A conducting surface in Lee-Wick electrodynamics

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Web End = A conducting surface in LeeWick electrodynamics

F. A. Barone1,a, A. A. Nogueira2,b

1 IFQ-Universidade Federal de Itajub, Av. BPS 1303, Pinheirinho, Caixa Postal 50, Itajub, MG 37500-903, Brazil

2 IFT-Rua Dr. Bento Teobaldo Ferraz, 271, Bairro: Barra-Funda, So Paulo, SP 01140-070, Brazil Received: 29 October 2014 / Accepted: 6 July 2015 / Published online: 22 July 2015 The Author(s) 2015. This article is published with open access at Springerlink.com

Abstract LeeWick electrodynamics in the vicinity of a conducting plate is investigated. The propagator for the gauge eld is calculated and the interaction between the plate and a point-like electric charge is computed. The boundary condition imposed on the vector eld is taken to be the one that makes, on the plate, the normal component of the dual eld strength to the plate vanish. It is shown that the image method is not valid in LeeWick electrodynamics.

1 Introduction

The simplest higher-order derivative gauge theory is the so called LeeWick electrodynamics [16], which is described by the Maxwell Lagrangian augmented by a higher-order derivative kinetic term. Since its proposal, the theory has been standing out by its classical as well as its quantum aspects, as it exhibits many interesting peculiarities. We can mention, for instance, the fact that in this electrodynamics the self-energy of a point charge is nite in 3 + 1 dimen

sions [712], a Dirac string can produce a magnetic eld [7], it stem a nite theory closely related to the PauliVillars regularization scheme [5,8,1318] where the divergences of the quantum electrodynamics are controllable [1921] and it exhibits classical dynamical stability [22]. These features have made the model a widely studied subject in a variety of scenarios, mainly regarding its extension to the Standard Model [14,16,18,2337].

In spite of all this interest, as far as the authors know, there is a lack in the literature regarding LeeWick electro-dynamics under the inuence of boundary conditions. That is a remarkable subject in any abelian gauge theory, since the experimental apparatuses commonly used to test electromagnetic phenomena are, usually, surrounded by conductors. In addition, it is also important for the investigation of situa

a e-mail: mailto:[email protected]

Web End [email protected]

b e-mail: mailto:[email protected]

Web End [email protected]

tions where it is possible to nd deviations from expected physical results in comparison to Maxwell electrodynamics. The presence of conductors can create suitable scenarios for this kind of search.

In Ref. [38] the presence of conducting surfaces was investigated in the context of the Casimir effect for LeeWick electrodynamics. In the work of Ref. [39], among other terms, a LeeWick type contribution (called the Uehling term) was inserted to compose an effective theory in order to calculate radiative corrections to the Casimir effect. In this paper we highlight that the real role of a conductor in this theory is not such a common issue and requires more cautious attention.

As is well known, this electrodynamics exhibits two modes, one massive and the other one massless. The propagator can then be split up into the sum of two parts, the rst one being just the usual Maxwell propagator and the second one the Proca propagator with an overall minus sign. Despite the presence of these ghosts modes, as is the case of PauliVillars regulators, the theory can be rendered unitary provided that the LeeWick particles decay. This fact means that the physical effects at tree level are trivial in most cases.

In this paper we show that this is not the case when the presence of a conducting plate is taken into account. The correction term that has to be added to the propagator in this case is not a simple subtraction of the corresponding Maxwell and Proca propagators coupled to the conductor. Due to this fact, we show that the physical phenomena are no longer as trivial as in the theory without the conducting surfaces.

Specically, in Sect. 2 we compute the propagator for the LeeWick gauge eld in the presence of a conducting plane. We employ quantum eld theory methods in order to obtain the functional generator for the gauge sector, since with this functional one can nd any physical quantity of interest of the theory. In Sect. 3 we calculate the interaction between a perfectly conducting plate and a point-like charge. We also compare the results with the standard Maxwell electrodynamics and show that in the LeeWick model the image method is no longer valid. In Sect. 4 we make a discussion regarding

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the two eld formalism of LeeWick theory in the presence of a conducting plate. That gives an alternative way to understand why the image method is not valid in LeeWick theory.Section 5 is devoted to our nal comments.

2 LeeWick propagator in the presence of a conducting plate

In this paper we work in a 3 + 1 dimensional Minkowski

spacetime with metric (+, , , ). The vector gauge eld

is designated by A and its corresponding eld strength by F = A A.

The electromagnetic sector of the so called LeeWick electrodynamics is described by the Lagrangian density [1 5,7]

LLW

14 F F

14m2 F F

( A)2

2 J A, (1)

where m is a parameter that has mass dimension, J is an external source, and is a gauge xing parameter. It is important to mention that there are other covariant gauge conditions for this theory [40,41].

