Path-integral quantization of Galilean Fermi fields

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Annals of Physics 323 (2008) 1191–1214 www.elsevier.com/locate/aop

Path-integral quantization of Galilean Fermi fields M. de Montigny a

a,b,*

, F.C. Khanna

a,c

, F.M. Saradzhev

a

Theoretical Physics Institute, University of Alberta, Edmonton, Alta., Canada T6G 2J1 b Campus Saint-Jean, University of Alberta, Edmonton, Alta., Canada T6C 4G9 c TRIUMF, 4004, Westbrook Mall, Vancouver, BC, Canada V6T 2A3 Received 26 June 2007; accepted 2 August 2007 Available online 14 August 2007

Abstract The Galilei-covariant fermionic field theories are quantized by using the path-integral method and five-dimensional Lorentz-like covariant expressions of non-relativistic field equations. First, we review the five-dimensional approach to the Galilean Dirac equation, which leads to the Le´vyLeblond equations, and define the Galilean generating functional and Green’s functions for positiveand negative-energy/mass solutions. Then, as an example of interactions, we consider the quartic self-interacting potential kðWWÞ2 , and we derive expressions for the 2- and 4-point Green’s functions. Our results are compatible with those found in the literature on non-relativistic many-body systems. The extended manifold allows for compact expressions of the contributions in (3 + 1) space–time. This is particularly apparent when we represent the results with diagrams in the extended (4 + 1) manifold, since they usually encompass more diagrams in Galilean (3 + 1) space–time.  2007 Elsevier Inc. All rights reserved. Keywords: Fermions; Galilean invariance; Path-integral quantization

1. Introduction Although its original successes lie in particle physics, quantum field theory has since then reached a much wider range of applications. Indeed, concepts such as perturbation *

Corresponding author. Address: Theoretical Physics Institute, University of Alberta, Edmonton, Alta., Canada T6G 2J1. E-mail addresses: [email protected] (M. de Montigny), [email protected] (F.C. Khanna), [email protected] (F.M. Saradzhev). 0003-4916/$ - see front matter  2007 Elsevier Inc. All rights reserved. doi:10.1016/j.aop.2007.08.002

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methods and Feynman diagrams, renormalization group procedures, spontaneous symmetry breaking, etc. (both at zero and finite temperature) have been interchangeably utilized by physicists working in particle physics as well as in condensed matter physics and statistical physics [1,2]. More modern field theories, such as conformal field theory, are applied in statistical physics, string theory, etc. However, a fundamental difference between particle physics and condensed matter (or statistical) physics is that the latter involves the non-relativistic regime, whereas highenergy physics involves relativistic kinematics. In fact, field theoretical models typically are constructed by taking into account various symmetries, such as Poincare´ space–time invariance. Recent achievements, such as Fermi condensates with ultra-cold potassium40 atoms [3], suggest that analogous procedures should be devised for Galilean-invariant systems. Recent interest in the Galilean symmetry (particularly in the plane) is due to its applications to Hall effect, anyons, Chern-Simons term, non-commutative geometry, etc [4]. This article is an extension to Fermi fields of a recent work where we have performed the path-integral quantisation of Galilean-invariant scalar fields [5]. It belongs to a series of papers whose general underlying program consists in using a formulation of Galilean covariance based on a relativistic framework in one higher dimension, which makes non-relativistic field theories similar to Lorentz-covariant theories [6–8]. In these articles, the extended manifold approach follows the lines of earlier investigations [9,10]. Similar approaches have been ubiquitous in physics [11,12]. Recently, it has been used in the study of fluid dynamics [13,14]. The occurence of the 2 + 1 Galilean group was observed [15] as the transverse motion to the direction of the infinite momentum frame, now better known as the light-cone frame, in a study of the perturbative behaviour in the limit of strong interaction processes. This has been suggested previously [16]. Later, this perspective was taken up in conjecturing an equivalence between eleven-dimensional M-theory and the N = 1-limit of the supersymmetric matrix quantum mechanics which describes D0 branes [17] (These authors actually consider the super-Galilei group, which admits 32 real super-generators.) Let us review the formalism briefly for our purposes. The algorithm henceforth consists in building action functionals by enforcing Lorentz covariance, as it is usually done with relativistic theories, except that Galilean kinematics is based on the so-called Galilean fivevectors (x, x4, x5). These vectors transform under Galilean boosts as x0 ¼ x  vx4 ; 0

x4 ¼ x4 ; 1 2 0 x5 ¼ x5  v  x þ jvj x4 ; 2

ð1Þ

where v is the relative velocity between the two reference frames. Altogether the kinematical transformations, which also include rotations and translations, form a fifteen-dimensional Lie algebra. This may be seen as the Poincare´ algebra in (4 + 1) space–time. Eleven of these fifteen generators form the extended Galilei group, where the central-extension parameter (the non-relativistic mass) is inherited from the generator of x5 translations. The transformation (1) for x 0 5 has occurred in various contexts [5–12]. In quantum mechanics, it is associated with the wave function’s phase which enforces invariance of the Schro¨dinger equation under Galilean transformations. Furthermore, it leads to a

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1193

superselection rule for mass conservation in Galilean covariant theories. Indeed, unlike the relativistic theories, new massive particles cannot be created in a Galilean framework. This would preclude Yukawa couplings except for massless particles with coupling to two massive particles. For massive particles, only the 4-particle coupling is allowed. The invariant scalar product is defined as A  B ¼ A4 B5 þ A5 B4  A  B; with the Galilean metric: 0 1 133 0 0 B C glm ¼ @ 0 0 1 A: 0 1 0

ð2Þ

This suggests that the non-relativistic time is a light-cone parameter of the Lorentz invariant theory on a manifold containing one additional space-like dimension [10,18]. Once we have constructed a Galilean covariant action functional, an appropriate embedding of the Galilean space–time into G ð4þ1Þ may be defined as  s ðx; tÞ,!xl ¼ ðx1 ; . . . ; x5 Þ  x; ct; ; c where c is a parameter with the dimensions of velocity, which will be specified below. The five-momentum   ot E ð3Þ pl ¼ iol ¼ ðir; i ; icos Þ ¼ p; ; mc ; c c where p4 ¼ p5 ¼ mc and p5 ¼ p4 ¼ Ec , suggests that the additional coordinate x5 ¼ cs is canonically conjugated to mc. From the relation os ¼ im, the phase factor of the wavefunction follows: WðxÞ  eims wðx; tÞ;

ð4Þ

which projects the fields from G ð4þ1Þ to (3 + 1)-dimensions. A different definition of dimensional reduction would lead to a Lorentz-covariant theory in (3 + 1)-dimensions [10,11]. Note that it is also possible to define WðxÞ  eims wþ ðx; tÞ þ eþims w ðx; tÞ; where w±(x, t) represent the positive- or negative-energy solutions, which makes evident the possibility of negative energy solutions [19]. This comes from the quadratic condition 2 ðos Þ ¼ m2 , and it is compatible with the embedding defined in Eq. (4) since additional terms with negative mass and negative energy can be included. Such a description is allowed by the symmetry ðct; csÞ ! ðct;  csÞ [20]. In (4 + 1)-dimensional Galilean theories, plpl = 2mE  p2 is an invariant, and the dynamics of Galilean covariant fields must be consistent with it. Let us take p l pl ¼ k 2 ; where k is a real constant that defines the invariant quantity. It leads to 2mE  p2 = k2, which is analogous to E2  p2c2, the invariant for Lorentz covariant fields that is equal to m2c4, thus defining m as the invariant quantity. This implies the dispersion relation:

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E¼

1 1 2 jpj2 þ k : 2m 2m

ð5Þ

Introducing a velocity parameter k c ¼ pffiffiffi ; 2m we cast the dispersion relation into the familiar form for non-relativistic energy with 2 1 jpj þ mc2 . Note that this equation, as well as the invariant 2mE  p2, is invariant E ¼ 2m under the changes m ! m;

E ! E;

so that in the (4 + 1)-manifold, one must reverse both m and E, but not each one independently. Henceforth, we will have to ensure that this is satisfied when we split the positiveand negative-energy/mass solutions. The constant k is the Galilean analogue of the Lorentzian rest mass. Since k can be absorbed within the energy E, its value is usually considered to be of no physical importance and taken to be zero. However, it may be possible to relate k to the chemical potential [5]. For Galilean Fermi fields, the dispersion relation (5) implies that the negative energy solutions are characterized by negative masses. The paper is organized as follows. In Section 2, we review the Le´vy-Leblond equations by means of the extended-manifold Dirac equation, and the positive- and negative-energy/ mass solutions, and we introduce the Galilean generating functional formalism. Appropriate embeddings associated with the virtual sources are defined. In Section 3, we establish the connection between the generating functional and the Green’s functions for both positive- and negative energy/mass solutions. In Section 4, we apply this formalism to the selfinteracting quartic potential. The 2- and 4-point functions are calculated. We distinguished between (3 + 1)- and (4 + 1)-manifold diagrams, the latter containing, in general, more diagrams in the reduced (3 + 1) space–time. Concluding remarks are in Section 5. 2. Free Dirac field 2.1. Five-dimensional Dirac equation Let us consider a free Dirac field W(x) defined on the five-dimensional manifold G ð4þ1Þ with Galilean metric, Eq. (2). Then a manifestly covariant Lagrangian for the Dirac field is given by $

L0 ¼ WðxÞðicl ol kÞWðxÞ;

ð6Þ

$

where a ob  12 ½aob  ðoaÞb. Both the field and its adjoint are anticommuting. The matrices cl in the extended space–time are four-dimensional and may be chosen as pffiffiffi !    a  ir 0 0 0 0 2 a 4 5 ; c ¼ ; c ¼ ; c ¼ pffiffiffi 2 0 0 ira 0 0 ra, a = 1, 2, 3 denoting the 2 · 2 Pauli matrices. They obey the usual anticommutation relations:

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fcl ; cm g ¼ 2glm : Let us apply the variational principle for the action integral with the free Lagrangian of Eq. (6) [21], Z I½W; W ¼ d5 xL0 ½WðxÞ; WðxÞ; ð7Þ R R l=2 where the integral over x5 is interpreted as dx5 ! liml ! 1 1l l=2 d x5 , and l is an arbitrary length. Then the Euler–Lagrange equations of motion for W(x) and its adjoint WðxÞ, respectively, are ðicl ol  kÞWðxÞ ¼ 0;

WðxÞðicl o l þ kÞ ¼ 0;

ð8Þ

where a o b ¼ ðoaÞb. The adjoint field is defined as WðxÞ ¼ Wy ðxÞc0 ; where 1 c0 ¼ pffiffiffi ðc4 þ c5 Þ ¼ 2



0 1

 1 : 0

Its reduction to (3 + 1)-dimensions is defined as 5  tÞ:  WðxÞ ¼ eimcx wðx;

ð9Þ

The first expression in Eq. (8), using Eqs. (3) and (4), reduces to pffiffiffi ððr  $Þ þ kÞw1 ðx; tÞ  2mcw2 ðx; tÞ ¼ 0; pffiffiffi E 2 w1 ðx; tÞ þ ððr  $Þ  kÞw2 ðx; tÞ ¼ 0; c

ð10Þ

where  wðx; tÞ ¼

w1 ðx; tÞ w2 ðx; tÞ



with w1(x, t) and w2(x, t) being two-component spinors. The two equations in Eq. (10) are analogous to the Pauli equations in the relativistic case. These Galilean wave equations describe non-relativistic Fermi fields in (3 + 1)-dimensions. If k = 0 then Eq. (10) coincides with the Le´vy-Leblond equations [24]. The wave equations for the adjoint Fermi fields have the same form and can be deduced from the second expression in Eq. (8) together with Eq. (9). In analogy with the relativistic theory, we find that the Fourier components of the Gal ilean Dirac fields satisfy (plpl  k2)W(p) = 0 and ðpl pl  k 2 ÞWðpÞ ¼ 0, which reduce to the Schro¨dinger wave equations. Then each component of the (3 + 1)-dimensional non-relativistic Fermi fields obeys the Schro¨dinger equation:  2  p k2 Ew1;2 ðpÞ ¼ þ w ðpÞ: 2m 2m 1;2 Thus the Schro¨dinger equation may be obtained either by first reducing the Dirac equation to the Le´vy-Leblond equations (10) with Eq. (4), or by first reducing Eq. (6) to the Lagrangian of the Schro¨dinger field.

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2.2. The positive- and negative-energy solutions and canonical quantization The Lagrangian, Eq. (6), and the Dirac equations given by Eq. (8) are invariant with respect to unitary transformations cl ! Scl S 1 ;

W ! SW;

where S is a 4 · 4-matrix. To construct the positive- and negative-energy/mass solutions explicitly, it is convenient to use a representation in which c0 is diagonal. This representation can be obtained from the one used in the previous section by performing the unitary transformation above with the choice   1 1 1 S ¼ pffiffiffi : 2 1 1 In particular, c0 becomes   1 0 0 : c ¼ 0 1

ð11Þ

It is important to point out that this form of c0 matrix does not imply chirality. In fact, there is no parity operator in five dimensions, hence no chirality. Only if we work in even dimensions, in this case six dimensions, can we find a parity operator, hence a chirality operator. Then the c-matrices are 8-dimensional. The details of this representation will appear elsewhere [28]. The matrices cl take the form       0 ira 1 1 1 1 1 1 a 4 5 ; c ¼ pffiffiffi : ð12Þ c ¼ ; c ¼ pffiffiffi ira 0 2 1 1 2 1 1 In what follows, we will use the representation defined by Eqs. (11) and (12). The plane-wave solutions for Eq. (8) are written in the usual form, Z 1 d5 p½uðrÞ ðpÞeipx þ vðrÞ ðpÞeipx ; WðrÞ ðxÞ ¼ 5 ð2pÞ where r = 1, 2 and the positive- and negative-energy spinors u(r)(p) = u(r)(p, E, m), v(r)(p) = v(r)(p, E, m) obey the equations ðcl pl  kÞuðrÞ ðpÞ ¼ 0;

ðcl pl þ kÞvðrÞ ðpÞ ¼ 0:

Taking the Dirac particle in the rest frame, p = 0, we find c0 uðrÞ ð0Þ ¼ uðrÞ ð0Þ;

c0 vðrÞ ð0Þ ¼ vðrÞ ð0Þ;

where uðrÞ ð0Þ  uðrÞ ð0; Ek ; mÞ; vðrÞ ð0Þ  vðrÞ ð0; Ek ; mÞ; and Ek 

k2 : 2m

ð13Þ

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1197

The representation with diagonal c0 is especially appropriate for describing particles at rest, the spinors u(r)(0), v(r)(0) being eigenvectors of c0 with eigenvalues +1 and 1, respectively. Let us define     1 0 ð1Þ ð2Þ n ð0Þ ¼ ; n ð0Þ ¼ ; 0 1 which satisfy the relation nyðrÞ ð0ÞnðsÞ ð0Þ ¼ drs : We write the spinors u(r)(0), v(r)(0) as !   0 nðrÞ ð0Þ ðrÞ ðrÞ u ð0Þ ¼ : ; v ð0Þ ¼ nðrÞ ð0Þ 0 In a moving frame, the spinors u(r)(p), v(r)(p) are expressed as uðrÞ ðpÞ ¼ d u ðcl pl þ kÞuðrÞ ð0Þ;

vðrÞ ðpÞ ¼ d v ðcl pl  kÞvðrÞ ð0Þ:

ð14Þ

These definitions are motivated by plpl = k2, so that (cl pl + k)(clpl  k) = p2  k2 = 0 and Eq. (13) is satisfied. The coefficients du, dv are computed from two conditions. First, the right- and left-hand sides of Eq. (14) must coincide for p = 0, that is, u(r)(p = 0) = u(r)(0) and v(r)(p = 0) = v(r)(0). Second, the orthonormality conditions  uðrÞ ðpÞuðsÞ ðpÞ ¼ drs ;

vðrÞ ðpÞvðsÞ ðpÞ ¼ drs ;

which can be checked for the p = 0 case, must be valid for non-zero p as well. These two conditions determine du and dv as  1=2 1 4Ek : d u ¼ d v ¼ 2k E þ 3Ek The general solution of the Galilean Dirac Eqs. (8) may be expanded in terms of the plane wave solutions as XZ 3 1 WðxÞ ¼ d p½aðrÞ ðpÞuðrÞ ðpÞeipx þ byðrÞ ðpÞvðrÞ ðpÞeipx ; ð15Þ 3=2 ð2pÞ r XZ 3 1  WðxÞ ¼ d p½ayðrÞ ðpÞ uðrÞ ðpÞeipx þ bðrÞ ðpÞvðrÞ ðpÞeipx ; ð16Þ 3=2 ð2pÞ r where a(r)(p) (a (r)(p)) and b(r)(p) (b (r)(p)) are destruction (creation) operators of particles and antiparticles, respectively. The fields are quantised by assuming that these operators obey the anticommutation relations: faðrÞ ðpÞ; ayðsÞ ðqÞg ¼ fbðrÞ ðpÞ; byðsÞ ðqÞg ¼ drs dðp  qÞ: All other anticommutation relations are zero.

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Using the non-relativistic ‘‘momentum-energy-mass’’ tensor oL  oL  Lglm ; T lm ¼ om W þ om W  oðol WÞ oðol WÞ we define the five-momentum of the Galilean Dirac field as Z Z pffiffiffi i  4 ol W  ol Wc  4 WÞ: P l ¼ d3 xdx5 2T 5l ¼ pffiffiffi d3 xdx5 ðWc 2 The charge operator is written as Z pffiffiffi 4  W: Q ¼ d3 xdx5 2Wc Substituting the expansions (15) and (16) into the expressions for Pl and Q and performing a normal ordering with respect to the vacuum state, we get aðkÞj0i ¼ bðkÞj0i ¼ 0

for all k and m;

giving, for the five-momentum and the charge operators, XZ 3 Pl ¼ d p  pl ½ayðrÞ ðpÞaðrÞ ðpÞ þ byðrÞ ðpÞbðrÞ ðpÞ; r

and Q¼

XZ

d3 p  ½ayðrÞ ðpÞaðrÞ ðpÞ  byðrÞ ðpÞbðrÞ ðpÞ:

r

This corroborates the point, mentioned earlier, that a (p) and a(p) are the creation and annihilation operators for particles of momentum p, mass m and charge +1, whereas the operators b (p) and b(p) correspond to antiparticles, which differ from the particles only by the sign of the charge, i.e. 1. 2.3. Galilean generating functional As in the usual path-integral formalism [26], the Galilean generating functional for the free field is given by the vacuum-to-vacuum transition amplitude with anticommuting virtual sources J(x) and J ðxÞ:  Z  Z Z Z 0 ½J ; J  ¼ DW DW exp i d5 x½L0 ½W; W þ WðxÞJ ðxÞ þ J ðxÞWðxÞ ; R R where DW and DW denote the functional integrations over WðxÞ and W(x), respectively. Here J(x) and J ðxÞ are anticommuting Grassmann virtual sources that we put equal to zero at the end. Let us define a new field W 0 : W0 ðxÞ ¼ WðxÞ  WJ ðxÞ; where WJ(x) satisfies the inhomogeneous equation of motion: ðicl ol  kÞWJ ðxÞ ¼ J ðxÞ:

ð17Þ

Then we can complete the square within the exponential and rewrite the generating functional as

M. de Montigny et al. / Annals of Physics 323 (2008) 1191–1214

Z 0 ½J ; J  ¼

Z

1199

 Z  Z i 5 5 0 0 d xðJ ðxÞWJ ðxÞ þ WJ ðxÞJ ðxÞÞ : DW DW exp i d xL0 ðW ; W Þ þ 2 0

0

Here we have changed the integration variables from W to W 0 , for which the Jacobian is unity. Denoting the integration over W 0 and W0 by Z0 [0], we observe that the generating functional becomes  Z  i 5 d xðJ ðxÞWJ ðxÞ þ WJ ðxÞJ ðxÞÞ : Z 0 ½J ; J  ¼ Z 0 ½0 exp 2 The field WJ(x) can be written as Z WJ ðxÞ ¼  d5 yS 1 ðx  yÞJ ðyÞ;

ð18Þ

where S1(x  y) is the free-field Green’s function, which satisfies ðicl ol  kÞS 1 ðx  yÞ ¼ ~ d5 ðx  yÞ;

ð19Þ

where we adopt a non-standard definition of the delta function: ~ d5 ðx  yÞ  ~ d5 ðx  y; mÞ þ ~ d5 ðx  y; mÞ;

ð20Þ

with ~ d5 ðx  y; mÞ being Dirac delta functions [5] in G ð4þ1Þ defined as 5 5 ~ d5 ðx  y; mÞ ¼ eimcðx y Þ d3 ðx  yÞdðx4  y 4 Þ:

Taking S1(x  y) as S 1 ðx  yÞ ¼ 

icl ol þ k Dðx  yÞ; 2k

ð21Þ

and introducing the ie prescription by replacing k2 with k2  ie, we bring Eq. (19) into the form 1 ðol ol þ k 2  ieÞDðx  yÞ ¼ ~ d5 ðx  yÞ; 2k

ð22Þ

which is the equation for the Feynman propagator for a free Galilean scalar field. In the next section, we will show that D(x  y) coincides with the Galilean Feynman propagator up to a constant factor.  J ðxÞ, The equation for the adjoint field W  J ðxÞðicl o l þ kÞ ¼ JðxÞ; W is solved by  J ðxÞ ¼  W

Z

d5 y JðyÞS 2 ðx  yÞ;

where S2(x  y) is a solution of S 2 ðx  yÞðicl o l þ kÞ ¼ ~ d5 ðx  yÞ:

ð23Þ

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The function S2(x  y) is related to D(x  y) by S 2 ðx  yÞ ¼

1 Dðx  yÞðicl o l  kÞ; 2k

so that S 2 ðy  xÞ ¼ S 1 ðx  yÞ: For the external source J, we factor out the coordinate x5 as follows: 5

