THE GENERATI NG FUNCTI ONAL FOR SCALAR FI ELD THEORI ES

Uni ver s i t à degl i St udi di Padova

DI PARTI MENTO DI FI SI CA E ASTRONOMI A “G. Gal i l ei ” Cor s o di l aur ea t r i ennal e i n Fi s i ca

Tes i di Laur ea

THE GENERATI NG FUNCTI ONAL FOR SCALAR FI ELD THEORI ES

Candi dat o: Lor enzo Ri cci

Rel at or e: Pr of . Mar co Mat one

1 Introduction

In 1931 Dirac published a paper concerning key similarities between the classical Hamilton-Jacobi theory and the transition amplitudes in quantum mechanics.

In particular, he derived the following relation

hq, t|Q, T i ∼ e

^¯hi

^R

^Tt

^dtL . (1.1) In 1948, Feynman developed Dirac’s suggestion and succeeded in deriving a new formulation of quantum mechanics, quite different from the standard one.

This formulation does not require the use of operators and of the Schrödinger equation to express the quantum mechanical amplitudes. The physical idea is that the probability amplitude to find a particle at the space-time point (Q, T ), knwowing that it was at (q, t), is given by the sum of all the possible paths between the two space-time points, each one contributing with an appropriate weight.

In this work it is firstly presented (Sec. 2) the Dirac original idea and then how Feynman developed his path-integral formulation of quantum mechanics.

Moreover, it is explained how this approach can be generalized to quantum field theory. In particular we focus on the case of a scalar field (Sec. 3), introducing the generating functional, a basic tool to compute Green’s functions without the use of Feynman diagrams. Finally, in Sec. 4 and 5, we illustrate two alternative representations of the generating functional, developed in the Ref.[2]. The first one is expressed as

W [J ] = T [φ c ] = N N 0

exp(−U 0 [φ c ]) exp  1 2

δ δφ c

∆ δ δφ c

 exp



− Z

V (φ c )

 , (1.2) where φ _c is defined as φ _c (x) = R d ^D yJ (y)∆(y − x). This dual representation is used to express Schwinger-Dyson equation, obtaining

 δ δφ c

+ e ^U

⁰

^[φ

^c

^] Z δV

δφ



∆ δ δφ c



e ^−U

⁰

^[φ

^c

^]



e

^12δφcδ

^∆

^δφcδ

e ^{− R V (φ}

^c

⁾ = 0 . (1.3) It is also possible to note the presence of a deep connection between the above dual representation and the Hermite polynomials. Then, we express T [φ _c ] in terms of “covariant” derivatives acting on 1

T [φ _c ] = N

N ₀ exp (−U ₀ [φ _c ]) exp



− Z

V (D ⁻ _φ

c

)



· 1 , (1.4)

where D ^± _φ (x) = ∓∆ _δφ ^δ (x) + φ(x). These “covariant” derivatives simplify the form of the equations. For example the Schwinger-Dyson equation becomes

 δ

δφ c (x) + Z δV

δφ

 D _φ ⁻

c

  exp



− Z

V (D _φ ⁻

c

)



· 1 = 0 . (1.5)

We also see how they make “more comfortable” some explicit calculations.

2 The path-integral

The Dirac formulation. The key initial idea, that led to the concept of path- integral, is due to Dirac who was looking for an alternative formulation of quan- tum mechanics provided by the Lagrangian. He believed that the Lagrangian formulation of classical dynamics is more fundamental than the Hamiltonian one for the following reasons. First of all the Lagrangian method allows to find the equations of motion, thanks to the stationary property of a certain action function. Secondly, the Lagrangian method could be easily expressed relativis- tically since the action is Lorentz invariant. For doing this, Dirac worked on the analogy between the classical theory of Hamilton-Jacobi and the transition amplitude in quantum mechanics. To show this analogy, let us consider a one- dimensional classical system of only one particle. Let q be the coordinate and p the momentum. H is the Hamiltonian of the system. The Hamilton variational principle

δ Z t

t

₀

dt

 p dq

dt − H(q, p, t)



= 0 , (2.1)

allows to describe the time evolution of q(t), p(t) by the Hamilton equations, expressed through the Poisson brackets as follows

dq

dt = {q, H} , dp

dt = {p, H} . (2.2) A canonical transformation is a transformation of q and p into new variables Q and P leaving Hamilton’s equations form invariant. It is well known that a function G(t, q, Q) called generating function exists, such that

p = ∂G

∂q , P = − ∂G

∂Q , H = H + ˜ ∂G

∂t , (2.3) where ˜ H is the Hamiltonian of the new system.

Denote S(t, q, Q) the special canonical transformation such as ˙ Q = ˙ P = 0. By (2.3), it follows that

H



q, p = ∂S

∂q , t



= − ∂S

∂t . (2.4)

Since

dS dt = ∂S

∂t + ∂S

∂q dq

dt = −H + p dq

dt , (2.5)

we have

S = Z t

t

₀

dt ⁰ L . (2.6)

Note that once S has been evaluated on the solution of the equation of motions, it can be interpreted as a functional of q(t) and q(t 0 ) = Q. In this way, the action is the generating function of the canonical transformation, transforming the system variables from t 0 to t.

Describing the same one-dimensional system in quantum mechanics it is possible

to introduce two independent representations for the system, |qi and |Qi, and

look for a hq|Qi connecting the two representations. If F is any function of the

dynamical variables, it will have a “mixed” representative hq| ˆ F |Qi and thanks to the completeness relation

Z

dq |qi hq| = 1 (2.7)

we get

hq| ˆ F |Qi = Z

hq| ˆ F |q ⁰ i hq ⁰ |Qi dq ⁰ = Z

hq|Q ⁰ i hQ ⁰ | ˆ F |Qi dQ ⁰ , (2.8) From these relations we obtain

hq|ˆ q|Qi = q hq|Qi , hq|ˆ p|Qi = −i¯ h ∂

∂q hq|Qi , (2.9) hq| ˆ Q|Qi = Q hq|Qi , hq| ˆ P |Qi = i¯ h ∂

∂Q hq|Qi . (2.10) However, since ˆ Q and ˆ q do not necessarily commute, if F = F [q, Q] the “mixed”

representative hq|F [ˆ q, ˆ Q]|Qi, may be not well defined. The generic function F = F [q, Q] is called well-ordered if it can be expressed as F [q, Q] = P

k f _k ¹ (q)f _k ² (Q).

Then if F is well-ordered, so the above “mixed” representative is well defined.

