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New Dirac delta function based methods with applications to perturbative expansions in

quantum field theory

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2014 J. Phys. A: Math. Theor. 47 415204

(http://iopscience.iop.org/1751-8121/47/41/415204)

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New Dirac delta function based methodswith applications to perturbativeexpansions in quantum field theory

Achim Kempf1, David M Jackson2 and Alejandro H Morales3

1Departments of Applied Mathematics and Physics, University of Waterloo, OntarioN2L 3G1, Canada2Department of Combinatorics and Optimization, University of Waterloo, Ontario N2L3G1, Canada3 Laboratoire de Combinatoire et dʼInformatique Mathématique (LaCIM) Université duQuébec à Montréal, Canada

E-mail: akempf@uwaterloo.ca

Received 10 April 2014, revised 7 September 2014Accepted for publication 9 September 2014Published 30 September 2014

AbstractWe derive new all-purpose methods that involve the Dirac delta distribution.Some of the new methods use derivatives in the argument of the Dirac delta.We highlight potential avenues for applications to quantum field theory andwe also exhibit a connection to the problem of blurring/deblurring in signalprocessing. We find that blurring, which can be thought of as a result of multi-path evolution, is, in Euclidean quantum field theory without spontaneoussymmetry breaking, the strong coupling dual of the usual small couplingexpansion in terms of the sum over Feynman graphs.

Keywords: Dirac delta, quantum field theory, Fourier transform, perturbativeexpansions, integration, deblurring, Laplace transform

1. A method for generating new representations of the Dirac delta

The Dirac delta distribution, see e.g., [1–3], serves as a useful tool from physics to engi-neering. Our aim here is to develop new all-purpose methods involving the Dirac deltadistribution and to show possible avenues for applications, in particular, to quantum fieldtheory (QFT). We begin by fixing the conventions for the Fourier transform:

g y g x x g x g y y( ) :1

2( ) e d , ( )

1

2( ) e d . (1)xy xyi i∫ ∫

π π= =∼ ∼ −

Journal of Physics A: Mathematical and Theoretical

J. Phys. A: Math. Theor. 47 (2014) 415204 (12pp) doi:10.1088/1751-8113/47/41/415204

1751-8113/14/415204+12$33.00 © 2014 IOP Publishing Ltd Printed in the UK 1

To simplify the notation we denote integration over the real line by the absence of integrationdelimiters. Assume now that g is a suitably well-behaved function. Then

x y( )1

2e d (2)xyi∫δ

π=

g yg y y

1

2

1

( )( ) e d (3)xyi∫π

=

( )gg y y

1

2

1

i

1

2( ) e d (4)

x

xyi∫π π

=− ∂

and therefore:

( )x

gg x( )

1

2

1

i( ). (5)

xδ

π=

− ∂∼

Here, g must be sufficiently well-behaved so that, in particular, g1 has an expansion as apower series, which then gives meaning to g1 ( i )x− ∂ as a series in derivatives.

By this method, each suitable choice of g in equation (5) yields an exact representation ofthe Dirac delta. For example, if we choose g to be a Gaussian

g x g x( ) : e , ( )1

e (6)x x2 22 2

σ= =∼σ σ− −

with 0σ > , then:

x( )1

2e e (7)x 2x2

2 2δπσ

= σ∂ −σ−

( )n

1

2 !e . (8)

n

xn

x

0

22

22∑πσ

=∂σ

σ

=

∞ −−

We obtain approximations of x( )δ by truncating the series of derivatives in equation (8). Forexample, when truncating the series after the first term, we obtain the standard Gaussianapproximation x( ) (2 ) e x1 2 22δ πσ≈ σ− − which converges to x( )δ as 0σ → in the weak limit.

