0

?2?

,

(2.49)

where we have used the initial condition f(t0) = x0. Substituting the expression of f(t) into eqs. (2.46), (2.47)

and (2.48), we can obtain solutions for R(t), K(t) and M(t). However, because of the limiting procedures in

eqs. (2.28) and (2.28), the solutions for small (t − t0) are enough for our purpose. For small (t − t0), f(t) can

be replaced by x0in eqs. (2.45), (2.46), (2.47) and (2.48). The solutions for S∗(t), R(t), K(t) and M(t) can

be given by

S∗(t) = −1

R(t) =m

8g2?

1

r−m

r3+C2

9

2m

1

720

x2

0− v2?2

6ωr + O(r3),

ω

r+C1

r5+C3

?ω3

r + O(r2),

(2.50)

2

(2.51)

K(t) =C2

28ω2r + O(r3),

r4+3ω

2m

(2.52)

M(t) =

C2

2

C2

r3+2ω

2

3

C3

r2+

ω2

24m

?

?

−9C1+ 7C2

?

C21

r+2ω2

9

C3

(2.53)

−

m

?

153C1− 56C2

?

C2+ 18g2

r + O(r2),

where r = t − t0, ω =

derived from eq. (2.42) straightforwardly

1

2mg2[3x2

0− v2], C2= C1+g2

ω2x0and C3is a constant. The expression for A(t) can be

A(t) = −1

2g2(x2

0− v2)r + O(r2).

(2.54)

2.2.2Derivation of the Lattice Lagrangian

We have obtained a series solution for the Hamilton-Jacobi equation in section 2.2.1. Then following the

limiting procedures in 2.1.4, we can derive the lattice Lagrangian. Because the additive properties of the

limiting, we can implement the limiting procedures term by term. The results are given by

lim

x0→xa

t0→ta

S∗(xb,tb) − S∗(xa,ta)

tb− ta

= −1

8g2?x2

a− v2)(xb− xa),(2.56)

a− v2?2,(2.55)

lim

x0→xa

t0→ta

1

tb− ta[A(tb)(xb− f(tb)) − A(ta)(xa− f(ta))] = −1

R(tb)(xb− f(tb))2− R(ta)(xa− f(ta))2?

lim

x0→xa

t0→ta

1

tb− ta

2g2(x2

lim

x0→xa

t0→ta

1

tb− ta

?

=m

2

(xb− xa)2

(tb− ta)2−g2

12

?3x2

= −1

a− v2?(xb− xa)2,(2.57)

24g2xa(xb− xa)3,(2.58)

?

K(tb)(xb− f(tb))3− K(ta)(xa− f(ta))3?

?

lim

x0→xa

t0→ta

M(tb)(xb− f(tb))4− M(ta)(xa− f(ta))4?

= −1

40g2(xb− xa)4. (2.59)

– 8 –

Page 10

Here we caution that we take the limit x0→xaat first, then take the limit t0→ta. In the limiting procedures

above, we have set the constants of integral C2and C3to be zeros, so C1= −g2

can be well defined; Otherwise the limits will be singular function of (t0− ta). The lattice Lagrangian is then

found to be

ω2x0, in order that the limits

ˆLlattice= −g2

8

??xb+ xa

2xa(xb− xa)3−g2

2

−xb− xa

2

?2

− v2

?2

−g2

2(x2

a− v2)(xb− xa) +m

2

(xb− xa)2

(tb− ta)2

(2.60)

−g2

40(xb− xa)4+ ··· .

Again for infinitesimal lattice spacing tb− ta= ǫ, eq. (2.60) can be further approximated as

m

2

ˆLlattice

tb→ta

− − − − →

xb→xa

(xb− xa)2

(tb− ta)2−g2

8

??xb+ xa

2

?2

− v2

?2

,

(2.61)

which is the expected lattice version for the double-well Lagrangian (2.29). We saw that the terms of power

higher than 3 disappear in the limit xb→ xa. For the double-well potential, we can verify this phenomenon

term by term. Thereafter, when we find series solutions for nonlinear theories, we always suppose that this

phenomenon holds. It seems to be plausible despite being difficult to prove.

3Examples of Scalar Field Theories

In this section, we discuss scalar field theories. We try to apply the discussions of section 2 to field theories.