Therefore, setting hq|Qi = e

^hi¯

^{U (q,Q)} into equations above we obtain hq|ˆ p|Qi = ∂U (q, Q)

∂q hq|Qi , hq| ˆ P |Qi = − ∂U (q, Q)

∂Q hq|Qi . (2.11) Finally supposing ^∂U _∂q and ^∂U _∂Q are well-order we find

ˆ p = ∂ ˆ U

∂q , P = − ˆ ∂ ˆ U

∂Q . (2.12)

So U is the analogue of the classical function S and in this way Dirac concluded that

hq, t|Q, T i ∼ e

^¯hß

^R

^Tt

^dtL . (2.13) The Feynman path-integral. As Dirac emphasized, the “∼” above is just a loose connection. As matter of fact, a “=” would not be correct in the previous relation as long as T − t is a finite time interval. Feynman started from (2.13) and he assumed it as an equality (up to a constant) only for an infinitesimal time interval:

hq _t ⁰ |q t+δt i = C exp



− i

¯

h δtL(q ⁰ _t , q t+δt )



. (2.14)

Now split the time interval T − t into N infinitesimal time intervals t a = t + a, N  = T − t, using the completeness relation (2.7), we find

hq _t ⁰ |q _T i = Z

dq ₁ dq ₂ · · · dq _{N −1} hq ⁰ _t |q ₁ i hq ₁ |q ₂ i · · · hq _{N −1} |q _T i , (2.15) which is an exact quantum mechanical relation. Replacing Eq.(2.14) into Eq.(2.15) we can conclude that

hq _t ⁰ |q t i = lim

N →∞ A ^N

Z ^{N −1} Y

i=1

dq i

!

e

^¯hi

^R

^Tt

^{dtL(q, ˙ q)} . (2.16)

Such an expression is not fully corrected, yet. It can be proved that to “exactly”

formulate the transition amplitude the action must be expressed through the Hamiltonian formalism, so we can finally define the transition amplitude as

hq ⁰ _t |q T i = Z

DqDp exp

 i

Z t T

dT

 p dq

dt − H(p, q)



. (2.17)

3 The path-integral for a scalar field

Let us apply the functional method of path-integral to the theory of a real scalar field φ(x). The Lagrangian density of the theory is the following

L(φ, ∂ µ φ) = L 0 (φ, ∂ µ φ) − V (φ) = 1

2 ∂ µ φ∂ ^µ φ + 1

2 m ² φ ² − V (φ) , (3.1) where V (φ) is the potential. The first step is to build the Hamiltonian density H, performing a Legendre transformation:

π(x) = ∂L

∂ (∂ ₀ φ) = ∂ ₀ φ ≡ ˙ φ , (3.2) H(π, φ, ~ ∇φ) = π ˙ φ − L = 1

2



π ² +  ~ ∇φ  ²

+ m ² φ ²



+ V (φ) , (3.3) where π(x) is the canonical momentum. Eq.(2.17), extended to field theory, defines the transition amplitude from φ a (0, x) to φ b (T, x) = T as

hφ _b (x)| e ^−iHT |φ _a (x)i = N Z

DφDπ exp

"

i Z T

0

d ⁴ x

 π ˙ φ − 1

2 π ² − 1

2 (∇φ) ² − V (φ)

 # , (3.4) where N is the normalization constant and φ(x), over which we integrate, has the following boundary conditions

φ(x) =

( φ(x) = φ _a (x) if x ⁰ = 0 ,

φ(x ) = φ b (x) if x ⁰ = T . (3.5) The integration over π is trivial, just completing the square on the exponent and integrating, we obtain

hφ b (x)| e ^−iHT |φ a (x)i = N ⁰ Z

Dφ exp

"

i Z T

0

d ⁴ xL

#

, (3.6)

where N ⁰ is a normalization constant. Hereafter we will give up the Hamiltonian formalism, and take Eq.(3.6) to define the Hamiltonian dynamics.

Although the relation (3.6) is a very elegant one, physicists are mostly concerned with computing quantities that can be measured, like cross sections and decay rates. These quantities can be related to the S-matrix, which can be computed from the connected correlation functions trough LSZ (Lehmann, Symanzik, Zim- mermann) reduction formula. Then, we need a formula to compute the corre- lation functions. Let us consider the following object

Z

Dφ(x)φ(x 1 )φ(x 2 ) exp

"

i Z T

−T

d ⁴ xL(φ)

#

, (3.7)

where φ(x) is constrained by the boundary condition φ(x) =

( φ(x) = φ a (x) if x ⁰ = −T ,

φ(x) = φ _b (x) if x ⁰ = T . (3.8) With some manipulation it can be proved that (3.7) is equal to

hφ b | e ^−iHT T {φ H (x 1 )φ H (x 2 )} e ^−iHT |φ α i , (3.9) where ¹ T is the time ordering operator and the operators φ _H are expressed through the Heisemberg picture. Eq.(3.7) can be extended to the case of cor- relation functions from t = −∞ to t = ∞; this quantity is very important in quantum field theory. Therefore, we have to take the limit for T → ∞(1 − i) which selects the lowest energy level, indicated with |Ωi. Eq.(3.7) becomes

hΩ| T φ _H (x ₁ )φ _H (x ₂ ) |Ωi = lim

T →∞(1−i)

R Dφφ(x 1 )φ(x 2 ) exp h i R T

−T d ⁴ xL i R Dφ exp h

i R T

−T d ⁴ xL i , (3.10) that is the desired formula. Higher correlation functions can be obtained, just inserting additional factors φ H and φ, respectively on the left-hand and on the right-hand sides of the previous equation. Another feature of Eq.(3.10) is that it is manifestly Lorentz invariant ad it preserves also all the symmetries the Lagrangian L may have.

3.1 The generating functional

Generalizing the above equations we can introduce the central object of this work: the generating functional W [J ] of the Green functions, defined as follows

W [J ] ≡ hΩ|Ωi _J hΩ|Ωi = N

Z

DφDπe ⁱ h ^{π ˙ φ−H+J φ} i

= N ⁰ Z

Dφ exp

 i



∂ µ φ∂ ^µ φ + 1

2 m ² φ ² − V (φ) + J φ



, (3.11) where hf (x 1 ) · · · f (x N )i ≡ R d ^D x 1 . . . d ^D x N f (x 1 , . . . , x N ) and N, N ⁰ are normal- ization constants.

This path-integral is not well-defined because of the oscillatory integrand; we can remedy to this problem introducing a damping term or working in the Eu- clidean space. In this work it will be often used the latter method. Therefore, setting the new variables x 0 = −i¯ x 0 , d ⁴ x = −id ⁴ x, ∂ ¯ µ φ∂ ^µ φ = − ¯ ∂ µ φ ¯ ∂ µ φ; then the generating functional in Euclidean space is the following:

W E [J ] = N E

Z

Dφ exp



−  1 2

∂ ¯ µ φ ¯ ∂ µ φ + 1

2 m ² φ ² + V (φ) − J φ



. (3.12) This object is very important since it allows to compute the Green functions, defined as the coefficients of the functional expansion

W [J ] =

∞

X

N =0

i ^N N !