As discussed in many works [15,7], the Lagrangian (1) exhibits gauge invariance and two distinct poles, in momenta space, for the corresponding propagator; a massless one and a massive one. This fact can be shown by considering that, in the Feynman gauge where = 1, the model (1) is equivalent

LLW

2 A 1 +

m2 A,

(2)

so that the corresponding Feynman propagator is given by

D(x, y) =

d4 p

(2)4

= 0 (5) where the sub-index indicates that the boundary conditions are taken on the plane x3 = a.

We shall compute the functional generator for the vector eld submitted to the boundary conditions (5) following the procedure proposed in Ref. [42], that is, carrying out the functional integral

ZC[J] = DAC ei

d4x(LLWJ A) (6)

where the sub-index C means that the integral is restricted only to eld congurations which satisfy the conditions (5). This restriction is achieved with the insertion of a delta functional that is non-vanishing only for eld congurations that satisfy the conditions (5) and integrating in all eld congu-rations, that is,

ZC[J] = DA[F3(x)|x

3 =a

1p2 m2

1 p2

p p

eip(xy) (3)

in the sense that

1 + m2

m2 D(x, y)

= 4(x y), (4)

where it is implicit, from now on, that there is a small imaginary part for the momentum square, p2 p2 + i.

Maxwell and LeeWick electrodynamics yield different dynamical equations for the gauge eld, but the dynamical equations for the charged particles and the Lorentz force are the same in both theories, namely, dppdt = qE + qv B,

where q and pp are, respectively, the charge and the spatial component of the particle momentum.

In this section we consider general aspects of LeeWick electrodynamics in the presence of a conducting plate. This is a non-trivial task since its inception, because we have to establish what is a conductor in this model. To answer this question we resort to the behavior of the electromagnetic eld in the presence of a conductor, according to the Maxwell theory.

A conducting surface in the Maxwell electrodynamics imposes a boundary condition on the gauge eld in such a way that the Lorentz force on the surface vanishes. It is achieved by taking as zero the component of the dual eld strength, normal to the surface. That is, if n is the normal four-vector to the conducting surface, we must have nF = 0 along it, where F = (1/2) F is

the dual to the eld strength, with standing for the Levi-Civita tensor ( 0123 = 1). Once in LeeWick electro-

dynamics the Lorentz force is the same as in Maxwell theory, the condition which makes the Lorentz force vanish in both theories must be exactly the same.

Here we consider the presence of a single perfectly conducting plate. Without loss of generality we take a coordinate system where the plate is perpendicular to the x3 axis, lying on the plane x3 = a, so its normal four-vector is

n = 3 = (0, 0, 0, 1) and the boundary condition for the

gauge eld A reads

F3(x)|x

d4x(LLWJ A). (7)

Now we use the functional Fourier representation

[F3(x)|x

3 =a

] ei

= DB exp i d4x(x3 a)B(x )F 3(x) (8)

3 =a

]

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Eur. Phys. J. C (2015) 75 :339 Page 3 of 11 339

where B(x ) is an auxiliary vector eld and the notation

x = (x0, x1, x2) is used for the coordinates parallel to the

plate.The auxiliary eld B(x) exhibits gauge symmetry,

B(x ) B(x ) + (k)(x ), (9)

which requires a cautious treatment of the integral (8). In the appendix we show that

[F3(x)|x

which can be calculated using the standard methods of quantum eld theory [7], and Z[J] is a contribution that does not

involve A,

Z[J] = DB exp i d4x(x3 a)I(x)B(x )

exp

d4xd4y(x3 a)(y3 a)B(x )B(y )

2 3 3( D(x, y)) +

2Q(x, y)

x y

(15)

3 =a

]

= N DB exp i d4x(x3 a)A(x) 3 B(x)

exp

i 2

d4xd4y(x3 a)B(x )

where we dened

I(x) = d4y 3

(y3 a)B(y ) , (10)

where is a gauge xing term, N is a constant which does not depend on the elds, and Q(x, y) is an arbitrary function that shall be chosen conveniently.

Substituting (10) into (7) we get

ZC[J] = N DADB ei

d4x(LLWJ A)

exp

2Q(x, y)

x y

x D(x, y) J(y) (16)

Notice that the integral (15) is Gaussian, so that it can be calculated exactly. For this task it is convenient to make the following choice:

Q(x, y) =

d4 p

(2)4

1p2 m2

1 p2

i d4x(x3 a)A(x) 3 B(x)

exp

eik(xy), (17)

and work in the gauge where = 1. Substituting (17) and

(16) into (15), using the fact that

d p3 2

1p2 m2

eip3x3 =

i2 ei |x3|

i 2

d4xd4y (x3 a)B(x )

d p3 2

1p2 eip3x3 =

i2L ei L|x3|a (18)

where =

(y3 a)B(y ) . (11)

In the rst exponential we have only the A eld and in the third one only the presence of B. The second exponential contains a coupling between A and B.