5

J ðxÞ ¼ eimcx jþ ðx; x4 Þ þ eimcx j ðx; x4 Þ; and similarly for JðxÞ. This factorization is motivated by the definition of the fields WJ(x)  J ðxÞ in Eqs. (18) and (23), respectively, so that we have, for instance: and W Z ðicl ol  kÞWJ ðxÞ ¼  d5 y ~ d5 ðx  yÞJ ðyÞ; Z 1 l=2 5 imcðx5 y 5 Þ 5 5 dy ½e þ eimcðx y Þ  ¼ lim l!1 l l=2 5

5

 ½eimcy jþ ðx; x4 Þ þ eimcy j ðx; x4 Þ ¼ J ðxÞ; in agreement with Eq. (17). Note that in the second line, we have eliminated y by integration and by using the definition of the delta function given in Eq. (20).  J ðxÞ defined by Eqs. (18) and (23), respectively, the generWith the fields WJ(x) and W ating functional takes the form  Z  5 5 Z 0 ½J ; J  ¼ Z 0 ½0 exp i d x d yJ ðxÞS 1 ðx  yÞJ ðyÞ : ð24Þ This is the generating functional of the Green’s function that characterizes the Dirac field. Note that Z 0 ½J ; J is written in terms of the propagator S1(x  y), which includes both particle and antiparticle contributions. 3. Green’s functions for particles and antiparticles Let us now turn to some properties of the Galilean propagators D(x  y) and S1(x  y). The Fourier transforms of these propagators are defined by expressions similar to the mass-shell condition: Z 1 ipðxyÞ  Dðx  yÞ ¼ d5 pDðpÞe 2p½dðp4  mcÞ þ dðp4 þ mcÞ; ð25Þ 5 ð2pÞ Z 1 S 1 ðx  yÞ ¼ ð26Þ d5 pS1 ðpÞeipðxyÞ 2p½dðp4  mcÞ þ dðp4 þ mcÞ: ð2pÞ5 By substituting Eq. (25) into Eq. (22) and using Z 1 ~ d5 peipðxyÞ 2p½dðp4  mcÞ þ dðp4 þ mcÞ; d5 ðx  yÞ ¼ ð2pÞ5 where the term between brackets is reminiscent of our non-standard definition of delta function in (4 + 1) dimensions, we find

M. de Montigny et al. / Annals of Physics 323 (2008) 1191–1214

 DðpÞ ¼

1201

2k ; pl pl  k 2 þ ie

so that we rewrite D(x  y) as " # pffiffiffi Z Z imcðx5 y 5 Þ imcðx5 y 5 Þ e e 2 5 4 4 : d3 p dp5 eipðxyÞip ðx y Þ 5 E Dðx  yÞ ¼ e  5 e 4 p þ c  i 2m p  Ec þ i 2m ð2pÞ c c Integrating over p5 with the change of variable p5 ! p5 þ Ec in the first integral, and p5 ! p5  Ec in the second one, and by using the following representation of the step function: Z 1 1 eixs ; dx hðsÞ ¼ limþ x þ ie e!0 2pi 1 we obtain Dðx  yÞ ¼

pffiffiffi 2DF ðx  yÞ;

where iDF ðx  yÞ  hðx4  y 4 ÞDðx  y; mÞ þ hðy 4  x4 ÞDðx  y; mÞ; DF(x  y) being the Galilean Feynman propagator for a free scalar field [20], and Z 1 d3 peipðxyÞ : Dðx  y; mÞ ¼ 3 ð2pÞ The positive- and negative-energy/mass contributions to D(x  y) can be written explicitly as pffiffiffi 5 5 5 5 Dðx  yÞ ¼ 2½eimcðx y Þ G0þ ðx  y; x4  y 4 ; mÞ þ eimcðx y Þ G0 ðx  y; x4  y 4 ; mÞ;

ð27Þ

where G0þ ðx  y; x4  y 4 ; mÞ 

i ð2pÞ

hðx4  y 4 Þ 3

Z

E

4 y 4 Þ

d3 peipðxyÞi c ðx

is the Schro¨dinger Green’s function of a scalar particle with mass m [5], and G0 ðx  y; x4  y 4 ; mÞ ¼ G0þ ðy  x; y 4  x4 ; mÞ: This ensures that Eq. (27) is compatible with our earlier statements about the splitting of positive versus negative energy and mass. With Eq. (27), S1(x  y) becomes pffiffiffi S 1 ðx  yÞ ¼ 2S F ðx  yÞ; where S F ðx  yÞ  

1 ðicl ol þ kÞDF ðx  yÞ 2k

is the Galilean Feynman propagator for a free Dirac field. If we substitute Eqs. (25) and (26) into Eq. (21), we find

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S1 ðpÞ ¼

cl p l þ k pl pl  k 2 þ ie

:

Using Eq. (27), we write S1(x  y) in the form: 5 y 5 Þ

S 1 ðx  yÞ ¼ eimcðx

S 1 ðx  y; x4  y 4 ; mÞ þ eimcðx

5 y 5 Þ

S 1 ðx  y; x4  y 4 ; mÞ;

ð28Þ

where S 1 ðx  y; x4  y 4 ; mÞ ¼

pffiffiffi 0 2S  ðx  y; x4  y 4 ; mÞ;

and S 0þ ðx  y; x4  y 4 ; mÞ ¼

1 4 4 c dðx  y 4 Þdðx  yÞ 2k Z i E 4 4  hðx4  y 4 Þ d3 pS þ ðp; mÞei½pðxyÞ c ðx y Þ 3 ð2pÞ

is the Schro¨dinger Green’s function of a Dirac particle of mass m with ! pffiffiffi 1 E þ 3Ek E  Ek  ic 2r  p pffiffiffi S þ ðp; mÞ ¼ : 4Ek E þ Ek  ic 2r  p E þ Ek For the negative-mass contribution, we have 1 4 4 c dðx  y 4 Þdðx  yÞ 2k Z i E 4 4  hðy 4  x4 Þ d3 pS  ðp; mÞei½pðxyÞ c ðx y Þ ; 3 ð2pÞ

S 0 ðx  y; x4  y 4 ; mÞ ¼ 

where S  ðp; mÞ ¼ 1  S þ ðp; mÞ: The propagator SF(x  y) can be defined in the canonical formalism as well. The expansions in Eqs. (15) and (16) yield an expression for the Feynman propagator as  h0jT ½WðxÞWðyÞj0i ¼ iS F ðx  yÞ; where T denotes the time ordering. This formula connects the path-integral and canonical formalism and proves their equivalence. Now let us define the Galilean one-particle Green’s function for free fields: ðiÞ2 d2 Z 0 ½J ; J  0 G ðx1 ; x2 Þ ¼ ; ð29Þ Z 0 ½0 dJ ðx1 ÞdJ ðx2 Þ J ¼0¼J

where Z 0 ½J ; J  is given in Eq. (24), thus leading to pffiffiffi G0 ðx1 ; x2 Þ ¼ i 2S F ðx1  x2 Þ:

ð30Þ

It is possible to calculate the average values of the translation generators in the Hilbert space given in Eq. (3), i.e. the observables corresponding to momentum, energy and mass in quantum mechanics:

M. de Montigny et al. / Annals of Physics 323 (2008) 1191–1214

b ¼i h Oi

Z

1203

d3 x dx5 lim½OG0 ðx; yÞ; y!x

where O denotes Pi, H = P4 or M =P5. We may generalize the one-particle Green’s function in Eq. (29) to the n-particle Green’s functions: G0 ðx1 ; . . . ; xn ; y 1 ; . . . ; y n Þ ¼ h0jT ðWðx1 Þ    Wðxn ÞWðy 1 Þ    Wðy n Þj0i; ðiÞ2n d2n Z 0 ½J ; J  ¼ : Z 0 ½0 dJ ðx1 Þ . . . dJ ðxn ÞdJ ðy 1 Þ . . . dJ ðy n Þ J ¼0¼J

ð31Þ

For instance, the 1-particle Green’s function is given in Eq. (29) and the 2-particle Green’s function is given as 1 d4 Z 0 ½J ; J  G0 ðx1 ; x2 ; y 1 ; y 2 Þ ¼ : ð32Þ Z 0 ½0 dJ ðx1 ÞdJ ðx2 ÞdJ ðy 1 ÞdJ ðy 2 Þ J ¼0¼J Explicit forms and perturbative expansion will be given later on in an interacting system with quartic interactions. For one-particle Green’s function, an equation similar to the Schwinger-Dyson equation is obtained with the self-energy defined explicitly.