D J ₁ · · · J N G ^{(N )} (1, . . . , N ) E

, (3.13)

1

See appendix A

G ^{(N )} (1, . . . , N ) = 1 i ^N

δ

δJ ₁ . . . δ δJ _N W [J ]

    _{J =0}

. (3.14)

Green’s functions G ^{(N )} in Minkowsky space are identified with correlation func- tions. However, we can still use W E to construct Green’s functions G ^{(N )} _E in Eu- clidean space, but in this case we have to relate them to G ^{(N )} through analytic continuation (Wick rotation), which presuppose no singularities are encountered in the process of contour rotation.

The Feynman propagator. Let us evaluate the generating functional for a free theory ( V = 0 ), working in the Minkowsky space and putting the damping term e ⁻

¹²

^φ for the convergence problem; at the end of the calculation we have to take the limit  → 0 ( > 0). Therefore the new generating functional is

W 0, ≡ N Z

Dφ exp

 i  1

2 ∂ µ φ∂ µ φ − 1

2 (m ² − i)φ ² − J φ



. (3.15) The standard method to compute this integral is to work in the momentum space, given the Fourier transform and anti-transform

F (p) = ˜ Z +∞

−∞

d ⁴ x

(2π) ² e ^−ip·x F (x) , F (x) = Z +∞

−∞

d ⁴ p

(2π) ² e ^ip·x F (p) . (3.16) ˜ Then introducing the new field ( ˜ φ is the Fourier transorm of φ(x))

φ ˜ ⁰ (p) = ˜ φ(p) + p ² − m ² + i  −1 J (p) , ˜ (3.17) the generating functional becomes

W ₀ [J ] = exp

"

− i 2

Z

d ⁴ p | ˜ J (p)| ² p ² − m ² + i

# Z

Dφ ⁰ e ⁱ h

¹₂

^∂

µ

φ

⁰

∂

^µ

φ

⁰

−

¹₂

(m

²

−i)φ

⁰²

i , (3.18) where Dφ ⁰ differs from Dφ only for an omitted multiplicative constant. Then the following relation is evident

W ₀ [J ] = W ₀ [0] exp

"

− i 2

Z d ⁴ p

J (p) ˜ ˜ J (−p) p ² − m ² + i

#

, (3.19)

using the Fourier anti-transform it follows that

W 0 [J ] = W 0 [0]e ⁻

²ⁱ

^hJ

¹

^∆

^{F 12}

^J

²

ⁱ , (3.20) where ∆ F 12 ≡ ∆ F (x 1 − x 2 ) is the Feynman propagator

∆ F (x − y) =

Z d ⁴ p (2π) ⁴

e ^{−ip·(x−y)}

p ² − m ² + i . (3.21) It could be more convenient to set

W 0 [J ] = e ^iZ

⁰

^{[J ]} , (3.22) where

Z ₀ [J ] =  i

2 J (x)∆ _F (x − y)J (y)



. (3.23)

3.2 Generating functional of connected Green functions

As in the free case seen above, we set

W [J ] = e ^{iZ[J ]} . (3.24)

The term Z[J ] plays a key role in quantum field theory since it is the generating functional of the connected Green functions. Now, we prove it with reference to [4]. Let G ^{(N )} c denote the N -point connected Green functions. The general G ^{(N )} contains σ K copies of G ^(K) ( K ≤ N ). Then G ^{(N )} may be expanded in the form

G ^{(N )} = X

{σ

₁

,σ

₂

,...,σ

_N

}

X

P

P h

G ⁽¹⁾ _c · · · G ⁽¹⁾ _c i

· · · h

G ^{(N )} _c · · · G ^{(N )} _c i

, (3.25)

where the occupation number σ i are constrained by 1σ 1 + . . . + N σ N = N . P denotes all possible distinct permutations of the N variables. Then

W [J ] = e ^{iZ[J ]} =

∞

X

N =0

i ^N N !

Z

d ^D x 1 · · · d ^D x N G ^N (x 1 , · · · , x N )J (x 1 ) · · · J (x N )

=

∞

X

N =0

i ^N X

{σ

1

,...,σ

_N

} N

Y

j=1

h R d ^D x 1 · · · d ^D x j G ^(j) c · · · G ^(j) c J (x 1 ) · · · J (x j ) i σ

j

σ j !(j!) ^σ

^j

.

(3.26) Noting that P ∞

N =0

P

{σ

1

,...,σ

N

} = P

σ

k

, where the summation on the right hand side has no restriction, we obtain

W [J ] =

∞

Y

j=1

∞

X

σ

j

=0

1 σ _j !

 i j!

Z

d ^D x 1 · · · d ^D x j G ^(j) _c (x 1 , . . . , x j )J (x 1 ) · · · J (x j )

 ^σ

j

= exp

∞

X

N =1

i ^N N !

Z

d ^D x 1 · · · d ^D x N G ^{(N )} _c (x 1 , . . . , x N )J (x 1 ) · · · J (x N ) ,

= exp (iZ[J ]) . (3.27)

4 Alternative representation for the generating functional

In this section we will introduce a different representation for the generating functional, working in D dimensional Euclidean space. Hereafter the subscript E will be omitted. First of all it is necessary to introduce some notations. For every even function or distribution G and for any functions or operators f ₁ and f ₂ , we set:

f ₁ Gf ₂ = hf ₁ (x)G(x − y)f ₂ (y)i , δ δf ₁ G δ

δf ₂ =

 δ

δf ₁ (x) G(x − y) δ δf ₂ (y)



.

(4.1)

The starting form of W [J ] is the following W [J ] ≡ e ^{−Z[J ]} = N

Z

Dφ exp



− Z

d ^D x  1

2 ∂ µ φ∂ µ φ + 1

2 m ² φ ² + V (φ) − J φ



, (4.2) where Z[J ] is the generating functional for connected Green functions and N is the normalization constant.

N =

Z

Dφ exp(−S[φ])

 −1

, (4.3)

where

S[φ] = Z

d ^D x  1

2 ∂ µ φ∂ µ φ + 1

2 m ² φ ² + V (φ)



. (4.4)

Schwinger representation. Suppose V (φ) can be expanded as

V (φ) =

∞

X

n=0

c ⁿ φ ⁿ . (4.5)

Using

δ

δJ (x) e ^hJφi = φ(x)e ^hJφi , (4.6) we have

W [J ] = N Z

Dφe ^{h−V (φ)i} e ⁻ h

¹2

∂

µ

φ∂

µ

φ+

¹₂

m

²

φ

²

−Jφ i

= N e ⁻ h ^V (

δJ^δ

)i Z

Dφe ⁻ h

¹2

∂

µ

φ∂

µ

φ+

¹₂

m

²

φ

²

−Jφ i

= N N 0

exp



− Z

V

 δ δJ



W 0 [J ] , (4.7)

where

N 0 =

Z

Dφ exp(−S 0 [φ])

 −1

. (4.8)

Expression (4.7) takes the name of Schwinger representation for the generating functional.