In order to decouple the elds A and B we perform the translation

A(x) A(x)

+ d4yD(x, y)(x3 a) 3 B(x), (12)

what brings the integral (11) into the form

ZC[J] = N ZLW[J] Z[J] (13) where ZLW[J] is the standard LeeWick functional generator

ZLW[J] = DA ei

d4x(LLWJ A)

= ZLW[0] exp

2Q(x, y)

x y

p2 m2 and L =

p2 (see the appendix),

dening the parallel momentum to the plate p =

(p0, p1, p2, 0) and the parallel metric

= 33 (19)

and carrying out some manipulations, one can write Eq. (15) in the form

Z[J] = Z[0] exp

i 2

d4xd4y J(x) D(x, y)J(y)

(20)

where we dened the function

D(x, y) =

d3 p (2)3

i 2

p p

1 L

exp[ip (x y )]

1L ei L|x3a|

1 ei |x3a|

i 2

d4xd4y J(x)D(x, y)J(y) ,

(14)

1L ei L|y3a|

1 ei |y3a| . (21)

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Substituting (14) and (20) into (13) we have the functional generator of the LeeWick gauge eld in the presence of a conducting plate

Z[J]C = Z[0] exp

i 2

d4xd4y J(x)(D(x, y)

+ D(x, y))J(y) . (22)

Notice that, from the above expression (22), one can identify the propagator of the theory in the presence of a conducting plate as

DC(x, y) = D(x, y) + D(x, y). (23)

As a matter of checking we point out that the gauge eld propagator under the boundary conditions (23) is really a Green function for the problem, in the sense that it is the inverse of the LeeWick operator, obtained from Eq. (2), that is,

1 + m2 m2 DC(x, y)

= 4(x y). (24) The above equation can be veried directly by using Eqs. (23), (4), and (21).

Moreover, any eld conguration obtained from (23) satises the boundary condition (5). It can be checked by calculating the eld generated by an arbitrary source

A(x) = d4yDC(x, y)J(y). (25) In terms of the gauge eld A, the boundary conditions (5)

read 3 A(x)|x

d4x d4y J(x) D(x, y)J(y) (27) In our case we take the source corresponding to a point-like steady charge placed at position b.

J(x) = q03(x b). (28)

Substituting (28) into (27), using the Fourier representation (23), carrying out the integrals in d3x, d3y, dx0, dk0 and dy0 and making some simple manipulations we obtain,

EPC =

q2 4

d2p (2)2

p2 + m2 p2

exp (

p2 R)

exp (

p2 + m2 R)

p2 + m2

3 0 = 0, so, by using (25) it can be

shown that the propagator must satisfy

3 DC(x, y)x

x3=0 =

, (29)

where we dened R = |a b|, which stands for the distance

between the plate and the charge. The sub-index PC means the interaction energy between the plate and a charge.

Expression (29) can be simplied by using polar coordinates, integrating out in the solid angle and performing the change of integration variable p = |p |/m,

EPC =

q223 m

0. (26)

With the aid of Eqs. (23), (4), and (21) it can be shown that the condition (26) is really satised.

At this point some comments are in order. The propagator (23) is composed by the sum of the free LeeWick propagator (3) with the correction (21) which accounts for the presence of the conducting plate. As exposed in the appendix, in the limit when m = the propagator (23) reduces to the same

one as that found with Maxwell electrodynamics in the presence of a conducting plate.

The free LeeWick propagator (3) is made up by the Maxwell propagator minus the Proca one. This fact makes most results of theory to be the composition of the corresponding ones obtained in the Maxwell and in the Proca theory. When the boundary condition (5) is involved this is no longer valid and the propagator (23) is not compound by the corresponding ones of the Maxwell and Proca theories with boundary conditions. This result suggests that the boundary

conditions mix up the modes of the LeeWick eld with and without mass (the photon and its LeeWick partner). Due to this fact, some physical phenomena of LeeWick electrodynamics in the vicinity of a conducting plate are not trivial and deserve investigation.

3 Particleplate interaction

In this section we consider the interaction between a point-like charge and a conducting plane. By using the arguments discussed in references [7,4345], we can show that the interaction energy between a conducting surface and an external source J(x) for the gauge eld is given by the integral

E =

0 d p p2[(p2 + 1) + p(p2 + 1)1/2]

e2pmR

p2 2

epmRep2+1m R

p(p2 + 1)1/2 +

e2p2+1m R

p2 + 1

(30)

Each contribution in the integral (30) can be calculated exactly. For the rst contribution we have

0 d p [(p2 + 1) + p(p2 + 1)1/2] exp (2pm R)

1 + 2(m R)2

4(m R)3 +

4m R [Y0(2m R) SH0(2m R)]

4(m R)2 [SH1(2m R) Y1(2m R)], (31)

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Eur. Phys. J. C (2015) 75 :339 Page 5 of 11 339

where Y and SH stand for the Bessel function of the second kind and the Struve function, respectively.