2. A representation of the Fourier transform through the Dirac delta

From equation (5) we obtain a representation of the Fourier transform:

( )g x g x( ) 2 i ( ). (9)xπ δ= − ∂∼

Let us test this representation of the Fourier transform by applying it to a basis of functions,namely the plane waves g x( ) e xwi= . Their Fourier transform must come out to beg y y w( ) 2 ( )π δ= +∼ , i.e., g x x w( ) 2 ( )π δ= +∼ . We have g ( i ) ex

w x− ∂ = ∂ . Therefore,equation (9) indeed holds:

x w x2 ( ) 2 e ( ). (10)w xπ δ π δ+ = ∂

J. Phys. A: Math. Theor. 47 (2014) 415204 A Kempf et al

2

Also, we can use the fact that g x x( ) : ( )δ= implies g x( ) 1 2π=∼ to obtain from equation (9):

( ) xi ( )1

2. (11)xδ δ

π− ∂ =

While equation (9) is an exact representation of the Fourier transform, we obtain potentiallyuseful approximations to the Fourier transform by replacing the Dirac delta in equation (9) byone of its approximations. The derivatives in the power series expansion of g ( i )x− ∂ can thenbe carried out successively. For example, within the path integral formalism of QFT,quantization is a Fourier transformation. In the case of a real scalar field, the generatingfunctional Z J[ ] of Feynman graphs (which we will here call ‘partition function’ even in thenon-Euclidean case) reads, see e.g., [4]:

Z J D[ ] e [ ]. (12)S J xi [ ] i dn∫ ∫ ϕ= ϕ ϕ+

Z J[ ] is, therefore, the Fourier transform of e Si [ ]ϕ . Using equation (9), we obtain

Z J N J[ ] e [ ], (13)S Ji [ i ] δ= δ δ−

where N is a formally infinite normalization constant. Equation (13) invites approximating thepartition function Z J[ ] by using any one of the approximations for the Dirac delta, such as asinc, Lorentzian or Gaussian approximation on the right-hand side (rhs). This then allows oneto evaluate the rhs of equation (13) explicitly, presumably best with the help ofdiagrammatics, similar to the usual use of Feynman rules. Recall that the usual pertubativeexpansion of Z J[ ] is not convergent and is at best asymptotic. Here, the freedom to choose aregularization of J[ ]δ should translate into some freedom to affect the convergence propertiesof the diagrammatics. We will return to this issue.

3. A second representation of Fourier transformation using the Dirac delta

The new representation of the Fourier transform, equation (9), features derivatives in theargument of the function that is to be Fourier transformed, g ( i )x− ∂ , which may be incon-venient in some circumstances. We now give a representation of the Fourier transform whichhas the derivatives in the Dirac delta instead:

( )g y y g x( ) 2 e i ( ). (14)xyx

iπ δ= ∂ −∼

In spite of appearances, the rhs does not depend on x. To prove equation (14), we first useequation (1) and then integrate by parts:

( )

( )

( )y g x

y g z z w

g z y z w

g z w y y z w

g z z g y

2 e i ( )

1

2e e 2 e i ( )d d

1

2( ) i e d d

1

2( ) ( )e d d

1

2( )e d ( ). (15)

xyx

zw wx zyz

zz w y wx

zy

zy

i

i i i

i ( ) i

i

i∫

∬

∬

∬

π δ

ππ δ

πδ

πδ

π

∂ −

= ∂ −

= − ∂ −

= + −

= = ∼

−

+ −

Let us test the representation of the Fourier transformation given by equation (14) by applyingit to the plane waves basis, g x( ) e xwi= . We have to show that equation (14) yields

J. Phys. A: Math. Theor. 47 (2014) 415204 A Kempf et al

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g y y w( ) 2 ( )π δ= +∼ . Indeed:

( )y w y

y w

2 e i e 2 e ( )e

2 ( ). (16)

xyx

xw xy xwi i i iπ δ π δ

π δ

∂ − = − −

= +

Applied to QFT, equation (14) yields:

Z J N J[ ] e (i )e . (17)J x Si d i [ ]n∫ δ δ δϕ= −ϕ ϕ

Any approximation of the Dirac delta which possesses a Taylor series expansion about theorigin can be used in equation (17) to yield another class of Feynman-like diagrammaticmethods for evaluating Z J[ ].