4.1 Dual representation for W [J ]

As we have seen the connection between the path-integral formalism and the operator one is the following

W [J ] = hΩ|Ωi _J

hΩ|Ωi , (4.9)

Note that we have

W [J ] = N h0|T exp

Z

(−V ( ˆ φ) + J ˆ φ)



|0i , (4.10)

where |0i is the free vacuum. Let us introduce the field φ _c (x), defined as φ c (x) =

Z

d ^D yJ (y)∆(y − x) , (4.11) which satisfies the following equation

−∂ ² + m ²
φ c (x) = J (x) . (4.12) Replacing φ by φ + φ c into Eq.(4.10), it follows that (up to a constant)

W [J ] = h0|T exp

Z

(−V ( ˆ φ + φ c )) + J ( ˆ φ + φ c )



|0i

= e ^−Z

⁰

^{[J ]} h0|T exp

Z

(−V ( ˆ φ + φ c ))



|0i . (4.13)

Note that, thanks to the Wick theorem (A.11), h0|T F [ ˆ φ + f ]|0i = h0| exp  1

2 δ δf ∆ δ

δf



: F [ ˆ φ + f ] : |0i

= exp  1 2

δ δf ∆ δ

δf



h0|F [f ]|0i

= exp  1 2

δ δf ∆ δ

δf



F [f ] . (4.14)

Finally, applying Eq.(4.14) to the right hand side of Eq.(4.13), we get W [J ] = exp(−Z ₀ [J ]) exp  1

2 δ δJ ∆ ⁻¹ δ

δJ

 exp



− Z

d ^D xV

Z

d ^D zJ (z)∆(z − x)



, (4.15) that, can be expressed through φ c as

W [J ] = T [φ c ] = N N 0

exp(−U 0 [φ c ]) exp  1 2

δ δφ c

∆ δ δφ c

 exp



− Z

V (φ c )

 , (4.16) where

U 0 [φ c ] = − 1

2 φ c ∆ ⁻¹ φ c , (4.17)

and ∆ ⁻¹ (x) = R d ^D p(p ² + m ² )e ^ipx .

4.2 Schwinger-Dyson equation in the dual representation

Here we shortly present the Schwinger-Dyson equation, then we will express this equation through the dual representation we have just introduced.

Schwinger-Dyson equation. This equation is the quantum equation of mo- tion for Green’s functions. In classical mechanics the equation of motion could be derived by imposing that the action has to be stationary under an infinites- imal variation

φ(x) → φ(x) + (x) . (4.18)

The appropriate generalization to quantum field theory is to consider this vari- ation as an infinitesimal change of variables

φ(x) → φ(x) + F [φ, x] , (4.19) which does not change the measure (Dφ = Dφ ⁰ ) and the value of the path- integral. F [φ, x] is an arbitrary functional of φ (we suppose it admits an expan- sion in powers of φ). The generating functional, expanded to the first order in

, becomes:

W  [J ] = Z

D



1 +   δF δφ

  1 − 

Z

d ^D x  δ hLi δφ − J φ

 F



exp (− hL − J φi) . (4.20) Collecting the terms proportional to , imposing that the path-integral does not change and using Eq.(4.6), we could find

Z d ^D xF

 δ

δJ , x   δ hLi δφ

 δ δJ



− J (x)



W [J ] = 0 . (4.21) If F = F (x), then (4.19) is just a translation of φ. The above equation reduces

to Z δ hLi

δφ

 δ δJ



− J



W [J ] = 0 , (4.22)

that could be expressed as



∆ ⁻¹ δ δJ (x) +

Z δV δφ(x)

 δ δJ



− J (x)



W [J ] = 0 , (4.23) where

∆ ⁻¹ δ δJ (x) ≡

Z

d ^D y∆ ⁻¹ (y − x) δ

δJ (y) . (4.24)

With the dual representation, the above equation becomes

 δ δφ c

+ Z δV

δφ



∆ δ δφ c



e ^−U

⁰

^[φ

^c

^] e

^12δφcδ

^∆

^δφcδ

e ^{− R V (φ}

^c

⁾ =

 δ δφ c

+ e ^U

⁰

^[φ

^c

^] Z δV

δφ



∆ δ δφ c



e ^−U

⁰

^[φ

^c

^]



e

^12δφcδ

^∆

^δφcδ

e ^{− R V (φ}

^c

⁾ = 0 . (4.25) As we will see there is a deep connection with the Hermite polynomials.

Relation with the Hermite polynomials. The standard representation of the “probabilistic” Hermite polynomials is given by

He n (x) = (−1) ⁿ e

^x22

D ⁿ e ⁻

^x22

. (4.26) Thanks to Eq.(B.7) the right hand side of (4.26) becomes

(−1) ⁿ e

^x22

D ⁿ e ⁻

^x22

= e ⁻

^D22

x ⁿ . (4.27) Replacing x with ix into (4.27) we get

e ⁻

^x22

D ⁿ e

^x22

= e

^D22

x ⁿ . (4.28)

Then, supposing f (x) can be expanded in power of x, we find

e ⁻

^x22

f (D)e

^x22

= e

^D22

f (D) . (4.29) This provides the following expansion

e

^D22

f (x) =

∞

X

n=0

(−i) ⁿ c n He n (ix) . (4.30) This equation can be used in quantum field theory. As matter of fact, (4.16) involves exp ¹ ₂ δ φ

c

∆δ φ

c

acting on φ c . So in the perturbative expansion there appear terms like

exp  1

2 δ φ

c

∆δ φ

c



φ ⁿ _c . (4.31)

Note that

δ φ

_c

∆δ φ

_c

φ ⁿ _c (x) = n(n − 1)∆(0)φ ⁿ⁻² _c (x) , (4.32) is the functional version of

∆(0)∂ ² _φ

_c

φ ⁿ _c = n(n − 1)∆(0)φ ⁿ⁻² _c , (4.33) thanks to (4.30) we obtain

exp  1

2 δ φ

_c

∆δ φ

_c



φ ⁿ _c (x) = (−i) ⁿ ∆

ⁿ²

(0)He n

 iφ _c (x)

∆

¹²

(0)



. (4.34) Eq.(4.34) suggests a connection of the Schwinger-Dyson equation with the Her- mite polynomials. We start from (4.27), through which we can find

e ^U

⁰

^[φ

^c

^] δ ⁿ

δφ ⁿ _c (x) e ^−U

⁰

^[φ

^c

^] =

" _n X

k=0

n k

 δ ^n−k

δφ ^n−k c (x) e ^−U

⁰

^[φ

^c

^]