For the third contribution to (30) we carry out the change in the integration variable u =

p2 + 1, as follows:

large quantity, for not so small distances R, this term is much smaller than the Coulombian one.

The interacting force between the plate and the charge is then given by

FPC =

EPC

0 d p p2[1 + p(p2 + 1)1/2] exp 2 p2 + 1m R

1 du[u(u2 1)1/2 + u2 1] exp (2um R)

q2 4

exp (2m R)

4(m R)3 (1 + 2m R),(32)

where K stands for the Bessel function.

The second contribution to (30) is calculated with the change of variable u = p + (p2 + 1)1/2,

1(2R)2 [1 + (m R) m R (m R)], (37)

where the prime denotes derivative of with respect to its argument. Notice that the force (37) is the usual Coulombian interaction between the charge and its image, placed a distance 2R apart, and a correction term m-dependent. The term inside brackets on the right hand side of Eq. (37) is always positive, so the expression (37) is always negative, which means that the force is attractive.

Let us compare the force (37) with the one obtained in LeeWick electrodynamics in 3+1 dimensions for the inter

action between two opposite charges, q and q, placed at a

distance 2R apart,

FCC =

q2 4

K0(2m R)2m R +

K1(2m R)

2(m R)2 +

0 d p 2p[(p2 + 1)1/2 + p] exp (p + p2 + 1)m R

= 2

1 du

u4 1

4u2 exp (um R)

= exp (m R)

2m R

1 (m R)2

1 (m R)3

1(2R)2 [1 exp(2m R) 2m R exp(2m R)].

(38)

Equation (38) can be obtained with the results of reference [7]. Specically, from Eq. (16) of reference [7] one can write the interaction energy between two opposite charges (q and

q). So that, taking the gradient of this energy (with an overall minus sign) with respect to the distance of the charges and setting it equal to 2R, we are taken to the result (38).

In the limit where R 0, both forces (37) and (38) are

nite, namely,

lim

FPC =

2(m R)Ei(1, m R) (33)

where Ei is the exponential integral function,

Ei(1, x) =

1 du

eux

x , x > 0. (34)

Substituting (31), (32), and (33) in (30) we have the interaction energy between a perfect conductor and a point-like steady charge,

EPC =

q2 24

q2 16

32m2, lim

FCC =

1R (1 + (m R)) (35)

where we dened the function,

(m R) = m R

K0(2m R) m R +

K1(2m R) (m R)2

q216 2m2. (39)

It is interesting to notice that the image method is not valid for LeeWick electrodynamics for the conducting plate condition (5). This fact can be seen from the interacting force between the plate and the charge (37). The deviation from the image method behavior can be seen from the difference between (37) and (38) normalized by the Coulombian force, for convenience,

(m R) = |

exp (2m R)

2(m R)3 +

exp (2m R)

(m R)2 (m R)Ei(1, m R)

2(m R)3 +

2m R [Y0(2m R) SH0(2m R)]

FPC| |FCC| [q2/(4)] [1/(2R)2]

. (40)

The results of our numerical analysis suggest that the behavior of (m R) exhibits the same shape as the one observed in Fig. 1 up to m R , with going to zero

monotonically, with negative values, as m R .

In the limit m R we have 0. In the interval

0 < m R < 0.85 we have > 0 and the modulus of

plate-charge interaction intensity is greater than the modulus of chargeimage interaction. For m R > 0.85 < 0, and

2(m R)2 [SH1(2m R) Y1(2m R)]

+exp (m R) 1

1 m R

. (36)

The result (35) is exact, but hard to be interpreted. The rst term on the right hand side is the plate-charge interaction obtained in Maxwell electrodynamics, where the image method is valid, and does not involve the mass parameter m. The second term falls when m R increases. Once m is a

2 (m R)2

2 (m R)3

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339 Page 6 of 11 Eur. Phys. J. C (2015) 75 :339

Fig. 1 Plot for (m R)

the chargeimage interaction dominates, in modulus. In most situations the modulus of the force between the plate and the charge is lower than the modulus of the force between the charge and its image, as one can see from Fig. 1. It is also interesting to notice that the discrepancy between the plate-charge interaction force and the chargeimage interaction exhibits a maximum around m R

= 0, 48, a minimum around

m R

= 3.85, and a zero around m R

= 0.85.

Once m must be a large quantity, only for very small values of the distance R the force between the plate and the charge would have a stronger intensity than it would be if the image method were valid. So, it is relevant to consider if the maxima, minima, and zero of the function can be really measured. Let us focus on the zero of , which occurs at R0

= 0.85/m. In recent work [4648] there are some

estimates for lower bounds of LeeWick mass. The lowest estimates give, in order of magnitude, m 10 GeV. Work

ing in the mks system, we can estimate in order of magnitude R0

= 1016, which is much smaller than the Compton wave

length C

= 1012. In this scale many other effects might be

taken into account, as the vacuum polarization, nite conductivity, rugosity, nite thickness of the plate, and many others. So the behavior of the function next to the plate observed in Fig. 1 is not feasible to reproduce in a laboratory.