To make these applications feasible, some new tools are required. These new tools are the covariant Hamilton

theories or the De Donder-Weyl theories for field theories, based on the multi-parameter generalization of

Hilbert’s independent integral. Taking scalar field theories for example, we will give a pedagogical introduc-

tions of the De Donder-Weyl theory. For more details, see [4–8]. Based on these covariant Hamilton-Jacobi

equations, we suggest an approach to derive discretized Lagrangians for field theories, similar to the procedures

in section 2.

3.1Linear Theories: Massive Scalar Theories

3.1.1 Covariant Hamilton-Jacobi Equations

For a massive free scalar field in four dimensional Minkowski space-time, its Lagrangian is given by

L =1

2ηµν∂µφ(x)∂νφ(x) −1

2m2φ(x)φ(x).

(3.1)

Here and hereafter we use the Lorentzian metric η = diag(1,−1,−1,−1). Following De Donder and Weyl, the

Legendre transformation is performed with the manifest Lorentz covariance

πµ=

∂L

∂(∂µφ)= ∂µφ(x),

(3.2)

which is different from the Legendre transformation only performed in the time derivative of a field; While

πµcan be regarded as the covariant conjugate momentum. Then the corresponding covariant Hamiltonian is

found to be

H = πµ∂µφ(x) − L =1

2πµπµ+1

2m2φ(x)φ(x).

(3.3)

From this covariant Hamiltonian, we can get the Hamiltonian canonical equations

∂µφ(x) =∂H

∂πµ

= πµ,

(3.4)

∂µπµ = −∂H

∂φ

= −m2φ.

(3.5)

– 9 –

Page 11

Obviously these canonical equations imply the following Euler-Lagrangian equation

∂µ∂µφ + m2φ = 0,

(3.6)

as we expected.

The covariant Hamilton-Jacobi equation or the De Donder-Weyl equation can de derived as follows. Sup-

posing the Lagrangian L is an independent integral of Hilbert, that is, L can be expressed as the total

derivative of a vector Sµ

L =dSµ

dxµ=∂Sµ

∂xµ+ ∂µφ(x)∂Sµ

∂φ,

(3.7)

which also equals to

∂Sµ

∂xµ+ ∂µφ(x)∂Sµ

∂φ

− L = 0.

(3.8)

Designating

πµ=∂Sµ

∂φ,

(3.9)

we then get the covariant Hamilton-Jacobi equation

∂Sµ

∂xµ+ H

?

φ(x),πµ=∂Sµ

∂φ

?

= 0.

(3.10)

As in classical mechanics, if we can find a complete solution for Sµfrom the Hamilton-Jacobi equation (3.10),

then we can get a solution for the Euler-Lagrange equation (3.6). However, there are some differences between

field theories and classical mechanics. The reasons are that a classical mechanical system only depends on a

single evolution parameter, the temporal variable t; While a system of fields depend on the variables of time

and space dimensions. Therefore, there are some integrability conditions involved for field theories. These

discussions can be understood as follows. If we find a solution for Sµ, then we can determine πµby eq. (3.9);

While on the other hand, πµis a conjugate momentum determined by the Legendre transformation (3.2), so

it can be expressed as the total derivative of fields. Therefore πµdetermined by eq. (3.9) is connected to the

total derivative of fields. The foregoing are the reasons why the integrability conditions appear. For the free

massive scalar field in this section, the integrability conditions are straightforwardly found to be

dπµ

dxν=dπν

dxµ,

(3.11)

or more specifically

∂πµ

∂xν+ ∂νφ(x)∂πµ

∂φ

=∂πν

∂xµ+ ∂µφ(x)∂πν

∂φ.

(3.12)

Here we noticed that πµis determined by eq. (3.9) but not eq. (3.2), that is, πµis derived from the functional

Sµ. So these integrability conditions are basically some restriction conditions on Sµ. For more detailed

discussions on integrability conditions, see [5, 8, 12].

3.1.2 Solutions of Covariant Hamilton-Jacobi Equations

For the Lagrangian (3.1) of a massive free scalar field, its De Donder-Weyl equation is given by

∂Sµ

∂xµ+1

2

∂Sµ

∂φ

∂Sµ

∂φ

+1

2m2φ(x)φ(x) = 0.