# δ ^k δφ ^k _c (x)

=

"

1

2 exp (δ φ

_c

∆δ φ

_c

)

n

X

k=0

n k



∆ ⁻¹ φ c
^n−k (x)

# δ ^k δφ ^k _c (x) .

(4.35) Using (4.34) we can express the Schwinger-Dyson equation for V = _n! ^λ φ ⁿ as follows

"

δ δφ c (x) +

n−1

X

k=0

λ(−i) ^k ∆

^k2

(0) (n − k − 1)!k! He k

 iφ _c (0)

∆

¹²

(0)

 

∆ δ δφ c

 n−k−1

(x)

#

e

^12δφcδ

^∆

^δφcδ

e ^{− R V (φ}

^c

⁾ = 0 . (4.36)

It interesting to consider a normal ordered potential : V (φ) := λ

n! : φ ⁿ := λ n! exp



− 1 2

δ δφ ∆ δ

δφ

 φ ⁿ , in this case (4.25) becomes

"

δ

δφ _c (x) + λ (n − 1)!

n−1

X

k=0

n − 1 k

 φ ^k _c (x)



∆ δ δφ _c

 ^n−k−1 (x)

#

e

^12δφcδ

^∆

^δφcδ

e ^{− R :V (φ}

^c

^): = 0 ,

(4.37)

that compared with (4.36) shows how the terms e ^±U

⁰

^[φ

^c

^] compensate the con-

tribution coming from the normal ordering regularization of the potential.

4.3 T [φ _c ] and normal ordered potentials

Let us consider only the case of a normal ordered potential, indicated with : V :.

It is useful to set D j = 1

2 δ δφ _c

_j

∆ δ

δφ _c

_j

, D jk = δ δφ _c

_j

∆ δ

δφ _c

_k

, D = 1 2

δ δφ _c ∆ δ

δφ _c . (4.38) The expression (4.16) becomes

T [φ _c ] = N N 0

exp(−U ₀ [φ _c ]) exp(D) exp



− Z

: V (φ _c ) :



. (4.39) Considering a generic functional F [φ], we want to express e ^D F [φ] as

exp(D) exp (F [φ]) = exp

∞

X

N =1

Q N

N !

!

, (4.40)

where {Q N } is a set of connected functionals, defined as Q N [φ] = e ^D F ^N [φ] 

conn

=

N

Y

i>j=1

e ^D

^ij

N

Y

i=1

e ^D

ⁱ

F [φ i ] 

conn φ

i

→φ . (4.41) The subscript “connect” means that at least one linkage operator (e ^D

^ij

) must be retained between each pairs of ² F [φ i ]. We need a similar decomposition; then we set

T [φ c ] = N N 0

exp −U 0 [φ c ] +

∞

X

k=1

Q k [φ c ] k!

!

, (4.42)

where now

Q N [φ c ] = e ^D

Z

: V (φ c ) :

 ^N     conn

. (4.43)

Note that, like U [φ _c ], the Q _N generate connected functions. Rescaling the potential by a constant µ we find

exp(D) exp



−µ Z

: V (φ c ) :



= exp

∞

X

K=1

µ ^k k! Q k [φ c ]

!

, (4.44)

and so

Q _k [φ _c ] = ∂ _µ ^k ln



exp(D) exp



−µ Z

: V (φ _c ) :



   _µ=0

. (4.45)

Now, the relation (B.1), expressed below through the appropriate variables, e ^D F [φ _c ]G[φ _c ] = e ^D

¹²

e ^D

¹

F [φ _c

₁

]e ^D

²

G[φ _c

₂

]


 _φ

c1

=φ

_c2

=φ

c

(4.46) allows to make some considerable simplification. To show this, let us calculate Q 1 and Q 2

Q 1 = −e ^D Z

: V := − Z

V (4.47)

2

This method is clearly exposed in [3].

and

Q ₂ = e ^D

¹²

− 1 



e ^D

¹

Z

: V [φ _c

₁

] :

  e ^D

²

Z

: V [φ _c

₂

] :



   _φ

c1

=φ

_c2

=φ

_c

= e ^D

¹²

− 1 

Z

V (φ c

₁

) Z

V (φ c

₂

)

    _φ

c1

=φ

_c2

=φ

c

. (4.48)

Then

Q _n [φ _c ] = (−1) ⁿ

n

Y

j>k

e ^D

^jk

n

Y

i=1

Z

V (φ _c

_i

)       _{c, φ}

c1

=φ

_c2

=...=φ

_c

, (4.49)

where the subscript c indicates that non connected terms must be discharged.

So the generating functional of the connected Green function can be expressed as follows

U [φ c ] = ln N N 0

+ U 0 [φ c ] +

∞

X

p=1

(−1) ^p+1 p!

n

Y

j>k

e ^D

^jk

n

Y

i=1

Z

V (φ c

_i

)       _{c, φ}

c1

=φ

_c2

=...=φ

c

. (4.50)

5 Generating functional and “covariant” deriva- tives

In this section we will express the generating functional through “covariant”

derivatives. The key expression is the following operator identity exp



− 1 2 IM I



F [δ I ] exp  1 2 IM I



= F [D M I ] , (5.1) where F is a functional, I and M are functions (or distributions), D M I (x) denotes the “covariant derivative”

D M I = δ

δI(x) + M I(x) . (5.2)

In our case we define

D _φ ^± (x) = ∓∆ δ

δφ (x) + φ(x) . (5.3)

It can be easily proved that these operators satisfy the following commutation relations

h D ⁻ _φ (x), D _φ ⁺ (y) i

= 2∆(x − y) , h

D _φ ^± (x), D _φ ^± (y) i

= 0 . (5.4) Another fundamental relation comes from the use of (5.2) into the operatorial version of (B.7)

exp  1

2 δ _I M ⁻¹ δ _I



F [M I] = F [D _{M I} ] · 1 . (5.5)

Thanks to the above relations, (4.16) becomes T [φ c ] = N

N ₀ exp (−U 0 [φ c ]) exp



− Z

V (D ⁻ _φ

c

)



· 1 . (5.6)

This new representation still simplifies the form of the Schwinger-Dyson equa- tion (4.25), reducing it to

 δ

δφ c (x) + Z δV

δφ

 D _φ ⁻

c

  exp



− Z

V (D _φ ⁻

c

)



· 1 = 0 . (5.7) Now we will try to calculate the Green function through covariant derivatives.

