VOLUME 88, NUMBER 18
6 MAY 2002
PHYSICAL REVIEW LETTERS
Ghost Fields in Classical Gauge Theories Wim Schoenmaker,* Wim Magnus, and Peter Meuris IMEC, Kapeldreef 75, B-3001 Leuven, Belgium (Received 1 February 2002; published 23 April 2002) It is shown that ghost fields, characterized as unphysical entities, are a valuable tool in finding numerical solutions of the Euler-Lagrange equations of gauge field theories. DOI: 10.1103/PhysRevLett.88.181602
Perturbative quantization of field theories with a local gauge symmetry requires the inclusion of ghost fields in order to evaluate the Fadeev-Popov determinant. The perturbative expansion of the S matrix leads to closed loops of ghost particles; i.e., the ghost particles do not appear in the physical in and out states of the S matrix. To summarize the situation, ghost fields are unphysical mathematical entities that are useful for the perturbative quantization of gauge theories. From this observation, one might conclude that ghost fields do only play a role in the quantization procedure. However, in this Letter we will argue that ghost fields, being identified as unphysical mathematical entities, play an equally important role in solving the classical field equations of a gauge theory. We will illustrate our arguments in classical electrodynamics; however, our conclusion remains valid for more general gauge theories. Furthermore, we emphasize that the “classical” ghost fields gain importance for finding nonperturbative solutions of the Euler-Lagrange equations. The action of the electromagnetic field is Z 1 Z 4 2 d xF 1 d 4 x jA , (1) S0 苷 2 4 where F 苷 dA. This action needs to be extended with a gauge-fixing term in order to eliminate the zero modes that are generated by the local gauge invariance. We may select the Lorentz gauge, 1 Z 4 d x共≠A兲2 . (2) Sgf 苷 2
PACS numbers: 11.15.Kc, 41.20.Gz, 84.32.Hh, 85.40.Ls
Finally, following Veltman [3], we use the field x to perform a Bell-Treiman (gauge) transformation, such that a ghost-source interaction term appears: Z Sgs 苷 d 4 x j≠x . (5) The ghost-source interaction is “invisible” provided that the source is physical, i.e., ≠j 苷 0. The Euler-Lagrange equations corresponding to the action S 苷 S0 1 Sgf 1 Sghost 1 Sgs are ≠2 An 2 ≠n 共≠A兲 2
1 n ≠ x 苷 jn , G0
2≠2 x 1 ≠A 苷 2G0 ≠j .
(6) (7)
Acting with ≠n on Eq. (6), we obtain 2≠2 x 苷 G0 ≠j. Therefore, if the source is physical, then ≠2 x 苷 0, and according to Eq. (7) this implies that the Lorentz gauge, ≠A 苷 0, is recovered. Since x has become a dynamical variable, we may identify x as a ghost field provided that it has no physical effect. In particular, the ghost field should not induce a shift of the source in Eq. (6). Therefore, the solution of Eq. (6) and (7) should lead to x being constant. The latter can be realized by choosing appropriate boundary conditions. As a consequence, the solution for A of the pair of Eqs. (6) and (7) will be identical to the solution of the usual equation, ≠2 An 苷 j n .