(3.13)

– 10 –

Page 12

We employ the “embedding method” introduced in section 2.2.1 to find its solution. In order to do that, we

need a solution of the Euler-Lagrange equation (3.6). We can find two types of solutions of equation (3.6):

I :

ϕ(x) = ϕ(z)cos

?m

√λr

?

, λ = kµkµ, r = kµ(x − z)µ,

(3.14)

II :˜ ϕ(x) = ˜ ϕ(z)

2

m√ξBeeeslJ(1,m

?

ξ), ξ = (x − z)µ(x − z)µ,

(3.15)

where kµand zµare constant vectors and we have used the initial conditions to normalize the solutions. These

two types of solutions can both be used in the “embedding method”. We take the type (I) solution for example.

Because eq. (3.13) only contains quadratical terms, we can suppose that Sµis given by

Sµ= fµ(x)

?

φ(x) − ϕ(x)

?2

+ hµ(x)

?

φ(x) − ϕ(x)

?

+ S∗µ(x).

(3.16)

Substituting this expression into the De Donder-Weyl equation (3.13), we can get a polynomial of [φ(x)−ϕ(x)].

Letting the coefficients of this polynomial to be zero, we get

?

?

?

φ(x) − ϕ(x)

?0

?1

?2

:

∂µS∗µ− ∂µϕ(x)hµ+1

2hµhµ+1

2m2ϕ2(x) = 0,

(3.17)

φ(x) − ϕ(x)

:

∂µhµ+ 2fµ[−∂µϕ(x) + hµ] + m2ϕ(x) = 0,

∂µfµ+ 2fµfµ+1

(3.18)

φ(x) − ϕ(x)

:

2m2= 0.

(3.19)

Similar to that in section 2.2.1, we can suppose

hµ = ∂µϕ(x),

dhµ

dxµ= −m2ϕ(x),

(3.20)

(3.21)

then eq. (3.18) gets satisfied. Obviously, Eqs. (3.20) and (3.21) are just the Hamiltonian canonical equations

(3.4) and (3.5). They are consistent because ϕ(x) is a solution of the Euler-Lagrange equation. While eq. (3.17)

is transformed to be

dS∗µ

dxµ

= ∂µϕ(x)hµ−1

2hµhµ−1

2m2ϕ2(x) = L∗?

ϕ(x)

?

,

(3.22)

whose right-hand side is the Lagrangian evaluated at ϕ(x). The solutions for fµ, hµand S∗µcan be given by

fµ(x) =1

2

m

√λcot

?m

?m

?2m

√λr

?

?

kµ,

(3.23)

hµ(x) = −m

√λsin

m

4√λsin

√λr

kµ,

(3.24)

S∗µ(x) = −

√λr

?

ϕ2(z)kµ,

(3.25)

where r has been defined by eq. (3.14).

3.1.3Derivation of the Discretized Lagrangian

In order to derive the discretized Lagrangian for field theories, we need a formula similar to eq. (2.26). However,

some differences emerge in field theories, because what we have in field theories is the vector functional Sµ. So

we need a multi-dimensional exploration for the formula (2.26). Inspired by the designation of the independent

integral of Hilbert, that is, eq. (3.7), we suggest the following definition for discretized Lagrangian

˜

Llattice =

3

?

i=0

Si(yi) − Si(xi)

yi− xi

,

(3.26)

– 11 –

Page 13

where

S1(y1) − S1(x1)

y1− x1

=

1

y1− x1

?

S1?x0,y1,x2,x3,φ(x0,y1,x2,x3)?− S1?x0,x1,x2,x3,φ(x0,x1,x2,x3)??

.

(3.27)

The definitions for other indices follow similarly. Obviously, in the limit of yi− xi→ 0, this definition is just

the independent integral of Hilbert (3.7). For the solutions in eqs. (3.23), (3.24) and (3.25), employing the

definition in (3.26), then taking the limits of zi→ xiand ϕ(z) → φ(x), we can get

ˆ

Llattice = lim

zi→xi

3

?

3

?

Here we have supposed the symmetrical lattice spacing, that is, yi−xi= ǫ. Furthermore, supposing infinites-

imal lattice spacing ǫ → 0 and φ(yi) − φ(xi) → 0, we get the final version of the discretized Lagrangian

3

?

Here we have defined the function

lim

ϕ(zi)→φ(xi)

1

ǫ

˜

Llattice

(3.28)

=

i=0

1

2

m

√λcot

?m

?m

√λǫki

?

ki?

ki?