4. Self-interacting quartic potential Now consider a Lagrangian which contains a non-trivial interacting potential: L ¼ L0 þ Lint ; where L0 is given by Eq. (6) and Lint is the interaction term that depends on W and W. With an arbitrary interaction, the generating functional is R R R DW DW expfiI  i d5 x½J ðxÞWðxÞ þ J ðxÞWðxÞg R R Z½J ; J  ¼ DW DW expðiIÞ with I given in Eq. (7), and where L0 is replaced by L. Following standard methods [26], we write the generating functional as  Z

 1 d 1 d Z½J ; J  ¼ N exp i d5 xLint ; ð33Þ Z 0 ½J ; J ; i dJ i dJ where N is a normalization factor, and Z0 is given by Eq. (24). We derive the Green’s functions from Eq. (31) by replacing Z 0 ½J ; J  with Z½J ; J . Consider an interaction Lagrangian in the form 2

 Lint ¼ gðWðxÞWðxÞÞ : When we expand Eq. (33) in powers of g, then Z½J ; J  becomes

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 d d d d Z½J ; J  ¼ N exp ig d x Z 0 ½J ; J ; dJ ðxÞ dJ ðxÞ dJ ðxÞ dJ ðxÞ    Z d d d d ¼ N 1  ig d5 z dJ ðzÞ dJ ðzÞ dJ ðzÞ dJ ðzÞ    Z g2 d d d d d d d d 5 5  d zd w dJ ðwÞ dJ ðwÞ dJ ðwÞ dJ ðwÞ dJ ðzÞ dJ ðzÞ dJ ðzÞ dJ ðzÞ 2 þ Oðg3 Þ Z 0 ½J ; J : Z

5



To zeroth order in g, we simply retrieve Z 0 ½J ; J . Henceforth, we shall utilize the short-hand notation Z hJ a ðxÞS ab ðx  yÞJ b ðyÞi  d5 x d5 yJ a ðxÞS ab ðx  yÞJ b ðyÞ; where Sab(x  y) ” (S1)ab(x  y), and a, b = 1,2,3,4 indicate the component structure of virtual sources. Thus we have Z 0 ¼ eihJ a ðxÞS ab ðxyÞJ b ðyÞi : To second order in g, we find   Z Z g2 d5 z d5 wT ðz; wÞ Z 0 ½J ; J; Z½J ; J ¼ N 1  ig d5 zT ðzÞ  2

ð34Þ

where T ðzÞ 

1 d d d d Z 0 ½J ; J ¼ TrðS 1 ð0ÞÞ2  ðTrS 1 ð0ÞÞ2 Z 0 ½J ; J dJ a ðzÞ dJ a ðzÞ dJ b ðzÞ dJ b ðzÞ  2iðS ab ð0Þ  dab TrS 1 ð0ÞÞhS bd ðz  vÞJ d ðvÞihJc ðuÞS ca ðu  zÞi þ hS bd ðz  vÞJ d ðvÞihJc ðuÞS cb ðu  zÞihS ad ðz  wÞJ d ðwÞihJc ðyÞSca ðy  zÞi;

and T ðz; wÞ 

1 d d d d ðT ðzÞZ 0 ½J; JÞ: Z 0 ½J ; J dJ a ðwÞ dJ a ðwÞ dJ b ðwÞ dJ b ðwÞ

The normalization factor in Eq. (34) is chosen in such way that Z[0] = 1,  1 Z Z g2 5 5 5 N ¼ Z 1 d ½0 1  ig d zT ðzÞ  zd wT ðz; wÞ ; 0 0 0 2 where T 0 ðzÞ  T ðzÞjJ ¼0¼J ;

T 0 ðz; wÞ  T ðz; wÞjJ ¼0¼J ;

excluding vacuum graphs from consideration. 4.1. 2-point function The 2-point function G(x1, x2) is given by Eq. (29) with Z 0 ½J ; J replaced by Z½J ; J. To zero-th order in g, G(x1, x2) is clearly the same as for the free field. To first order in g, we obtain

M. de Montigny et al. / Annals of Physics 323 (2008) 1191–1214

Gab ðx1 ; x2 Þ ¼ G0ab ðx1 ; x2 Þ  2g

Z

d5 zG0aa ðx1 ; zÞðS ab ð0Þ  dab TrS 1 ð0ÞÞG0bb  ðz; x2 Þ:

1205

ð35Þ

Using the definition of S1(x  y) given in Eq. (26), yields the positive and negative-mass contributions to S ab ð0Þ,

1 4 5  Z Z ðc p þ kÞ  12 c5 ab 1 3 5 2mc d p dp S ab ð0; mÞ ¼ : e 3 p5  Ec þ i 2m ð2pÞ c Taking into account the identity, 1 1 ¼ P  ipdðxÞ; x  ie x where e > 0 and P denotes the principal value, non-diagonal elements of S ab ð0; mÞ and S ab ð0; mÞ can be shown to be equal in magnitude and opposite in sign, so that they cancel each other, and S ab ð0Þ ¼ S ab ð0; mÞ þ S ab ð0; mÞ is diagonal, S ab ð0Þ ¼ 14dab TrS 1 ð0Þ;

ð36Þ

where i TrS 1 ð0Þ ¼ 2TrS 1 ð0; mÞ ¼  pffiffiffi 2 2p3

Z

d3 p

is a divergent quantity which can be made finite with a cutoff. This serious divergence problem may be resolved by including a momentum-dependant vertex function. For example, if the vertex function decreases rapidly with momentum, then this integral can be convergent and finite. Such a form may be anticipated for any realistic formulation of the problem. With Eq. (36), the first-order 2-point function can be rewritten as ð1Þ

Gab ðx1 ; x2 Þ ¼ G0ab ðx1 ; x2 Þ þ Gab ðx1 ; x2 Þ; where ð1Þ Gab ðx1 ; x2 Þ

 R1 ð0Þ

Z

d5 zG0aa ðx1 ; zÞG0ab ðz; x2 Þ

and R1 ð0Þ 

3g TrS 1 ð0Þ: 2

It is represented diagrammatically in Fig. 1. The first diagram shows the 2-point function for a free Galilean Dirac field. In the second diagram, R1(0) is represented by a closed loop with one vertex on it. The second order in g contribution to the 2-point function is

Fig. 1. Diagrams for the first order 2-point function for the quartic potential.