In addition, for experimental setups, the distance scale relevant to investigate the interaction between classical objects are of micrometer order. So, taking the estimate of m

10 GeV, we have m R

= 1011 in mks, which is in the range

where the function goes to zero with negative values.The estimate of m 10 GeV is among the lowest ones.

Other estimates [48] suggest a lower bound for m in order of TeV, which makes the phenomena related to the Lee

Wick theory in the presence of a conducting plate even more negligible.

4 The two eld formalism

We showed that when we consider the LeeWick eld in the presence of a conducting surface, the gauge eld must be

submitted to the boundary conditions (5) and the corresponding propagator (23) cannot be decomposed as the Maxwell propagator minus the Proca one, each one submitted to the conditions (5) separately. As an immediate consequence, the image method is not valid for the LeeWick eld.

In order to understand why this decomposition is no longer valid in the presence of a conducting plate, let us review how we can describe the LeeWick Lagrangian (1) without boundary conditions (without the presence of conductor) as the Maxwell Lagrangian minus the Proca one. Following a path integral approach, we can establish LeeWick theory by a functional generator with an alternative Lagrangian of two coupled elds, A and S, as follows:

Z2[J] = DADS exp i d4x

A S

4 A A

4 SS

2 AS J A . (41)

It is important to notice that the external source couples strictly to the A eld, and not to the S eld.

Integrating out the functional (41) on S and preforming some manipulations, we are taken to

Z2[J] = ZS[0]ZLW[J], (42) where ZLW[J] is the standard functional generator for the

LeeWick theory dened in (14) and

ZS[0] = DS exp

i 2

4 SS (43)

is the generating functional for S without any external source. Once ZS[0] does not depend on the external source,

it can be absorbed into a renormalization overall multiplicative constant and has no physical effect. So, from Eq. (42), we can write

Z2[J] = ZLW[J]. (44) It proves that without the presence of a conducting plate the two eld functional (41) is completely equivalent to (14), the functional generator obtained from the LeeWick Lagrangian (1) with a single eld.

Now we perform, in the functional generator (41), the following change in the integrating eld variables:

A = U + V , S = U V

U =

2(A + S), V =

d4x

2(A S). (45)

The Jacobian of the transformation (45) is a (divergent) constant which does not depend on the elds, thus it does not affect the functional generator because its contribution can be absorbed into an overall multiplicative constant. So

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Eur. Phys. J. C (2015) 75 :339 Page 7 of 11 339

DADS

= DUDV and the functional (41), after some manip

ulations, reads

Z2[J] = DU exp i d4x

14UU JU

DV exp i d4x

14 VV

2 V V JV ,

(46)

where we dened

U = U U, V = V V . (47)

Notice that (46) can be decomposed as the product of a Maxwell functional generator for the U eld with a Proca functional generator with an overall minus sign in the Lagrangian for the V eld. The external source J couples to both elds, U and V . It proves that (41), and so (14), is completely equivalent to a Maxwell functional generator minus a Proca functional generator with an overall minus sign in the Lagrangian.

Now, let us consider the two eld theory (41) in the presence of a conducting plate. The Lorentz force must vanish on the plate, as discussed in Sect. 2. Once only the A eld couples to the external source J, the conditions which make the Lorentz force on the plate vanish is given by Eq. (5); it is imposed only on the A eld. In this case we have to restrict the functional integral over eld congurations which satisfy (5). This constraint over the eld conguration can be attained by inserting the delta functional (8) in the integral (41), in the same manner as we have done in Sect. 2,

Z2,C[J] = DADS [F3(x)|x

d3x d3y B(x )

2Q(x, y)

x y

x3=a

y3=a

B(y )

exp

d4x

14 VV

2 V V JV

exp

14UU JU , (51)

where Q(x, y) is an arbitrary function we will choose conveniently and is a gauge parameter.

From now on, we shall work in the gauge where = 1.

Before we start to solve the functional integral (51), it is important to notice that it exhibits just one single auxiliary eld B coupled to both elds, U and V .

The rst and second exponentials in (51) couple, respectively, the elds U and V to the auxiliary eld B. We can

decouple these elds in the same way as we have done in Sect. 2, with the translations

U(x) U(x)

+ d4yD(M)(x, y)(y3 a) 3 B(y ) V(x) V(x)

d4yD(P)(x, y)(y3 a) 3 B(y ), (52)

where we dened the Maxwell and Proca propagators, respectively, as follows:

D(M)(x, y) =

d4 p (2)4

d4x

3 =a

]

exp

d4x

A S A S

4 A A

4 SS +

2 AS J A , (48)

where Z2,C[J] stands for the two eld functional generator

in the presence of a conducting plate.