(8)
where G0 is a constant of dimension 关length兴2 to match the dimensions. We have chosen x dimensionless. The Lagrange multiplier field can be elevated to a dynamical variable, as was done by Lautrup [1] and Nakanishi [2] for obtaining a covariant quantization procedure of electrodynamics in Hilbert spaces with an indefinite metric. Then we should add to the action the ghost contribution, 1 Z 4 Sghost 苷 2 d x x≠2 x . (4) 2G0
We will now argue that including the dynamical ghost field in the equations of motion is beneficial, despite the fact that an additional degree of freedom and an additional equation must be considered. Whereas Eq. (8) can be formally solved using Green functions, there are many circumstances where the sources depend on the field A, as is the case for non-Abelian gauge theories as well as for many problems in magnetohydrodynamics and electrical engineering. The Green function has only limited value and, in order to find nonperturbative solutions, a computational approach is preferred. The setup of the computation is based on a series of straightforward restrictions due to memory limitations: (i) Only a finite volume of space-time, V, can be submitted to the calculation; (ii) only a finite number of grid nodes can be allocated; and (iii) the grid nodes are separated from each other by
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© 2002 The American Physical Society
An alternative way to introduce the gauge fixing is by adding a Lagrange multiplier field x, such that 1 Z 4 d x x≠A , Sgf 苷 (3) G0
0031-9007兾02兾 88(18)兾181602(4)$20.00
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PHYSICAL REVIEW LETTERS
a finite distance. Since the gauge field A represents a connection in the differential-geometrical meaning, it is a one-form and, on a computational grid, its values should be assigned to the links, connecting neighboring grid points. Whereas Wilson [4] and later R authors [5] assign the group elements U共ᐉ兲 苷 expi ᐉ Adx toRthe links of the grid, we assign the Lie-algebra element ᐉ Adx to the links. It should be emphasized that we attempt to solve the Euler-Lagrange equations that take the form of partial differential equations for the gauge potentials. Therefore, the Lie-algebra variables are the key ingredients of the classical theory. Since we determine the gauge potential (and not the field strengths), our method is different from the finite-integration technique (FIT) [6]. Our method also differs from the finite-integration implicit-time domain (FI2TD) [7], in its regularized version as given by Clemens and Weiland [8,9], since that scheme lacks the use of the ghost field. We consider a rectangular grid in D dimensions having N nodes in each direction. The grid has Mnodes 苷 N D nodes and Mlinks 苷 DN D 关1 2 共1兾N兲兴 links. As a conse-
quence, there are Mlinks link variables for which we must construct algebraic equations. Setting Am 苷 共f, A兲 and j m 苷 共r, j兲, we obtain the following set of Euler-Lagrange equations: ∂ µ ≠2 A 1 ≠f =3=3A1 1 1= =x 2 j 苷 0 , ≠t 2 ≠t G0 (9) ∂ µ 1 ≠x ≠A 2 2 r 苷 0, (10) 2=2 f 2 = ? ≠t G0 ≠t 2
≠2 x ≠r ≠f 1 = ? A 1 G0 1 =2 x 1 ≠t 2 ≠t ≠t 1 G0 = ? j 苷 0 .
(11)
In order to discretize Eq. (9), we consider a link ᐉ in the grid. This link is in general an edge of four spacelike cubes in the grid. With DSᐉ being the spacelike surface element perpendicular area of the link, the discretization of Eq. (9) starts with a “smearing” of the continuous equation over the surface element: ∂∏ ∑ µ µ ∂ Z Z Z ≠2 A ≠f 1 2 2 2 1 d x eᐉ ? 共= 3 = 3 A兲 1 d x eᐉ ? d x eᐉ ? 1= =x 2 j 苷 0 , (12) ≠t 2 ≠t G0 DSᐉ DSᐉ DSᐉ
where eᐉ is a unit vector in the direction of the link ᐉ. After application of Stokes’ theorem, the first term of Eq. (12) can be decomposed as a sum over the links of the spacelike facets (plaquettes) that have the link ᐉ as edge [10]. Therefore, 13 Z X dx 2 eᐉ ? 共= 3 = 3 A兲 ⯝ Wᐉ,; A;, (13) DSᐉ
;苷1