To do this, note the expression below δ

δJ (x) exp (−U 0 [φ c ]) = exp (−U 0 [φ c ]) D _φ ⁻

c

(x) , (5.8)

so the N -point Green function is δ ⁿ W [J ]

δJ (x ₁ ) . . . δJ (x _N ) = exp(−U ₀ [φ _c ])D _φ ⁻

c

(x ₁ ) . . . D _φ ⁻

c

(x _N ) exp



− Z

V (D _φ ⁻

c

)



· 1

= exp(−U 0 [φ c ])



− Z

V (D ⁻ _φ

c

)

 D _φ ⁻

c

(x 1 ) . . . D ⁻ _φ

c

(x N ) · 1 . (5.9) The above representation makes easier the explicit calculation as we will see later.

Another feature of this representation concerns the case of a normal ordered potential. According to the Wick theorem (A.11) and to (5.1), we can write

: F [φ] := F [D ⁺ _φ

c

] · 1 . (5.10)

For example let us try to calculate : φ ⁴ :. We use the notation ∆(x ₁ − x ₂ ) = ∆ ₁₂ and φ(x _i ) = φ _i .

: φ ² (x) : =

2

Y

k=1

D _φ (x _k ) · 1      x

1

=x

2

=x

=

2

Y

k=1

φ(x _k ) + ∆ ₁₂      x

1

=x

2

=x

= φ ² (x) − ∆(0) ,

: φ ³ (x) : =

3

Y

k=1

D φ (x k ) · 1      x

_k

=x

=

3

Y

k=1

φ(x k ) − ∆ 12 φ 3 − ∆ 13 φ 2 − ∆ 23 φ 1

     x

_i

=x

= φ ³ (x) − 3∆(0)φ(x) , : φ ⁴ (x) : =

4

Y

k=1

D φ (x k ) · 1      x

_k

=x

=

4

Y

k=1

φ k − ∆ 12 φ 3 φ 4 − ∆ 13 φ 2 φ 4 − ∆ 14 φ 2 φ 3 −

− ∆ ₂₃ φ ₁ φ ₄ − ∆ ₂₄ φ ₁ φ ₃ − ∆ ₃₄ φ ₁ φ ₂ + ∆ ₁₂ ∆ ₃₄ + ∆ ₁₃ ∆ ₂₄ + ∆ ₂₃ ∆ ₁₄ | _x

k

=x

= φ ⁴ (x) − 6∆(0)φ ² (x) + 3∆ ² (0) . (5.11)

Since D _φ ⁺ and D ⁻ _φ differ only by the sign of ∆(x − y), we can easily get the expression of Q n

k D ⁻ _φ from (5.11). For example let us calculate T [φ c ] for V (φ) =

λ

4! , to the first order in λ. Using Eq.(5.6) and thanks to (5.11) T [φ _c ] = N

N 0

exp (−U ₀ [φ _c ])

 1 − λ

4!

Z

d ^D xD ⁻ _φ

c

4 (x) + ...



· 1 ,

= N

N ₀ exp (−U ₀ [φ _c ])

 1 − λ

4!

Z

d ^D x φ ⁴ _c (x) + 6φ ² _c (x)∆(0) + 3∆ ² (0)

 + . . .

 . (5.12) The generating functional of connected Green functions. Let us em- ploy this representation to Z[J ]. With reference to section 4.3 we will consider a generic potential V [φ] which can be expanded in powers of φ (V is not normal ordered as in the previous case). Then the generating functional is

T [φ _c ] = N

N ₀ exp (−U ₀ [φ _c ]) exp  1 2

δ δφ _c ∆ δ

δφ _c

 exp



− Z

V (φ _c )



= N N 0

exp −U 0 [φ c ] +

∞

X

k=1

Q _k [φ _c ] k!

!

= exp (−U [φ c ]) , (5.13) where Q N are connected functionals. Rescaling the potential with µ, we could obtain

Q k [φ c ] = ∂ _µ ^k ln



exp(D) exp



−µ Z

V (φ c )



   _µ=0

. (5.14)

Introducing the covariant derivatives and thanks to the following relations exp



± 1 2

δ δφ ∆ δ

δφ

 exp



− Z

V (φ c )



= exp



− Z

V (D ^∓ _φ

c

)



· 1 , (5.15) exp(D)F [φ]G[φ] = F [D ⁻ _φ ]G[D ⁻ _φ ] · 1 , (5.16) we are able to make some relevant simplifications. Let us try to compute Q ₁ and Q ₂

Q 1 = −e ^D Z

V = − Z

V  D _φ ⁻

c

 · 1 , (5.17)

Q ₂ = e ^D Z

V Z

V = Z

V  D ⁻ _φ

c

 Z V 

D ⁻ _φ

c

 · 1 . (5.18) So generalizing, we can express the generating functional as follows:

U [φ c ] = ln N N 0

+ U 0 [φ c ] +

∞

X

p=1

(−1) ^p+1 p!

Z V 

D ⁻ _φ

c

  p

· 1      c

, (5.19)

where the subscript c means that terms non connected by at least one propagator must be discharged. Eq.(5.19) can be expanded to the case of normal ordered potential : V (φ) : as

U [φ c ] = ln N N 0

+ U 0 [φ c ] +

∞

X

p=1

(−1) ^p+1 p!

Z V 

D ⁺ _φ

c

 · 1     _φ

c

=D

⁻_φc

! ^p

· 1      c

= ln N

N ₀ + U 0 [φ c ] +

∞

X

p=1

(−1) ^p+1 p!

Z V 

D ⁺ _φ

c

 · 1     _φ

c

=D

⁻

φc

! p−1

· Z

V (φ c )       _c

.

(5.20)

Now, we calculate G ⁽²⁾ c and G ⁽⁴⁾ c for _4! ^λ φ ⁴ theory to the second order in λ, to show how this alternative representation works. Eq.(5.19) gives expression to the generating functional. Using the following notations: ∆(x i − x j ) ≡ ∆ ij , φ c (x i ) ≡ φ i , φ c (y) ≡ φ y , the generating functional, expanded to the first order in λ is

U [φ c ] = ln N N 0

+ U 0 [φ c ] + λ 4!