It is important to point out that in Eq. (48) the integral is performed over all A-eld congurations. The constraint (5) is attained by the delta functional [F3(x)|x

3 =a

]. There is

no constraint on the S-eld.

Integrating out the functional (48) on the S variable, and performing some manipulations, we are taken to

Z2,C[J] = ZS[0]ZC[J], (49) where ZC[J], dened in (22), is the LeeWick functional

generator with boundary conditions (5) and ZS[0] is the func

tional generator (43) for the eld S with no boundary conditions and no external source. As discussed previously, in

this case ZS[0] has no physical effect and we can write from

Eq. (49)

Z2,C[J] = ZC[J]. (50) It proves that the boundary conditions can be imposed in both approaches, namely, the one with two elds, the other with one single eld. The physical results are the same in both cases.

Now we perform the change of integrating variables (45) in the two eld functional with a conducting plate (48), in order to know how the boundary conditions (5) are implemented in terms of the elds U and V . For this task we substitute Eq. (10) in (48), similarly to what we have done in Sect. 2, and we use the denitions (45), which leads to

Z2,C[J] = DUDV DB

exp

i d4x(x3 a) 3U(x) B(x )

exp

i d4x(x3 a) 3V(x) B(x )

exp

i 2

1p2 eip(xy)

123

339 Page 8 of 11 Eur. Phys. J. C (2015) 75 :339

D(P)(x, y) =

d4 p (2)4

1m2 p2

p p m2

eip(xy).

(53)

With these considerations we can rewrite the functional generator (51), after some manipulations, in the form

Z2,C[J] = Z2,C[J]

DU exp i d4x

14UU JU

DV exp i d4x

The wrong boundary condition

In spite of not being the true boundary condition imposed on the elds by the presence of a conducting surface, we could ask what kind of theory we would have by imposing, to each eld A and S, a boundary condition similar to (5). Before we answer this question, we point out, once more, that the matter source J couples just with the A eld, so is this eld which produces the Lorentz force on the charged particles. On a conducting surface, the Lorentz force must vanish, which is attained by imposing boundary conditions only on the A-eld and not on the S-eld.

In order to impose the conditions (5) to the elds A and S, we must insert two delta functionals inside the integral (41), as follows:

Z2,NC[J] = DADS[F3(x)|x

14 VV

2 V V JV

(54)

where we dened

Z2,C[J] = DB exp

i 2

d4xd4y

2Q(x, y)

x y

3 =a

][S3(x)|x3

]

exp

d4x

3 3

2[D(M)(x, y) D(P)(x, y)] x y

A S A S

2 AS

4 A A

(x3 a)(y3 a)B(x )B(y )

exp

d4x(x3 a)B(x )

d4y 3 J(x)

x (D(M)(x, y)

D(P)(x, y))

4 SS J A , (58)

where S = S and the sub-index NC stands for

boundary conditions which do not represent a conducting surface.

Each delta functional in (58) has an integral representation like the one in Eq. (10), namely,

[F3(x)|x

. (55)

Choosing the function Q(x, y) as the one dened in Eq. (17), using denitions (53), and solving the functional integral in Eq. (55), which is quadratic in B, it can be shown that Z2,C[J] gives the same result found in Eq. (20) for Z (up

to an overall multiplicative constant which does not depend on J), it is,

Z2,C[J] Z[J]

= Z2,C[0]e

i 2

= N DB exp i d4x(x3 a)A(x) 3 B(x )

exp

i 2A

3 =a

]

d3 xd3y B(x )

2QA(x, y) x y

x3=a

y3=a

B(y )

[S3(x)|x

3 =a

]

d4xd4y J(x) D(x,y)J(y). (56)

Substituting (56) in (54) and integrating out on U and V it can be shown that the two eld functional generator with the presence of conducting plate (54) is equal to the right hand side of (22), namely

Z2,C[J] = Z2,C[0] exp

i 2

= N DC exp i d4x(x3 a)S(x) 3C(x )

exp

i 2S

d4x d4y C(x )

2QS(x, y) x y

x3=a

y3=a

C(y )

d4xd4y J(x)(D(x, y)

+ D(x, y))J(y) . (57)

It proves again that, with the boundary conditions (5), the two eld formalism leads to the same results as the ones obtained from the formalism with just one single eld.

(59)

where A and S are gauge parameters and QA(x, y) and QS(x, y) are arbitrary functions we shall choose conveniently.

At this point one comment is in order. When we insert the integrals (59) into (58) we shall have two auxiliary elds, B and C. In Eq. (51) we have just one auxiliary eld.