where ⯝ denotes the conversion to the algebraic equation, and Wᐉ,; are geometrical weight factors. The second integral in Eq. (12) can also be discretized as a sum over links provided that the potential f is assigned to links in the time direction, ∂∏ ∑ 2 µ 1 1 Z X X X ≠ A ≠f 2 ⯝ d x eᐉ ? W A 共t 兲 1 1 = Wᐉ,k,i fk 共tn1i 兲 . (14) ᐉ,tn1i ᐉ n1i ≠t 2 ≠t DSᐉ i苷21 k苷n0 ,n1 i苷0 In here, n0 and n1 are the begin and end points of the link, where Wı; and Vjk are sparse matrices. The discretizaᐉ, and Wᐉ,k,i are geometrical factors of the space-time grid. After identifying f 苷 A0 , the discretization of Eq. (14) tion of Eq. (10) can be achieved by considering spacelike volume elements DVi around each node, j: corresponds to the circulations around the timelike facets ∂ µ that have the link ᐉ as an edge. The third term in Eq. (12) Z 1 ≠x ≠ 3 2 共= ? A兲 2 2 r 苷 0. d x 2= f 2 can be discretized by projection of the variables on the link ≠t G0 ≠t DVj ᐉ. For that purpose, we assign the ghost variables to the (18) nodes. By setting jᐉj the length of the link, ᐉ, we obtain The first term in (18) can be evaluated as Z DSᐉ Z X 共xn1 2 xn0 兲 . d 2 x eᐉ ? =x ⯝ (15) jᐉj d 3 x共2=2 f兲 ⯝ Wj,j1kˆ fj1kˆ 共tn 兲 . (19) DSᐉ Furthermore, the current of the link ᐉ can be evaluated by defining Jᐉ 苷 j ? eᐉ and constant, resulting in Z d 2 x eᐉ ? j ⯝ DSᐉ Jᐉ . (16) DSᐉ
Collecting all algebraic equations corresponding to Eq. (12), one obtains NX links ;苷1
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Wı; A; 1
Nnodes 1 X Vjk x k 2 Jı 苷 0 , G0 k苷1
(17)
DVj
kˆ
The second term of (18) becomes ∂ X µ Z ≠ 3 d x 2 共= ? A兲 ⯝ ≠t DVj ᐉ
X i苷21,0
Wᐉ,i Aᐉ 共tn1i 兲 , (20)
where the sum over ᐉ runs over all spatial links that have the node j as the begin point. The sum of (19) and (20) corresponds to all circulations around the timelike facets that have the link assigned to fj 共tn 兲 苷 A0ᐉ as an edge. Finally, the remaining terms can be evaluated by assigning r 181602-2
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FIG. 1. The sparseness pattern for link-to-link coupling is generated by the “common-wing” property of the links.
to the links in the time direction and by the finite difference ≠x approximation ≠t 苷 关xj 共tn 兲 2 xi 共tn21 兲兴兾Dt. Again collecting all algebraic equations corresponding to (10), one obtains Nnodes NX links 1 X Wı;0 A; 1 Vjk0 x k 2 rı 苷 0 , (21) G 0 k苷1 ;苷1 where the superscript zero reminds us that timelike directions are considered. In summary, the discretization of Am 苷 共f, A兲 ! Aᐉ 苷 Am nˆ m leads to the following set of algebraic equations: NX Nnodes links 1 X Wı; A; 1 Vjk x k 2 jı 苷 0 , (22) G 0 ;苷1 k苷1 where the sparseness of W is determined by the facets having both the links ı, ; as edges as is depicted in Fig. 1, and the sparseness of V is determined by the condition that j, k should be the begin and end points of the link ı. The discretization of Eq. (11) gives rise to a matrix equation as given below: NX NX links nodes 共Tı; A; 1 G0 Sı; j; 兲 苷 0 . Ujk x k 1 (23) k苷1
;苷1
The sparseness of the matrix U is determined by nearest-neighbor coupling, whereas the sparseness of the matrices T and S for each node, j, is determined by all links leaving the node j. This is depicted in Fig. 2. Equations (22) and (23) can be combined into a single equation, M ⴱ A 1 J 共A兲 苷 0, where ∑ ∏ ∑ ∏ A W V 兾G0 M 苷 , A苷 , T U x ∏ ∑ (24) 2j . J 共A兲 苷 G0 Sj The dimension of the sparse matrix M is dim共M兲 苷 共Nlinks 1 Nnodes 兲 3 共Nlinks 1 Nnodes 兲. Furthermore, the matrix, M, is also regular. It should be noted that the matrix W is singular since it contains the zero modes of the underlying gauge invariance. The singularity of W , or the interdependency of the rows, is compensated by the gauge condition; i.e., without use of the ghost field, the Euler-Lagrange equations are given by ∑ ∏ ∑ ∏ W j共A兲 ⴱ 关A兴 苷 . (25) T 0 181602-3
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PHYSICAL REVIEW LETTERS
FIG. 2. The sparseness pattern for node-to-node coupling and node-to-link coupling is generated by the four-dimensional “star.”