φ(yi) − φ(xi)

?2

−

i=0

1

ǫ

m

√λsin

√λǫki

?

φ(yi) − φ(xi)

?

−

3

?

i=0

1

ǫ

m

4√λφ2(xi)sin

?2m

√λǫki

?

ki.

Llattice= lim

yi→xi

lim

φ(yi)→φ(xi)

ˆ

Llattice=

i=0

sgn(i)1

2

1

ǫ2

?

φ(yi) − φ(xi)

?2

−1

2m2?φ(y) + φ(x)

2

?2

.

sgn(i) =

?

1,i = 0,

−1,i = 1,2,3.

(3.29)

Obviously, eq. (3.29) is the lattice version of the Lagrangian (3.1) as we expected. The appearance of the

sgn(i) is because we work on Minkowski space-time with a Lorentzian metric. For infinitesimal ǫ → 0, we can

make the replacement

φ(yi) − φ(xi)

ǫ

ǫ→0

− − − → ∂iφ(x),

(3.30)

then the lattice Lagrangian will approximate to the continuous one

Llattice

ǫ→0

− − − →1

2ηµν∂µφ(x)∂νφ(x) −1

2m2φ(x)φ(x),

(3.31)

which is of course the Lagrangian (3.1) we began with.

Moreover, we should mention that the Lagrangian (3.29) is not the only lattice Lagrangian we can obtain

by following the foregoing procedure. The reason is that we can obtain more solutions besides the solutions

in equations (3.23), (3.24) and (3.25). Actually, we can get two more solutions of eq. (3.19) for fµ

fµ

1(x) = −1

2

m

√λtanh

m

√ξ

?m

√λr

?

kµ,

(3.32)

fµ

2(x) = −1

2

BesselY(2,m√ξ)

BesselY(1,m√ξ)(x − z)µ,

(3.33)

where r and ξ have been defined in eqs. (3.14) and (3.15). Associated with solutions (3.24) and (3.25),

eqs. (3.32) and (3.33) both construct new solutions for Sµ. Following the limiting procedures as we just did

above, we can derive two new Lagrangians from these two new solutions. They are given by

fµ

1: L1 = −1

fµ

2m2φ(x)φ(x),

2ηµν∂µφ(x)∂νφ(x) −1

(3.34)

2: L2 = −1

2m2φ(x)φ(x).

(3.35)

– 12 –

Page 14

The first new one has no kinetic term because fµ

new one has the minus kinetic term, that is, a ghost kinetic term because fµ

We might expect the integrability condition introduced in (3.11) to kill these two new solutions, but they both

satisfy the integrability condition. We will see that this phenomenon happens for linear theories universally.

So far we can not figure out they could correspond to two new sectors of quantum theories we can derive from

solutions of the De Donder-Weyl equations, or they mean the procedure we have employed is incomplete so

we need some more criteria to select the physical solution.

1does not contribute in the limit of ǫ → 0; While the second

2yields the minus kinetic term.

3.2Nonlinear Theories: Scalar Theories with λφ4Potential

In this section, we discuss nonlinear field theories. These discussions are in conjunction with that of linear

theories. The differences are we can get exact solutions for linear theories, but we can only get series solutions

for nonlinear theories.

3.2.1De Donder-Weyl Equation and Its Solution

We consider a scalar field theory of λφ4potential in four dimensional Minkowski space-time, its Lagrangian is

given by

L =1

2ηµν∂µφ(x)∂νφ(x) −1

2m2φ(x)φ(x) −λ

4!φ4(x) − Λ,

(3.36)

where we include a density of vacuum energy Λ for generality. Performing the Legendre transformation

πµ=

∂L

∂(∂µφ)= ∂µφ(x),

(3.37)

we get the covariant Hamiltonian

H =1

2πµπµ+1

2m2φ(x)φ(x) +λ

4!φ4(x) + Λ.

(3.38)

The corresponding Hamiltonian canonical equations are

∂µφ(x) =∂H

∂πµ

= πµ,

(3.39)

∂µπµ = −∂H

∂φ

= −m2φ −λ

3!φ3(x).

(3.40)

They imply the following Euler-Lagrangian equation

∂µ∂µφ + m2φ +λ

3!φ3(x) = 0.