4. Integration in terms of the Dirac delta

Let us now use our two new representations of the Fourier transform to obtain two newrepresentations of integration. To this end, we use the fact that integration is the zero-frequency case of the Fourier transform: g x x g( ) d 2 (0)∫ π= ∼ . From our first representationof the Fourier transform, equation (9), we therefore obtain:

( )g x x g x( ) d 2 i ( ) . (18)xx 0

∫ π δ= − ∂=

From our second representation of the Fourier transform, equation (14), we obtain anotherrepresentation of integration:

( )g x x g x( ) d 2 i ( ). (19)x∫ π δ= ∂

We can also see this directly, using equation (2):

( )g x g x w g x w w g w w2 i ( ) e ( ) d ( ) d ( ) d . (20)xw x∫ ∫ ∫πδ ∂ = = − =− ∂

Applied to the quantum field theoretic path integral, equation (12), the equations (18) and (19)yield:

Z J N[ ] e [ ] (21)S J xi [ i / ] / d

0

n∫ δ ϕ= δ δϕ δ δϕ

ϕ

− +

=

Z J N[ ] [i ]e . (22)S J xi [ ] i dn∫δ δ δϕ= ϕ ϕ+

Here, N is a formally infinite normalization. Note that in equation (22) the dependence on ϕdrops out in the same way that x drops out in equations (14) and (19).

5. Explicit examples, and applications to the Laplace transform

In this section, we show that the new methods indeed add to the toolbox of practical tech-niques. To this end, we give examples for the Fourier transform and for integration over thereal line, over bounded intervals and over semi-bounded intervals. As an example of thelatter, we obtain a new method for evaluating the Laplace transform.

J. Phys. A: Math. Theor. 47 (2014) 415204 A Kempf et al

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5.1. Fourier transform

For a simple application of the new Fourier transform method given in equation (9), let uscalculate the Fourier transform f

∼of f x x( ) cos ( )e x2= − :

f x x y( )1

2cos ( )e e d (23)y yxi2∫

π=∼ −

now using equation (9):

( ) x2 cos i e ( ) (24)( )x

i x2

π δ= − ∂ − − ∂

( ) x2

e e e ( ) (25)x x x2π δ= +∂ −∂ ∂

now expressing δ (x) through equation (7) with σ = 2:

( )1

2 2e e e (26)x 4x x

2= +∂ −∂ −

( )1

2 2e e (27)x x( 1) 4 ( 1) 42 2= +− + − −

x1

2 ee cosh ( 2). (28)x

1 442= −

5.2. Integration

Using equation (18), we can evaluate straightforwardly, for example, the following integral:

( )x

xx e x

sin ( )d 2

1

2ie

1

i( ) (29)

x x 0

x x∫ π δ= −− ∂

∂ −∂

=

( ) x ce e ( ( ) ) (30)x 0x xπ Θ= − +∂ −∂

=

x c x c( ( 1) ( 1) ) (31)x 0π Θ Θ= + + − − − =

. (32)π=Here, Θ is the Heaviside function and c is an integration constant. Similarly, one readilyobtains, e.g., x x xsin ( ) / d 3 /85∫ π= , x x xsin ( ) / d2 2∫ π= , tx x x t(1 cos ( ))/ d | |2∫ π− = or

x x xcos ( )e dx24e

2

1 4∫ = π− .Recall that the relationship between equations (9) and (18) expresses integration as the

zero frequency case of the Fourier transformation. Notice that this implies that when anyintegral is performed in the above way, i.e., through equation (18), then the Fourier transformof the integrand is also immediately obtained, simply by dividing by 2π and by not settingx = 0. For example, equations (29)–(31) immediately also yield for f x x x( ) sin ( )= thatf x x x( ) 2 ( ( 1) ( 1))π Θ Θ= + − −∼

.We remark that while the above integrals may also be integrated by other means, these

tend to require a non-trivial search, such as a search for a suitable contour or for a suitableauxiliary integrating factor. The evaluation of the integrals with the new methods is morestraightforward in the sense that the new methods allow one to evaluate the integralsessentially through the more direct task of the taking of derivatives.