Z  D _φ ⁻

c

 ⁴

· 1     _c

. (5.21)

But we have already calculated  D ⁻ _φ

c

 ⁴

· 1, and so we have

 D _φ ⁻

c

 ⁴

· 1 = φ ⁴ _x + 6φ ² _x ∆ xx + 3∆ ² _xx . (5.22) Now, we proceed to calculate the generating functional to the second order in λ

D _φ ⁻

y

· φ ⁴ _x + 6φ ² _x ∆ xx + 3∆ ² _xx

= φ ⁴ _x φ _y + 4φ ³ _x ∆ _xy + 6φ ² _x φ _y ∆ _xx + 12φ _x ∆ _xx ∆ _xy + 3φ _y ∆ ² _xx , (5.23)

 D ⁻ _φ

y

 2

· φ ⁴ _x + 6φ ² _x ∆ _xx + 3∆ ² _xx
= φ ⁴ _x φ ² _y + 8φ ³ _x φ y ∆ xy + φ ⁴ _x ∆ yy + 12φ ² _x ∆ ² _xx + 6φ ² _x φ ² _y ∆ xx + 12φ x φ y ∆ xx ∆ xy + 6φ ² _x ∆ _xx ∆ _yy + 12φ _x φ _y ∆ _xx ∆ _xy + 12∆ _xx ∆ ² _xy + 3φ ² _y ∆ ² _xx + 3∆ ² _xx ∆ _xy ,

(5.24)

 D ⁻ _φ

y

 3

· φ ⁴ _x + 6φ ² _x ∆ xx + 3∆ ² _xx
= φ ⁴ _x φ ³ _y + 12φ ³ _x φ ² _y ∆ xy + 3φ ⁴ _x φ y ∆ y y + 36φ ² _x φ y ∆ ² _xy + 12φ ³ _x ∆ xy ∆ yy + 24φ _x ∆ ³ _xy + 18φ ² _x φ _y ∆ _xx ∆ _yy + 36φ _x ∆ _xx ∆ _xy ∆ _yy + 6φ ² _x φ ³ _y ∆ _xx + 36φ x φ ² _y ∆ xx ∆ xy + 36φ y ∆ xx ∆ ² _xy + 3φ ³ _y ∆ ² _xx + 6φ y ∆ ² _xx ∆ yy + 3φ y ∆ ² _xx ∆ xy .

(5.25) Our final task is to compute G ⁽²⁾ c and G ⁽⁴⁾ c . Let us start with the 2-point Green’s function and to find it we use the following relation:

G ⁽²⁾ _c (x 1 , x 2 ) = − δ ² Z[J ] δJ 1 δJ 2

    _{J =0}

= − Z

d ^D y 1 d ^D y 2 ∆ x

1

y

1

∆ x

2

y

2

δ ² U [φ _c ] δφ c (y 1 )δφ c (y 2 )

    _φ

c

=0

. (5.26) It is not necessary to explicitly calculate 

D ⁻ _φ

y

 ⁴

· φ ⁴ _x + 6φ ² _x ∆ xx + 3∆ ² _xx
be- cause many of its terms do not contribute to the 2-point Green’s function. We just consider the terms of the generating functional which meet our purpose.

Then, we have to discharge the disconnected terms and the terms that do not contain φ ² _y or φ ² _x or φ x φ y . We get

U [φ c ]| ₂ = ln N N 0

+ U 0 [φ c ] + λ 4!

Z

d ^D xφ ⁴ _x + 6φ ² _x ∆ xx + 3∆ ² _xx



− λ ² 2(4!) ²

Z

d ^D xd ^D y 96φ _x φ _y ∆ ³ _xy + 144φ _x φ _y ∆ _xx ∆ _yy ∆xy + 72φ ² _x ∆ _yy ∆ ² _xy + 72φ ² _y ∆ _xx ∆ ² _xy

+o(λ ² ) ,

(5.27)

where the subscript 2 means that this expression can just lead to G ⁽²⁾ c . There- fore, by (5.26) applied to (5.27), we obtain:

G ⁽²⁾ _c (x ₁ , x ₂ ) = ∆ _x

₁

_x

₂

− λ 2 Z

d ^D x∆ _x

₁

_x ∆ _xx ∆ _x

₂

_x + λ ² 6

Z

d ^D xd ^D y∆ _x

₁

_x ∆ ³ _xy ∆ _x

₂

_y + λ ²

4 Z

d ^D xd ^D y∆ _x

₁

_x ∆ ² _xy ∆ _yy ∆ _x

₂

_x + λ ² 4

Z

d ^D xd ^D y∆ _x

₁

_x ∆ _xx ∆ _xy ∆ _yy ∆ _x

₂

_y + o(λ ² ) . (5.28) Finally we work on G ⁽⁴⁾ c , that can be expressed as follows:

G ⁽⁴⁾ _c = − δ ⁴ Z[J ] δJ 1 . . . δJ 4

    _{J =0}

= − Z

d ^D y ₁ · · · d ^D y ₄ ∆ _x

₁

_y

₁

· · · ∆ x

4

y

4

δ ⁴ U [φ _c ] δφ c (y 1 ) . . . δφ c (y 4 )

    _φ

c

=0

. (5.29) Following a similar procedure, we obtain the generating functional to calculate 4-point Green’s function (to the second order in λ) and it is expressed as

U [φ c ]| ₄ = ln N N 0

+ U 0 [φ c ] − λ 4!

Z

d ^D xφ ⁴ _x + 6φ ² _x ∆ xx + 3∆ ² _xx



+ λ ² 2(4!)

Z

d ^D xd ^D y 72φ ² _x φ ² _y ∆ ² _xy + 48φ _x φ ³ _y ∆ _xx ∆ _xy + 48φ ³ _x φ _y ∆ _yy ∆ _xy
+ o(λ ² ) . (5.30) We obtain

G ⁽⁴⁾ _c (x ₁ , x ₂ , x ₃ , x ₄ ) = −λ Z

d ^D x∆ _x

₁

_x ∆ _x

₂

_x ∆ _x

₃

_x ∆ _x

₄

_x + λ ²

6 Z

d ^D xd ^D y ∆ ² _xy [∆ _x

₁

_x ∆ _x

₂

_x ∆ _x

₃

_y ∆ _x

₄

_y + ∆ _x

₁

_x ∆ _x

₃

_x ∆ _x

₂

_y ∆ _x

₄

_y + ∆ _x

₁

_x ∆ _x

₄

_x ∆ _x

₂

_y ∆ _x

₃

_y ]
+ + λ ²

2 Z

d ^D xd ^D y (∆ _yy ∆ _xy [∆ _x

₁

_x + ∆ _x

₂

_x + ∆ _x

₃

_x + ∆ _x

₄

_y + cyclic permutations]) + o(λ ² ) , (5.31)

and it is the result we expected to find (for example see [7]).