(3.41)

In eq. (3.38), designating

πµ=∂Sµ

∂φ,

(3.42)

we get its De Donder-Weyl equation

∂µSµ+1

2

∂Sµ

∂φ

∂Sµ

∂φ

+1

2m2φ(x)φ(x) +λ

4!φ4(x) + Λ = 0.

(3.43)

Following the procedures in section 3.1.2, we employ the “embedding method” to find the solution of

eq. (3.43). As in section 3.1.2, we need a solution of the Euler-Lagrange equation, which can be given by

ϕ(x) = ϕ(z)JacobiDN

?√λϕ(z)

2√3σ

r,k

?

, k =

?

2 +

12m2

λϕ2(z), r = pµ(x − z)µ, σ = pµpµ.

(3.44)

– 13 –

Page 15

Here pµand zµare constant vectors, and we have used ϕ(x)|z= ϕ(z) to normalize the solution. According to

the “embedding method”, we suppose a series solution for Sµ

Sµ= S∗µ(x) + Pµ(x)[φ(x) − ϕ(x)] + Rµ(x)[φ(x) − ϕ(x)]2

+ Kµ(x)[φ(x) − ϕ(x)]3+ Mµ(x)[φ(x) − ϕ(x)]4+ Nµ(x)[φ(x) − ϕ(x)]5+ ··· .

(3.45)

Substituting this expression into eq. (3.43), we get a series expression of [φ(x) − ϕ(x)]. Supposing the coeffi-

cients of this series to be zeros term by term, we get

[φ(x) − ϕ(x)]0: ∂µS∗µ− ∂µϕ(x)Pµ+1

[φ(x) − ϕ(x)]1:

[φ(x) − ϕ(x)]2: ∂µRµ+ 3Kµ[−∂µϕ(x) + Pµ] + 2RµRµ+1

[φ(x) − ϕ(x)]3:

[φ(x) − ϕ(x)]4: ∂µMµ+ 5Nµ[−∂µϕ(x) + Pµ] +9

2PµPµ+1

2m2ϕ2(x) +λ

4!ϕ4(x) + Λ = 0,

(3.46)

∂µPµ+ 2Rµ[−∂µϕ(x) + Pµ] + m2ϕ(x) +λ

6ϕ3(x) = 0,

(3.47)

2m2+λ

4ϕ2(x) = 0,

(3.48)

∂µKµ+ 4Mµ[−∂µϕ(x) + Pµ] + 6RµKµ+λ

6ϕ(x) = 0,

(3.49)

2KµKµ+ 8MµRµ+λ

24= 0,

(3.50)

where terms of power higher than 4 are omitted for convenience. By supposing the self-consistent canonical

equations

∂µϕ(x) = Pµ,

∂µPµ = −m2ϕ −λ

(3.51)

3!ϕ3(x),

(3.52)

eq. (3.47) gets satisfied, and eq. (3.46) is transformed to be

∂µS∗µ= ∂µϕ(x)Pµ−1

2PµPµ−1

2m2ϕ2(x) −λ

4!ϕ4(x) − Λ = L∗?

ϕ(x)

?

.

(3.53)

Eqs. (3.48), (3.49) and (3.50) are simplified to be

∂µRµ+ 2RµRµ+1

2m2+λ

4ϕ2(x) = 0,

(3.54)

∂µKµ+ 6RµKµ+λ

6ϕ(x) = 0,

(3.55)

∂µMµ+9

2KµKµ+ 8MµRµ+λ

24= 0.

(3.56)

The exact solutions of these equations are difficult to derive. However, the behavior of solutions around small

r is enough for our purpose. For small r, we can replace ϕ(x) with ϕ(z) in eqs. (3.54), (3.55) and (3.56). Then

we can get the following series solutions for Rµ, Kµand Mµ

Rµ=

?1

?C2

?9

2

r3+1

C2

2

r5+C3

+ω2

24

1

r−1

6ωr

2ωC2

r4+3ω

?

pµ+ O(r3),

(3.57)

Kµ=

r

+C1

8ω2r

C2

2

r3+2ω

rpµ+2ω2

?

pµ+ O(r3),

(3.58)

Mµ=

223

C3

r2

?

pµ

(3.59)

?

−9C1+ 7C2

?

C21

9

C3pµ−

??17

80C1−

7

90C2

?

C2+

1

120

λ

σ

?

rpµ+ O(r2),

– 14 –