J. Phys. A: Math. Theor. 47 (2014) 415204 A Kempf et al

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5.3. Integrals over semi-bounded intervals

Our integration methods can also be applied to integrals over semi-bounded intervals. Webegin with integrals over the interval [0, )∞ and then extend this to the interval a[ , )∞ wherea is real. From equation (18), we have:

( ) ( )f x x x f x x f x( ) d ( ) ( )d 2 i i ( ) . (33)x xx0 0

∫ ∫ Θ π Θ δ= = − ∂ − ∂∞

−∞

∞

=

But, from equation (9):

( ) x xi ( )1

2( ). (34)xΘ δ

πΘ− ∂ = ∼

Moreover,

x xx

( )2

( )1

2

i, (35)PPΘ π δ

π= +∼

where PP denotes the ‘principal value of’. Combining these three, we conclude that:

( )f x x f xx

( ) d i ( )i

. (36)xx0 0

PP⎜ ⎟⎛⎝

⎞⎠∫ π δ= − ∂ +

∞

=

For integration over the interval a[ , )∞ , where a is real, this then also yields:

( )f x x f x a x f a xx

( )d ( )d i ( )i

. (37)a

xx0 0

PP⎜ ⎟⎛⎝

⎞⎠∫ ∫ π δ= + = − ∂ + +

∞ ∞

=

The following example of an application of equation (36) will now show that the Dirac deltaneeds to be defined also in the upper complex plane. Assume a is real. Then:

x xx

e d e ( )i

(38)ax a

x0

i

0

x PP⎜ ⎟⎛⎝

⎞⎠∫ πδ= +

∞− − ∂

=

aa

(i )1

. (39)⎜ ⎟⎛⎝

⎞⎠π δ= +

Comparing with the known result aπ for Re a( ) 0> and ‘undefined’ for Re a( ) 0< , weconclude that the definition of the Dirac delta needs to be extended to the upper half ofthe complex plane: x( ) 0δ = when Im x( ) 0> , while x( )δ has to remain undefined whenIm x( ) 0< . The definition of x( )δ on the real line remains unchanged, of course.

5.4. Laplace transforms

We now apply the above results to the Laplace transform. Equation (36) yields, for a 0> , thefollowing new representation of the Laplace transform:

( )f x x f xx

e ( )d i e ( )i

(40)axx

a

x0

i

0

x PP⎜ ⎟⎛⎝

⎞⎠∫ πδ= − ∂ +

∞− ∂

=

( )f x ax a

i ( i )i

i. (41)x

x 0

PP⎜ ⎟⎛⎝

⎞⎠πδ= − ∂ + +

+ =

J. Phys. A: Math. Theor. 47 (2014) 415204 A Kempf et al

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Let us test the new representation of the Laplace transform given in equation (41) by applyingit to the basis of monomial functions xn, where n is any non-negative integer:

( )x x x ax a

x a n x a

e d i ( i )i

i

( i) ( i ) i( i) ( 1) !( i ) ,

ax nx

n

x

n n

x

n n nx

0 0

( )

0

10

PP⎜ ⎟⎛⎝

⎞⎠∫ πδ

π δ

= − ∂ + ++

= − + + − − +

∞−

=

=

− −=

where x x( ) : ( )nxn( )δ δ= ∂ . Recalling that x( )δ as well as its derivatives vanish when Im x( ) 0,>

yields indeed:

x x n ae d ! . (42)ax n n

0

1∫ =∞

− − −

We now use equation (41) to evaluate an example of a more non-trivial Laplace transform:

( )

( )

( ) ( )

( )

wx

xx

x ax a

c x a

a w a w

a w

a w

w

a

sin ( )e d

1

2ie e

1

i

( i )i

i

i

2e e ( log ( i ))

i

2(log (i ) log (i ))

i

2log

i

itan .

ax w w

x

x

w w

x

0

i i i i

0

0

1

x x

x x

⎜ ⎟

⎜ ⎟ ⎜ ⎟

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎛⎝

⎞⎠

∫

πδ

= −− ∂

+ ++

= − + +

= + − −

= +−

=

∞− − ∂ − − ∂

=

∂ −∂

=

−

5.5. Integrals over a bounded interval

Finally, let us consider examples of integrals over a finite interval a b[ , ] of the real line. Tothis end, we notice that, from equations (9) and (18), integrals over a function f, weighted by afunction g, can be written in the following form:

( ) ( )f x g x x f g x( ) ( )d 2 i i ( ) , (43)x x x 0∫ π δ= − ∂ − ∂=

( )f g x2 i ( ) . (44)xx 0

π= − ∂ ∼=

Let us choose for g the characteristic function of the interval a b[ , ]

g xx a bx a b

( ) :1 if [ , ],0 if [ , ],

(45)⎧⎨⎩= ∈

∈

which has the Fourier transform:

( )g xx

( )1

2 ie e . (46)bx axi i

π= −∼

J. Phys. A: Math. Theor. 47 (2014) 415204 A Kempf et al

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Thus, equation (44) yields this new representation of integration over a finite interval:

( )( )f x x fx

( )d i1

ie e . (47)

a

b

xbx ax

x

i i

0∫ = − ∂ −

=

Let us test it by applying it to the basis of monomial functions xn. Indeed, equation (47)yields:

( )( )

( )

x xx

b x

k

a x

k

b a

n

d i i1

e e

i i(i )

!

(i )

!

1.

a

bn

xn bx ax

x

xn

k

n k k k k

x

n n

i i

0

1

1 1 1

0

1 1

⎛⎝⎜

⎞⎠⎟

∫

∑

= − − ∂ −

= − − ∂ −

= −+

=

=

+ − −

=+ +

Notice that, for example, Fourier series coefficients x xe da

b n xyi∫ then also immediately follow.Notice also that a change of variables in equation (47) yields for the anti-derivative:

f x x f y c( )d ( )(e 1) . (48)x

yxy

y 0∫ ′ ′ = ∂ − +

=

6. Blurring and deblurring using the Dirac delta

For a concrete application of the new methods let us now apply one of our formulas above,equation (5), in the context of an important problem in scientific [5] and engineering signalprocessing [6], namely to the problem of deblurring signals such as images. The process ofblurring is the mapping of a signal, f, into a blurred signal, fB, through a convolution. In onedimension:

f y f x g x y x( ) : ( ) ( ) d . (49)B ∫= −

For example, in the case of so-called Gaussian blurring, the blurring kernel reads:

g xa

( ) :2

e . (50)xGauss

a2

2

π= −

While analytic Fourier methods for deblurring are well known in the signal processingcommunity, we can use the above results to describe generic deblurring in a new and verycompact way. Namely, our claim is that if g is such that the blurring by g is invertible, thenthe deblurring can be implemented through an operator Dg which can be written asD g: 1 ( 2 (i ))yg π= ∂∼ , i.e.,:

( )D f y

gf y f y( )

1

2 i( ) ( ). (51)

yg B Bπ

=∂

=∼

Here, we assume that g1 ∼ possesses a power series expansion about the origin. To proveequation (51), we use equation (5):

J. Phys. A: Math. Theor. 47 (2014) 415204 A Kempf et al

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( ) ( )( )

( )

gf y f x

gg x y x

z x y x z

f y zg

g z z

f y z z z f y

1

2 i( ) ( )

1

2 ˜ i( ) d

use: : , d d ,

( )1

2 i( ) d

( ) ( ) d ( ). (52)

y y

y z

z

B ∫

∫

∫

π π

π

δ

∂=

∂−

= − = ∂ = −∂

= +− ∂

= + =

∼

∼

For example, in the case of the Gaussian blurring kernel of equation (50), we readily find ourdeblurring operator:

Dn a

e1

! 2. (53)

n

yn

Gauss

0

2

a y1

22

⎛⎝⎜⎜

⎞⎠⎟⎟∑= =

−∂− ∂

=

∞

Let us recall that the exponentiation of a single derivative, ea x∂ , has a simple interpretation as atranslation operator, f x f x ae ( ) ( )a x = +∂ . Equation (53) shows that the operator obtained byexponentiating the second derivative possesses an interpretation as a deblurring operator. Letus consider applying the series expansion on the rhs of equation (53) term by term on ablurred function fB(y). In this way, we obtain a series expansion of the deblurring operation.Intuitively, a blurred function is one in which fine details have been suppressed. The lowestorder term, 1, reproduces the blurred image. The higher order terms add derivatives of thefunction with suitable positive and negative prefactors. The derivatives enhance fine details ofthe function. Eventually, by adding more and more terms of the series of derivatives inequation (53), the original image is restored. While this picture is accurate, it must be usedwith caution because simply adding derivatives would not suffice to sharpen an image. Theparticular coefficients and, in particular, the minus sign in equation (53) are providing crucialcancellations in the series. To see this, notice that the inverse of the deblurring operator,DGauss

1− , which is the same as DGauss except for the minus sign, is blurring rather thandeblurring. Indeed, the special case of Gaussian blurring is also the case of the heat equationwith the blurring operator being the heat kernel. Heat kernel methods, see e.g., [7], then alsolink up with spectral methods, see e.g., [8].