Using the gauges where A = S = 1/2, taking the func

tions QA(x, y) = QS(x, y) equal to the one of Eq. (17),

substituting Eqs. (59) in (58), and performing the change of

123

Eur. Phys. J. C (2015) 75 :339 Page 9 of 11 339

eld variables (45) and

B+ = B + C, B = B C

B =

2(B+ + B), C =

DP(x, y) =

d4 p

(2)4

1p2 m2

p p

eip(xy)

DM(x, y) =

d4 p

(2)4

2(B+ B), (60)

whose Jacobian does not depend on any eld, we have

Z2,NC[J] = ZM,C [J]ZP,C[J] (61) where

ZM,C [J] = DU exp i d4x

1p2 eip(xy) (65)

and we dened the functions

DP(x, y) =

d3 p (2)3

i 2

p p

ei |x3a|ei |y3a|eip (x y )

DM(x, y) =

14UU JU

DB+ exp i d4x(x3 a)U(x) 3 B+(x )

exp

i 2

d3 p (2)3

i 2L

p p

(62)

is the functional generator for the Maxwell eld U submitted to the boundary condition U3 = 0 on the conducting surface

x3 = 0 and

ZP,C[J]

= DV exp i d4x

14 VV

x3=a

y3=a

B+(y )

ei L|x3a|ei L|y3a|eip (x y ), (66)

which give the corrections for the above propagators due to the boundary conditions V 3 = 0 and U3 = 0, respectively.

Substituting (64) in (61) we nally have

Z2,NC[J]

= exp

i 2

d3 xd3y B+(x )

2Q(x, y) x y

d4xd4y J(x)[(DM(x, y) + DM(x, y))

(DP(x, y) + DP(x, y))]J(y) . (67)

From (67) one can identify the propagator of the model as composed by the Maxwell propagator in the presence of a conducting plane minus a Proca propagator (with an overall minus sign) submitted to the condition V 3 = 0 on the con

ducting plane. In this case, it is not difcult to show that the image method is valid and all physical phenomena of the theory, with the wrong boundary conditions, can be decomposed as the corresponding ones obtained in the Maxwell Theory in the presence of a conducting plate minus the ones obtained in the Proca theory, with the condition V 3 = 0 on the plate.

5 Conclusions and nal remarks

In this paper we developed LeeWick electrodynamics in the presence of a single perfectly conducting plate. The boundary condition imposed on the gauge eld is the one which makes, on the plate, the dual eld strength normal to the plate vanish. That is justied because it is the condition which makes the Lorentz force on a given electric charge placed on the plate vanish, which is in accordance with the notion of a perfect conductor. Using functional methods of quantum eld theory, we calculated the gauge eld propagator and showed that it is not the composition of the Maxwell propagator minus the Proca one, both in the presence of a conducting plate. This fact makes the physical phenomena in LeeWick electrodynamics unusual, even classically.

With the obtained propagator any quantity, quantum or classical, related to the gauge eld in LeeWick electrody-

2 V V JV

DB exp i d4x(x3 a)V(x) 3 B(x )

exp

i 2

d3 xd3y B(x )

2Q(x, y) x y

x3=a

y3=a

B(y )

(63)

is the functional generator for the Proca eld V , with an overall minus sign in the Lagrangian, submitted to the boundary condition V 3 = 0 on the conducting surface x3 = 0.

Both integrals (62) and (63) can be calculated following the same procedure as employed in the previous sections. Working in the Lorenz gauge for the Maxwell eld, the results are

ZM,C [J] = exp

i 2

d4xd4y J(x)(DM(x, y)

+ DM(x, y))J(y)

ZP,C[J] = exp

d4xd4y J(x)(DP(x, y)

DP(x, y))J(y) (64)

where DP(x, y) and DM(x, y) stand for the free (without boundary conditions) Proca and Maxwell propagators, respectively,

i 2

123

339 Page 10 of 11 Eur. Phys. J. C (2015) 75 :339

namics could be computed. We calculated the interaction energy between a stationary point charge and a conducting plate. The counterpart of this result in Maxwell or Proca electrodynamics is the plate-charge interaction, which leads to the well known image method. From our results it is shown that in LeeWick electrodynamics the image method is not valid anymore.

We made a discussion regarding the two eld formalism for LeeWick electrodynamics, in the presence of a conducting plate, and showed that the physical results are the same as the ones obtained from the one eld formalism. We also discussed, in the two eld formalism, what the boundary conditions would be which make the theory equivalent to the Maxwell electrodynamics with the presence of the conducting plate minus the Proca Theory with boundary conditions.We showed that this boundary conditions are not the true ones imposed by a conducting surface.

LeeWick electrodynamics in the presence of two conducting parallel plates can be developed with the same methods employed in this paper. It is another interesting situation where we can study true quantum phenomena, as the Casimir effect, for instance [49].