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M. de Montigny et al. / Annals of Physics 323 (2008) 1191–1214 ð2Þ Gab ðx1 ; x2 Þ

g2 ¼ 2

Z

d2 d zd w  ; ðT ðz; wÞ  2T 0 ðzÞT ðwÞÞ dJ a ðx1 ÞdJ b ðx2 Þ J ¼0¼J 5

5

that is the sum of three terms ð2Þ Gab ðx1 ; x2 Þ

¼g

2

Z

d5 zd5 w½4G0ad ðx1 ; wÞG0da ðw; zÞCab ðz; wÞG0bb  ðz; x2 Þ

þ 3iTrS 1 ð0ÞG0aa ðx1 ; zÞCab ðz; wÞG0bb  ðz; x2 Þ

ð37Þ

9 2 þ ðTrS 1 ð0ÞÞ G0aa ðx1 ; zÞG0ab ðz; wÞG0bb  ðw; x2 Þ: 4 The corresponding diagrams are shown in Fig. 2. The function Cab ðz; wÞ in Eq. (37) is defined as Cab ðz; wÞ  G0ad ðz; wÞG0db ðw; zÞ þ dab G0cd ðz; wÞG0dc ðw; zÞ;

ð38Þ

being represented by a closed loop with two vertices. Each of the vertices can have up to two external lines. However, for the first diagram in Fig. 2, one leg of the vertex at w is joined to a leg of the vertex at z creating an internal line, so that we have one external line at each vertex. At the second diagram, two legs of the vertex at w are joined together, producing R1(0). From Eq. (38), we deduce the following relations for Cab ðz; wÞ: Caa ðz; wÞ ¼ 3G0cd ðz; wÞG0dc ðw; zÞ and Cab ðz; zÞ ¼ 

3 2 d  ðTrS 1 ð0ÞÞ : 16 ab

The second diagram in Fig. 1 and the first two in Fig. 2 are one-particle irreducible; they cannot be disconnected by cutting through any one internal line. The third diagram in Fig. 2 is a chain of two first-order one-particle irreducible graphs. ð2Þ To simplify the expression for Gab ðx1 ; x2 Þ, Eq. (37), we can factor the fifth coordinate 0 out of the functions G and C and then perform integrations over z5 and w5. The function Cab ðz; wÞ is factorized as follows: Cab ðz; wÞ ¼ C0ab ðz; w; z4 ; w4 ; mÞ þ e2imcðz þ e2imcðz

5 w5 Þ

ðÞ

5 w5 Þ

Cab ðz; w; z4 ; w4 ; mÞ;

ðþÞ

Cab ðz; w; z4 ; w4 ; mÞ ð39Þ

Fig. 2. Diagrams for the g2-order of the 2-point function for the quartic potential. The third diagram is not oneparticle irreducible. The contribution of second diagram vanishes.

M. de Montigny et al. / Annals of Physics 323 (2008) 1191–1214

1207

where C0ab ðz; w; z4 ; w4 ; mÞ  ½G0ad ðz; w; z4 ; w4 ; mÞG0db ðw; z; w4 ; z4 ; mÞ þ G0ad ðz; w; z4 ; w4 ; mÞG0db ðw; z; w4 ; z4 ; mÞ þ dab ½G0cd ðz; w; z4 ; w4 ; mÞG0dc ðw; z; w4 ; z4 ; mÞ þ G0cd ðz; w; z4 ; w4 ; mÞG0dc ðw; z; w4 ; z4 ; mÞ

ð40Þ

is that part of Cab ðz; wÞ which does not oscillate in the fifth coordinates, while ðþÞ

Cab ðz; w; z4 ; w4 ; mÞ  G0ad ðz; w; z4 ; w4 ; mÞG0db ðw; z; w4 ; z4 ; mÞ þ dab G0cd ðz; w; z4 ; w4 ; mÞG0dc ðw; z; w4 ; z4 ; mÞ and ðÞ

ðþÞ

Cab ðz; w; z4 ; w4 ; mÞ  Cab ðz; w; z4 ; w4 ; mÞ: Integrating both parts of Eq. (39) over w5 (or z5) and using the limit Z 1 l=2 5 2imcw5 sinð2mclÞ ¼ 0; liml!1 dw e ¼ liml!1 l l=2 mcl valid for nonzero values of m, we obtain Z dw5 Cab ðz; wÞ ¼ C0ab ðz; w; z4 ; w4 ; mÞ;

ð41Þ

i.e. the oscillating parts of Cab ðz; wÞ do not contribute to the integral. Using the expressions given by Eqs. (28) and (39), we find that the product G0ad ðx1 ; wÞG0da ðw; zÞCab ðz; wÞG0bb  ðz; x2 Þ; in the first term of the right-hand side of Eq. (37), has the following non-oscillating parts in z5 and w5: 5

5

eimcðx1 x2 Þ G0ad ðx1 ; w; x41 ; w4 ; mÞG0da ðw; z; w4 ; z4 ; mÞ 4 4  C0ab ðz; w; z4 ; w4 ; mÞG0bb  ðz; x2 ; z ; x2 ; mÞ 5

5

þ eimcðx1 x2 Þ G0ad ðx1 ; w; x41 ; w4 ; mÞG0da ðw; z; w4 ; z4 ; mÞ 4 4  C0ab ðz; w; z4 ; w4 ; mÞG0bb  ðz; x2 ; z ; x2 ; mÞ;

so that the corresponding integral can be written as sum of positive- and negative-mass contributions. Let us represent G0ad ðx1 ; w; x41 ; w4 ; mÞ by a line in the (3 + 1) space–time, with an arrow pointed in the direction in which the particle is moving, i.e. from x1 to w, and G0ad ðx1 ; w; x41 ; w4 ; mÞ by a line again running from x1 to w and carrying an arrow in the opposite direction, as in Fig. 3. When the Green’s function G0ab , defined in (4 + 1)-dimensions, is reduced to (3 + 1) Galilean space–time, it contains two parts, one for positive energy and the other for negative energy. If we represent the (4 + 1)-dimensional Green’s function by a simple line, and the (3 + 1)-dimensional Green’s functions by a line containing an arrow, then the relation between the (4 + 1)-dimensional Green’s function and the positive- and negative-energy/

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Fig. 3. Green’s functions in (3 + 1) dimensions for (a) positive and (b) negative energy/mass.

mass contributions in (3 + 1) space–time diagrams is represented as in Fig. 4. The symmetry of the two-point Green’s function under interchange of x1 and w is obvious. For instance, the total contribution of the first one-particle-irreducible second-order diagram (in (4 + 1) space–time) in Fig. 2 contains four diagrams after reduction to (3 + 1) space–time. These diagrams are shown in Fig. 5. Although the number of diagrams increases after the reduction to (3 + 1)-dimensions, a clear interpretation in terms of particles and antiparticles becomes possible. The first two diagrams with external lines being particles represent the second-order positive-mass contribution to the 2-point function, while two others with external lines being antiparticles represent the second-order negative-mass contribution. Performing the integration over w5 in the second term of the right-hand side of Eq. (37) as well and using Eq. (41), this gives Z Z Z d5 wCab ðz; wÞ ¼ d3 w dw4 C0ab ðz; w; z4 ; w4 ; mÞ: It is clear from the expression for C0ab ðz; w; z4 ; w4 ; mÞ, Eq. (40), that the positive- and negative-mass Green’s functions contribute separately. Calculating these contributions, we have Z Z 1 ðc4 Þab TrS 1 ð0Þ; d3 w dw4 G0ad ðz; w; z4 ; w4 ; mÞG0db ðw; z; w4 ; z4 ; mÞ ¼  8mc that results in Z Z d3 w dw4 C0ab ðz; w; z4 ; w4 ; mÞ ¼ 0: Therefore, the second diagram in Fig. 2 does not contribute. Let us write the sum of one-particle-irreducible graphs for 2-point function in all orders in g as Z d5 zd5 wG0aa ðx1 ; zÞRab ðz; wÞG0bb  ðw; x2 Þ;

Fig. 4. Positive- and negative-energy/mass contributions to the Green’s function through dimensional reduction.