APPENDIX A

A Wick theorem

Let us consider the 2-field correlation function expressed through the operatorial formalism ³

h0|T {φ(x 1 )φ(x 2 )} |0i . (A.1) We would like to rewrite it in a form that it is easy to evaluate and that can also be expanded to the case of more than two fields. First of all, we can write the field as following

φ(x) = φ ⁺ (x) + φ ⁻ (x) , (A.2) where

φ ⁺ =

Z d ³ p (2π) ³

1 p2E p

a _p e ^−ip·x ; φ ⁻ =

Z d ³ p (2π) ³

1 p2E p

a ^† _p e ^+ip·x . (A.3)

This decomposition is very useful, because thanks to a p e ^−ip·x and a ^† _p follows

φ ⁺ (x) |0i , h0| φ ⁻ (x) = 0 . (A.4)

A term like a ^† _p a ^† _q a _k a _l is said to be normal ordered and has a vanish vacuum expectation value. Let us define the normal ordering symbol N () whose action is to make into normal order the operators it contains. We introduce, now, one more quantity, the contraction of two field, defined as follows:

φ(x)φ(y) =

( [φ ⁺ (x), φ ⁻ (y)] for x ⁰ > y ⁰ ,

[φ ⁺ (y), φ ⁻ (x)] for y ⁰ > x ⁰ . (A.5) This quantity is exactly the Feynman propagator

φ(x)φ(y) = ∆(x − y) , (A.6)

Now, supposing x ₀ > y ₀ , the time-ordered product is

T φ(x)φ(y) = φ ⁺ (x)φ ⁺ (y) + φ ⁺ (x)φ ⁻ (y) + φ ⁻ (x)φ ⁺ (y) + φ ⁻ (x)φ ⁻ (y)

= φ ⁺ (x)φ ⁺ (y) + φ ⁻ (y)φ ⁺ (x) + φ ⁻ (x)φ ⁺ (y) + φ ⁻ (x)φ ⁻ (y) + φ ⁺ (x), φ ⁻ (y) . (A.7)

The relation between the time-ordering and the normal-ordering is the following T {φ(x)φ(y)} = N {φ(x)φ(y)} + φ(x)φ(y) = N {φ(x)φ(y)} + h0|φ(x)φ(y)|0i .

(A.8) The generalization to many arbitrary field takes the name of Wick’s theorem and it is the following (for example, see [6])

T {φ(x ₁ )φ(x ₂ ) . . . φ(x _n )} = N {φ(x ₁ )φ(x ₂ ) . . . φ(x _n ) + all possible contractions} . (A.9) This rule admits a functional form that could make easier the explicit calcula- tions. For example (A.8) can be written as

T (φ(x ₁ )φ(x ₂ )) =

 1 ± 1

2 Z

d ^D y ₁ d ^D y ₂ δ

δφ(y 1 ) ∆(y ₁ − y ₂ ) δ δφy 2



N (φ(x ₁ )φ(x ₂ )) . (A.10)

3

The fields are operator expressed through the Heisemberg picture

APPENDIX A

The equation above could be generalized to the case of many arbitrary fields (an excellent reference is [8]); the result is the following

T F [ ˆ φ] = exp  1 2

δ δ ˆ φ ∆ δ

δ ˆ φ



: F [ ˆ φ] : . (A.11)

APPENDIX B

B Some relations

In this appendix we detail the proofs of some relations used in this work.

B.1

Let φ _i (x _i ), ∆(x _i , x _j ), J _i (x _i ) be functions or distributions, let F and G be func- tionals. The relation we want to prove is the following

e ^D F [φ]G[φ] = e ^D

¹²

[(e ^D

¹

F [φ ₁ ])(e ^D

²

G[φ ₂ ])] 

 _φ

1

=φ

₂

=φ , (B.1) where

D j = − i 2

Z δ

δφ j

∆ δ δφ j

, D ij = −i

Z δ δφ i

∆ δ δφ j

. (B.2)

Suppose F [φ], G[φ] can be expanded in powers of φ, then the starting point is the relation below

F [φ] = F



−i δ δJ



· exp

 i

Z J φ

    _{J =0}

. (B.3)

Generalizing this equations to the product of two functionals and considering that e ^D commutes with F −i _δJ ^δ  , we get

e ^D F [φ]G[φ] = F  1 i

δ δJ ₁

 G  1

i δ δJ ₂



e( ⁻

ⁱ²

^R

^δφδ

^A

^δφδ

)e ^{[i R φ(J}

¹

^+J

²

^)] 

   _J

1

=J

₂

=0

. (B.4) Now, noting that

exp



− i 2

Z δ δφ A δ

δφ

 exp

 i

Z J φ



= exp

 i

Z

J AJ + i Z

J φ



, (B.5)

we obtain e ^D F [φ]G[φ] =

F · G · exp  i 2

Z

J 1 ∆J 1 + i 2

Z

J 2 ∆J 2 + i Z

J 1 ∆J 2



· exp

 i

Z

φ(J 1 + J 2 )

    _J

1

=J

₂

=0

,

(B.6)

that can be rearranged to Eq.(B.1) .

B.2 APPENDIX B

B.2

Let I and L be functions, let F be a functional and let M be an even function or distribution. Then, we have

exp



− 1 2 IM I



F [δI] exp  1 2 IM I



=  1

2 δ I M ⁻¹ δ I



F [M I] . (B.7) We are will not prove it, but we will prove its more general operatorial version.

We define

F [M I] ≡ exp δ I M ⁻¹ δ I
F [M I] . (B.8) The operatorial version of Eq.(B.7) is

exp



− 1 2 IM I



F [δ _I ] exp  1 2 IM I



= exp (−IM I) F [δ _I ] exp (LM I)   _L=I .

(B.9) It immediately follows that if we let the right hand side of the above equation acting on 1, we obtain the Eq.(B.7). Indeed

exp (−IM I) F [δ I ] exp (LM I) 

 _L=I · 1 = F [δ I ] 

 δ

I

≡M I = F [M I] . (B.10) Then, let us demonstrate Eq.(B.9). To start with we introduce the Laplace transform

L{f }(s) ≡ Z ∞

−∞

dte ^−st f (t) , (B.11)

through which we can express F [M I] as follows F [M I] = e

¹²

^δ

^I

^M

⁻¹

^δ

^I

Z

DJ e ^{IM J} F [M J ] = ˆ Z

DJ e

¹²

J M J +IM J F [M J ] . ˆ (B.12) then we have

F [δ I ] = Z

DJ e

¹²

^{J M J +J δ}

^I

F [M J ] . ˆ (B.13) Now, we will have the Eq.(B.9) acting on a generic functional G[M I]

e ⁻

¹²

^{IM I} F [δ I ]e

¹²

^{IM I} G[M I] = e ⁻

¹²

^{IM I} Z

DJ e ^{J δ}

^I

F [M J ]e ˆ

¹²

^{IM I} G[M I]

= e ⁻

¹²

^{IM I} Z

DJ ˆ F [M J ]e

¹²

(J +I)M (J +I) G[M (I + J )]

= e ^{−IM I} Z

DJ e

¹²

^{J M J} F [M J ]e ˆ ^{LM (I+J )}     _L=I

G[M (I + J )]

= e ^{−IM I} Z

DJ e

¹²

^{J M J +J δ}

^I

F [M J ]e ˆ ^{LM I}     _L=I

G[M I]

= e ^{−IM I} F [δ _I ]e ^{LM I} 

 _L=I G[M I] . (B.14)

and so we have proved Eq.(B.9).

References

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[3] H.M. Fried, “Green’s Functions and Ordered Exponentials”, Cambridge Uni- versity Press (2002) .

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2016)