7. A connection between deblurring and Euclidean QFT

We will now show that Euclidean quantum field theoretical path integrals naturally containGaussian deblurring operators. These deblurring operators possess a series expansion whichthen yields a series expansion of the partition function Z J[ ]. To see this, let us consider theexample of Euclidean ϕ4 theory with m 02 > and 0λ > , i.e., without spontaneous symmetrybreaking:

Z J D[ ] e [ ]. (54)x x m x x J x xd ( )( ) ( ) ( ) ( ) ( )n 12

2 4∫ ∫ ϕ= ϕ Δ ϕ λϕ ϕ− − + − +

The deblurring operator originates in the kinetic term of the action: we replace theoccurrences of x( )ϕ in the quadratic term in the action by J x( )δ δ and we move theexponentiated quadratic term in front of the path integral, to obtain:

J. Phys. A: Math. Theor. 47 (2014) 415204 A Kempf et al

9

( )Z J

D

[ ] e

e [ ]. (55)

x m

x x J x x

d

d ( ) ( ) ( )

nJ x J x

n

12 ( )

2( )

4∫∫

∫ ϕ

=

×

Δ

λϕ ϕ

− − +

− +

δδ

δδ

Since the Laplacian, Δ− , is a positive operator, the term before the path integral is a Gaussiandeblurring operator and the resulting expansion is convergent. The deblurring operator isdiagonal in the momentum representation and deblurs every mode J(p) by a different amount,given by the scale p m2 2+ . Notice that, in the case of QFT on Minkowski space, the extraimaginary unit in the exponent of the term before the path integral would not allow one tointerpret this term as a deblurring operator. We will discuss the related issue of Wick rotationfurther below. What, then is the physical meaning of this deblurring expansion in EuclideanQFT? It is the strong coupling expansion which was first introduced in [9]. To see this, wechange variables in equation (55) from J x( )i to R x J x( ) : ( )i i

1 4λ= − and carry out theremaining hypergeometric integrals to obtain

( )Z R c e

e , (56)

xR x

mR x

x P R x

1 4 d( ) ( )

d log( ( ( )))

n

n

12

2⎡⎣ ⎤⎦ ∫

∫

λ =

× ′

δδ Δ δ

δ− +

′

λ

where we recognize the expansion in powers of1 λ . Here, c is a constant that is independentof R and

P r r F r

F r

( )1

43

4

( (3 4))5

4,

3

2,

1

256

2 21

2,

3

4,

1

256, (57)

2 20 2

4

0 24

⎜ ⎟⎜ ⎟

⎜ ⎟

⎛⎝

⎞⎠

⎛⎝⎜

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎞⎠⎟

ΓΓ

π

=

+

where F0 2 is a generalized hypergeometric function. P(r) can easily be Taylor expanded, sothat the perturbative expansion in 1 λ in equation (56) can be carried out.

In contrast, in order to derive the Feynman rules for the small coupling expansion,starting from equation (54), one normally replaces occurrences of x( )ϕ in the interaction termby J x( )δ δ and then moves the interaction term in front of the path integral:

( )Z J

D

[ ] e

e [ ]. (58)

x J x x

x x m x x J x x

d ( ( )) ( )

d ( ) ( ) d ( ) ( )

n

n n

4

12

2∫∫

∫ ∫ ϕ

=

×

λ δ δ

ϕ Δ ϕ ϕ

−

− − + +

As is well known, see e.g., [11, 12], this step is analytically not justified. Indeed, since theEuclidean path integral is of the form

z j x( ) e d (59)ax x jx2 4∫= λ− − +

any expansion in λ about 0λ = must have a vanishing radius of convergence. This is becausez(j) diverges for all negative λ. As a consequence, equation (59) yields a perturbativeexpansion in λ which is divergent. The reason why the small coupling expansion isnevertheless useful is of course that the expansion happens to be asymptotic, i.e., itapproaches the correct value to some extent before diverging.