Acknowledgments F. A. Barone and A. A. Nogueira are very grateful to CNPq and Capes (Brazilian agencies) for nancial support. The authors thank to F. E. Barone for reviewing the paper and for very pertinent comments.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/

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Web End =ons.org/licenses/by/4.0/ ), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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Appendix A: Delta functional

In this appendix we obtain expression (10). First, due to the gauge invariance (9), we have to extract the innite gauge volume from the integral (8). For this task we follow the FaddeevPopov trick, xing the covariant gauge

F[B(x )] = B(x ) = f (x ), (68) where f (x ) is an arbitrary function. The corresponding

FaddeevPopov determinant does not depend on the eld B(k)(x ) and, therefore, does not contribute to the integral.

In this way, we can write (8) in the form

[nF(x)|a] DB[F[B(x )] f (x )]

exp

a convergent functional of f (x ) and integrating over f (x ),

as follows:

[nF(x)|a] = N D f DB [F[B(x )] f (x )]

exp

i d4x(x3 a)A(x)n B(x)

exp

i 2

d4xd4y(x3 a) f (x )Q(x, y) f (y )(y3 a) ,(70)

where Q(x, y) is an arbitrary function, N is a constant and is an arbitrary gauge constant.

Performing the functional integral in f and two integration by parts in Eq. (70) we are taken to Eq. (10).

Appendix B: Integrals (18)

In this appendix we compute the integrals (18). For this task, it is enough to nd just the rst of Eq. (18) because the second one is a special case of the rst one, with m = 0.

Using the fact that there is a small negative imaginary part for the momentum square, as discussed just below Eq. (4), we have

d p3 2

1p2 m2

eip3x3 = lim

d p3 2

1p2 (p3)2 + i

eip3x3

= lim

d p3 2

1(p3)2 L2 i

eip3x3. (71)

Now, we solve the integral in the second line of the above equation by using the residue theorem and take the limit 0. The result is the rst of Eq. (18).

Appendix C: The limit m

In this appendix we calculate the limit m of the

propagator (23). Once the parameter m is present only in

D(x, y), it is enough to consider only this term. First we note that the correction to the propagator (21) can be written as

D(x, y) = OI(x, y), (72) where we dened the differential operator O =

and the integral

I =

d3 p (2)3

i 2

1 p2

1 L

1 eip (x y )

d4x(x3 a)B(x)F 3(x) . (69)

Now we integrate by parts the argument of the exponential and use the t Hooft trick, multiplying both sides of (69) by

ei L|x3a|

ei |x3a|

ei L|y3a|

ei |y3a|

(73)

123

Eur. Phys. J. C (2015) 75 :339 Page 11 of 11 339

Now we make a Wick rotation only in the parallel coordinates, in the integral (73), with k4 = ip0, k1 = p1,

k2 = p2, X4 = ix0, X1 = x1, and X2 = x2, and

dene the Euclidean 3-vectors k = (k1, k2, k4), X =

(X1, X2, X4) (and similarly for y ), which leads to

I =

d3k

(2)3

eik4(X4Y4)

2k2

eik1(X1Y1)eik2(X2Y2)1 ik2

1 ik2+m2

ik2

k2|x3a|

1 ik2 + m2

ek2+m2|x3a|

ik2

ek2+m2|y3a| .

(74)

Taking the limit m in the above expression,

lim

m I =

d3k

(2)3

k2|y3a|

1 ik2 + m2

2k2 eik4(X4Y4)

eik1(X1Y1)eik2(X2Y2) e

k2|x3a|e

k2|y3a|

ik2

. (75)

Performing an inverse Wick rotation back to Minkowski space on Eq. (75), substituting the result into Eq. (72) and acting with the operator O, we obtain

lim

m D

(x, y) =

d3 p (2)3

i 2

p p

eip (x y ) ei L(|x3a|+|y3a|)

L , (76)

which is the propagator for the Maxwell eld in the presence of a perfectly conducting plate placed at position x3 = a

[42].

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Lee-Wick electrodynamics in the vicinity of a conducting plate is investigated. The propagator for the gauge field is calculated and the interaction between the plate and a point-like electric charge is computed. The boundary condition imposed on the vector field is taken to be the one that makes, on the plate, the normal component of the dual field strength to the plate vanish. It is shown that the image method is not valid in Lee-Wick electrodynamics.

Details

Title

A conducting surface in Lee-Wick electrodynamics

Author

Barone, F A; Nogueira, A A

Pages

1-11

Publication year

2015

Publication date

Jul 2015

Publisher

Springer Nature B.V.

ISSN

14346044

e-ISSN

14346052

Source type

Scholarly Journal

Language of publication

English

DOI

https://doi.org/10.1140/epjc/s10052-015-3563-x

ProQuest document ID

1707494702

SIF and Springer-Verlag Berlin Heidelberg 2015

A conducting surface in Lee-Wick electrodynamics

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