M. de Montigny et al. / Annals of Physics 323 (2008) 1191–1214

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Fig. 5. Contributions in (3 + 1) space–time of the leftmost diagram in Fig. 2.

with two external lines and one self-energy subgraph Rab ðz; wÞ. To order g2, we have Rab ðz; wÞ ¼ dab dð5Þ ðz  wÞR1 ð0Þ þ 4g2 G0ad ðw; zÞCdb ðz; wÞ þ Oðg3 Þ: Then the complete 2-point function is given by a sum of chains of one, two, and more of these subgraphs connected with the free field propagators Z Z Gab ðx1 ; x2 Þ ¼ G0ab ðx1 ; x2 Þ þ d5 zd5 wG0aa ðx1 ; zÞRab ðz; wÞG0bb ðw; x Þ þ d5 zd5 w  2 Z  ÞG0db ð  G0aa ðx1 ; zÞRab ðz; wÞG0bd  d5zd5 w zÞRdd ðz; w w ; x2 Þ þ    ;  ðw;  Z ð42Þ ¼ G0ab ðx1 ; x2 Þ þ d5 zd5 wG0aa ðx1 ; zÞRab ðz; wÞGbb  ðw; x2 Þ: Introducing the Fourier transform Rab(p) and Gab(p) of the functions Rab(x1,x2) and Gab(x1,x2) in the same way as the Fourier transforms of the Galilean propagators in Eqs. (25) and (26), we rewrite Eq. (42) in (4 + 1)-dimensional momentum space as iGab ðpÞ ¼ S ab ðpÞ þ S aa ðpÞRab ðpÞGbb  ðpÞ; that results in the exact expression for Gab(p): 1

Gab ðpÞ ¼ ½iS 1 1 ðpÞ  RðpÞab : This expression is similar to the case of many-body systems where the exact expression of the 2-point function depends on self-energy. 4.2. 4-point function The 4-point function G(x1, x2; y1, y2) is given by Eq. (32) with Z 0 ½J ; J replaced by Z½J ; J. To find its irreducible part, we can use a generating functional W ½J ; J, which generates only connected Feynman diagrams or connected Green’s functions. It is related to Z½J ; J as W ½J ; J ¼ i ln Z½J ; J: We define the irreducible or connected 4-point function as

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 1 ; x2 ; y 1 ; y 2 Þ ¼ Gðx

1 d4 W ½J ; J  ; W ½0 dJ ðx1 ÞdJ ðx2 ÞdJ ðy 1 ÞdJ ðy 2 Þ J ¼0¼J

ð43Þ

 1 ; x2 ; y 1 ; y 2 Þ and the complete 4-point functhat gives us the following relation between Gðx tion G(x1, x2; y1, y2):  abcd ðx1 ; x2 ; y 1 ; y 2 Þ ¼ iGabcd ðx1 ; x2 ; y 1 ; y 2 Þ þ i½Gad ðx1 ; y 2 ÞGbc ðx2 ; y 1 Þ G  Gac ðx1 ; y 1 ÞGbd ðx2 ; y 2 Þ:

ð44Þ

To order g0, the complete 4-point function contains only reducible parts, G0abcd ðx1 ; x2 ; y 1 ; y 2 Þ ¼ G0ad ðx1 ; y 2 ÞG0bc ðx2 ; y 1 Þ  G0ac ðx1 ; y 1 ÞG0bd ðx2 ; y 2 Þ:

ð45Þ

This represents the Hartree-Fock part of the 4-point function. All remaining parts  0 defined in Eq. (44), we find include interaction among the particles. Using G 0  that Gabcd ðx1 ; x2 ; y 1 ; y 2 Þ ¼ 0. The diagrams corresponding to Eq. (45) are shown in Fig. 6. The irreducible parts appear in the first order in g, Z  ð1Þ ðx1 ; x2 ; y 1 ; y 2 Þ ¼ 2g dz5 ½G0 ðx1 ; zÞG0 ðz; y 2 ÞG0  ðx2 ; zÞG0 ðz; y 1 Þ G  abcd a a ad bb bc  ððd; y 2 Þ $ ðc; y 1 ÞÞ; where (d, y2) M (c, y1) means that there is an additional term in the square brackets, which can be obtained from the first one by replacing (d, y2) with (c, y1) and vice versa. These parts are represented diagrammatically in Fig. 7. The second order in g contribution to the irreducible 4-point function is  ð2Þ ðx1 ; x2 ; y 1 ; y 2 Þ ¼ i½Gð1Þ ðx2 ; y 1 ÞGð1Þ ðx1 ; y 2 Þ  Gð1Þ ðx1 ; y 1 ÞGð1Þ ðx2 ; y 2 Þ G abcd bc ad bd ac 4 2 Z g d d5 zd5 w  þi 2 dJ a ðx1 ÞdJb ðx2 ÞdJ c ðy 1 ÞdJ d ðy 2 Þ  ðT ðz; wÞ  2T 0 ðzÞT ðwÞÞjJ ¼0¼J

Fig. 6. Diagrams for g0-order of the 4-point function for the quartic potential.

M. de Montigny et al. / Annals of Physics 323 (2008) 1191–1214

1211

Fig. 7. Diagrams for g1-order of the irreducible 4-point Green’s function for the quartic potential.

that can be rewritten as  ð2Þ ðx1 ; x2 ; y 1 ; y 2 Þ ¼ 2g G abcd

Z

ð1Þ

d5 w½G0aa ðx1 ; wÞG0ad ðw; y 2 ÞG0bb ðx2 ; wÞGbc  ðw; y 1 Þ ð1Þ

þ G0aa ðx1 ; wÞG0ad ðw; y 2 ÞGbb ðx2 ; wÞG0bc  ðw; y 1 Þ ð1Þ

þ G0aa ðx1 ; wÞGad ðw; y 2 ÞG0bb ðx2 ; wÞG0bc  ðw; y 1 Þ ð1Þ

þ Gaa ðx1 ; wÞG0ad ðw; y 2 ÞG0bb ðx2 ; wÞG0bc  ðw; y 1 Þ Z 0 0  4ig2 d5 zd5 w½G0aa ðx1 ; wÞG0bb ðx2 ; zÞCb c; a d ðz; wÞGcc ðz; y 1 ÞG dd ðw; y 2 Þ 0 þ G0aa ðx1 ; wÞG0ad ðw; zÞG0dd ðz; y 2 ÞG0bb ðx2 ; wÞG0b c ðw; zÞGcc ðz; y 1 Þ

 ððd; y 2 Þ $ ðc; y 1 ÞÞ: Some diagrams representing these processes are shown in Fig. 8. Part (a) corresponds to the third line of the previous equation; there are three more similar diagrams with self-energy loop R1(0) on one of the remaining three legs. Part (b) represents line 6, and part (c) corresponds to line 5 of the equation above. The function Cb c; a d ðz; wÞ is defined as 0 0 Cb c; c ðz; wÞ  a d Cb a d ðz; wÞ  Gb ac ðw; zÞ þ d d ðz; wÞG 0 0 þ db  c C c Gcd ðz; wÞGdc ðw; zÞ; a d ðw; zÞ  3d a d db

ð46Þ

being represented by a closed loop with two vertices and four external lines, as in Fig. 8(c). This function has the following symmetry property Cb c; c ðw; zÞ: a d ðz; wÞ ¼ C a d;b

Fig. 8. Diagrams for g2-order of the irreducible 4-point Green’s function for the quartic potential.