Finally, let us recall that the weak and the strong coupling expansions in Euclidean andMinkowski space QFT are obtained by expressing either the interaction terms or the kineticterms of the action through functional differentiations respectively. Equation (22), which we

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derived in Minkowski space, opens up a new possibility, in which neither part of the action isexpressed in terms of functional derivatives. Instead, a leading Dirac delta is expressed interms of functional derivatives. In principle, the Dirac delta in equation (22) can be replacedby any one of its representations, e.g., in terms of sequences of Gaussians or sinc functions.The resulting expansion of Z[J] should be neither a weak nor a strong coupling expansion andits properties should therefore be very interesting to determine.

8. Summary and outlook

We obtained new representations of the Dirac delta in equation (5), of the Fourier transform inequations (9), (14), of integration in equations (18), (19) and of the Laplace transform inequation (41). Notice that, since our new methods are meant to serve as all-purpose methods,we intentionally did not use Wick rotation in their derivation (or anywhere else), because thatwould have required restriction to special cases in which poles are absent in certain regions ofthe complex plane. In particular, we did not use Wick rotation in our derivation of the newrepresentation of the Laplace transform in equation (41).

The new all-purpose methods should find applications in a range of fields from physics toengineering. In particular, the new methods could be useful in practical calculations of Fouriertransforms, Laplace transforms and integrals in general, as we showed in section 5 by givingexplicit examples. An advantage of the new methods is that they do not involve a search for asuitable contour or a suitable integrating factor. Instead, the new methods essentially rely ondifferentiation, which is more straightforward.

The new methods could also be useful in formal calculations, by enabling one to deriveand represent relationships in a succinct new way. Further, the Dirac deltas in the newrepresentations of Fourier and Laplace transforms and of integration may sometimes beusefully replaced by approximations of the Dirac delta. The new methods therefore offernumerous new ways in which integrals and Fourier and Laplace transforms, such as thoseoccurring in QFT, can be regulated or approximated.

In particular, with equations (13), (17), (21), (22) we obtained new ways to representquantum field theoretical partition functions. More representations of partitions functions canbe derived with the new methods, for example, by using the new representation of the Laplacetransform, equation (41). Notice that our new representations of partition functions offer aformal alternative to using path integration, since they involve only functional differentiationsand the Dirac delta. The new techniques invite application in formal calculations, such asderivations of Dyson–Schwinger equations and Ward or Slavnov–Taylor identities. This thenalso leads, for example, to the question of how the effects of a non-trivial path integralmeasure will manifest themselves within the new integral-free approach. In particular, whenthe partition function is expressed as a path integral, anomalies arise from a non-trivialtransformation behavior of the path integralʼs measure, as Fujukawaʼs method shows [13].How do anomalies arise in the new representations of the partition function that do notinvolve path integration? Further, the new representations of the partition functions invite theexploration of the resulting perturbative expansions of the partition functions. This includes,in particular, also those expansions that arise when replacing the Dirac delta by a regularfunction that approximates the Dirac delta and that possesses a power series expansion. Ofinterest in this context are then, for example, the convergence properties and the physicalinterpretation of these expansions of Z J[ ].

In order to illustrate that the new methods possess concrete applications, we then appliedequation (5) in section 6 to obtain a compact and transparent expression, namely

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equation (51), for a generic deblurring operator for signals. We also showed that deblurringoperators naturally occur in Euclidean QFT. In particular, we identified the large couplingexpansion as a deblurring expansion. Blurring, being a convolution, can be viewed as a sumover ‘paths’. In a sense, therefore, in QFT the sum over paths that occurs in the phenomenonof (Gaussian) blurring is the strong coupling version of the small coupling sum over ‘paths’given by the the usual sum over Feynman graphs.

A very interesting problem is the development of the functional analytic and distributiontheoretic underpinnings of the new uses of the Dirac delta, in particular, when the latter hasderivatives in its argument, and also the study of the algebraic and combinatorial ramificationsfor operations in rings of formal power series. For the necessary combinatorial background,see e.g., [10].

Acknowledgements

AK and DMJ acknowledge support from NSERCʼs Discovery program. AHM was supportedby a CRM-ISM postdoctoral fellowship.

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