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 ¼ c and summing over b,  this gives us Taking b 0 0 Cbb;  a a d ðw; zÞ  8d a d Gcd ðz; wÞGdc ðw; zÞ: d ðz; wÞ ¼ 3C

In a similar way, we obtain 0 0 Cb c; c ðz; wÞ  8db c Gcd ðz; wÞGdc ðw; zÞ: a a ðz; wÞ ¼ 3Cb

The polarisation part arise from the 4-point function and provides a sum of the loops to arbitrary order. The equivalence to the case of non-relativistic many-body systems interacting by two-particle interactions is obvious. Here we have parts that may be considered for particles and anti-particles. 5. Concluding remarks This paper is the continuation of our previous works on quantization of Galileancovariant field theories: path-integral quantization of complex scalar fields in Ref. [5] and the canonical quantization of both scalar and Fermi fields in Ref. [20]. The main purpose of this approach is to exploit relativistic tensorial techniques for applications to nonrelativistic many-body systems. It is also interesting to compare Lorentzian and Galilean theories. An example of a rather unexpected similarity is that the non-zero spin is also predicted within a Galilean framework coherently defined [24,25]. The presence of antiparticles is another example. However, there is no creation of particle and antiparticle pairs. It may be emphasized that the antisymmetrisation of the 4-point functions for fermions is also clearly respected. In addition to many familiar dissimilarities, some deserve to be emphasized, such as the existence of two Galilean formulations of electrodynamics [22,23]. We have discussed the Dirac equation on a (4 + 1) manifold and its reduction to the Le´vy-Leblond equations [24], and the coexistence of positive- and negative-energy/mass solutions [19]. While doing so, a representation of the Dirac matrices different from what is used earlier is presented, as well as the related spinors. After discussing the Galilean generating functional and Green’s functions for particles and antiparticles, we compute the 2and 4-point functions for the self-interacting quartic potential. From this study, we find that the following observations on the use of a (4 + 1)-dimensional Galilean space–time are in order. There exists a mass superselection rule, which prevents the creation of massive particles. This makes Yukawa coupling to massive particles irrelevant, because only couplings that involve at least four particles are allowed whereas Yukawa coupling to massless particles like the photon is possible. An open question concerns the parity operator in Galilean field theories. The Clifford algebra theory asserts that there is no parity operator analogous to c5 in any odd-dimensions. We are currently investigating the possibility to embed the Galilean space–time into a (5 + 1)-dimensional Minkowski manifold, for which a parity operator exists with natural 8-dimensional Dirac gamma matrices [27]. This extension will allow us to study the Galilean analogue of the Nambu-Jona-Lasinio model [28]. In Poincare´ covariant field theories in (3 + 1) dimensions, plpl = E2  p2c2 is an invariant and is equal to m2c4. This mass is an invariant quantity. However, in Galilean covariant field theories in (4 + 1) dimensions, we have that plpl = 2mE  p2 is an invariant that is set equal to a constant k2. It is important to emphasize that m appears as central charge in the Galilean algebra and this leads to the definition of the five-momentum as

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ðp; E=c; mcÞ. Thus the renormalization process would affect the invariant k in the Galilean covariant theory. Finally, with the set up of the functional form for the path-integral approach, this would allow us to write down the transition amplitudes, hence the cross-sections, with the usual process of combining the square of the transition amplitude and the necessary phase space. The formulation as presented here has established contact with the usual perturbation theory for non-relativistic systems. However, it is important to emphasize that a covariant Galilean field theory is compatible with the idea of particles with energy E and mass +m, and antiparticles with energy E and mass m. Acknowledgment We acknowledge partial support by the Natural Sciences and Engineering Research Council of Canada. References [1] A.A. Abrikosov, L.P. Gorkov, I.E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics, Dover, New York, 1963; E.M. Lifshitz, L.P. Pitaevskii, Statistical Physics, Part 2: Theory of the Condensed State, Landau and Lifshitz Course of Theoretical Physics, vol. 9, Pergamon Press, Oxford, 1980; A.M. Tsvelik, Quantum Field Theory in Condensed Matter Physics, Cambridge University Press, New York, 2003. [2] A.L. Fetter, J.D. Walecka, Quantum Theory of Many-Particle Systems, McGraw-Hill, New York, 1971. [3] M. Greiner, C.A. Regal, D.S. Jin, Nature 426 (2003) 537. [4] J. Lukierski, P.C. Stichel, W.J. Zakrzewski, Phys. Lett. A 357 (2006) 1; M.A. del Olmo, M.S. Plyushchay, Ann. Phys. (NY) 321 (2006) 2830; P.A. Horvathy, L. Martina, P.C. Stichel, Phys. Lett. B 564 (2003) 149; P.A. Horvathy, M.S. Plyushchay, Nucl. Phys. B 714 (2005) 269; P.A. Horvathy, L. Martina, P.C. Stichel, Phys. Lett. B 615 (2005) 87; R. Jackiw, V.P. Nair, Phys. Lett. B 480 (2000) 237; R. Jackiw, V.P. Nair, Phys. Lett. B 551 (2003) 166; C. Duval, P.A. Horvathy, Phys. Lett. B 479 (2000) 284; C. Duval, P.A. Horvathy, J. Phys. A: Math. Gen. 34 (2001) 10097; C. Duval, P.A. Horvathy, Phys. Lett. B 547 (2002) 306. [5] L. Abreu, M. de Montigny, F.C. Khanna, A.E. Santana, Ann. Phys. (N.Y.) 308 (2003) 244. [6] M. de Montigny, F.C. Khanna, A.E. Santana, E.S. Santos, J.D.M. Vianna, Ann. Phys. (N.Y.) 277 (1999) 144. [7] M. de Montigny, F.C. Khanna, A.E. Santana, E.S. Santos, J.D.M. Vianna, J. Phys. A: Math. Gen. 33 (2000) L273; M. de Montigny, F.C. Khanna, A.E. Santana, E.S. Santos, J. Phys. A: Math. Gen. 34 (2001) 8901; M. de Montigny, F.C. Khanna, A.E. Santana, Int. J. Theor. Phys. 42 (2003) 649. [8] M. de Montigny, F.C. Khanna, A.E. Santana, J. Phys. A: Math. Gen. 36 (2003) 2009. [9] Y. Takahashi, Fortschr. Phys. 36 (1988) 63; Y. Takahashi, Fortschr. Phys. 36 (1988) 83. [10] M. Omote, S. Kamefuchi, T. Takahashi, Y. Ohnuki, Fortschr. Phys. 37 (1989) 933. [11] G. Pinski, J. Math. Phys. 9 (1968) 1927; D.E. Soper, Classical Field Theory, Wiley and Sons, New York, 1976, Sect. 7.3; C. Duval, G. Burdet, H.P. Ku¨nzle, M. Perrin, Phys. Rev. D 31 (1985) 1841; C. Duval, G.W. Gibbons, P. Horva´thy, Phys. Rev. D 43 (1991) 3907; H.P. Ku¨nzle, C. Duval, in: U. Majer, H.J. Schmidt (Eds.), Semantical Aspects of Spacetime Theories, BI-Wissenschaftsverlag, Mannheim, 1994, p. 113; C. Duval, P. Horva´thy, L. Palla, Ann. Phys. (NY) 249 (1996) 265.

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