However, this matrix formulation is not square and not suitable for iterative solution methods that are indispensable for solving large sparse systems. Thus, the inclusion of the ghost field opens the gateway to exploit these iterative methods. Furthermore, the parameter G0 allows one to adapt the condition number of the matrix such that convergence properties improve. As far as the boundary conditions are concerned, in order not to generate unphysical sources due to oscillations in the ghost field, the largest spatial size Lmax 苷 max兵Lx , Ly , Lz 其 should not exceed the temporal size Lt 苷 1兾nmax ; i.e., Lmax nmax , 1兾2. In order to illustrate our method, we will compute the magnetic energy of a microelectronic on-chip spiral inductor of two windings with time-independent voltage conditions at the ends. The thickness of the wire is 2 3 2 mm2 . An under path with height of 2 mm is used to connect the inner winding. The structure is depicted in Fig. 3. The time independence implies that a three-dimensional problem must be solved, yet the problem nicely illustrates how the use of a ghost field converts a singular problem into a regular one. The full set of equations is 1 =x 2 j 苷 0, 2=2 f 2 r 苷 0 , G0 =2 x 1 = ? A 1 G0 = ? j 苷 0, j 苷 sE , (26)
=3=3A1
E 苷 2=f,
B 苷 = 3 A,
and are discretized as described above. The time independence also implies that the Poisson equation decouples and as a consequence only the ghost-field equation and
FIG. 3. On-chip spiral inductor with under path to inner winding.
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PHYSICAL REVIEW LETTERS
TABLE I. Energy calculations of the spiral inductor.
25
130
20
110
Electric energy (J) E1elec E2elec
Magnetic Energy (J) magn E1 magn E2
217
1.11 3 10 9.89 3 10218
3.15 3 10212 3.24 3 10212
90 70 10
50 30
O共10214 兲, and is in the range of the numerical accuracy in double precision. The method of this Letter can be generalized to the non-Abelian theories. The number of ghost fields is equal to the dimension of the Lie algebra; i.e., for each gauge field, there is defined a ghost variable. The generalized form of Eqs. (6) and (7) are :
10 0.5 0.5
10
20
Dm F mn 2
25
FIG. 4. Magnetic field strength in the plane of the spiral inductor (nT).
the curl-curl equation need to be solved simultaneously. The boundary condition xj≠V 苷 0 suffices to guarantee that only physical solutions are obtained. The boundary condition Aj≠V 苷 0 corresponds to B⬜ ≠V 苷 0. Fi⬜ nally, we set E≠V 苷 0 except for the contacts of the wire for which we apply Dirichlet’s boundary conditions: fjC1 苷 f1 and fjC0 苷 0. With a voltage difference of 0.1 V at the contact and a resistivity of 1028 Vm, the current is 0.414 A. The spiral inductor is placed in a simulation domain with the size of 25 3 25 3 16 mm3 . The grid-cell size is 1 mm3 . In Fig. 4, the strength of the magnetic field is shown in the plane of the spiral inductor. Since we have calculated the potentials, the electric and magnetic energy can be obtained numerically in two 1R elec different ways. We can evaluate E dy rf as 苷 1 2 V R 1 well as E2elec 苷 2 e0 V dy E 2 . For the magnetic energy, magn magn 1R 苷 2 V dy J ? A as well as E2 苷 we may use E1 R 1 2 m dy B . In Table I, we collected the data for the 2 0 V spiral inductor. The inductance of the spiral inductor is 3.78 3 10211 H. The solution for the ghost field x ⯝ 2
≠ xa ≠xc 1 gcabc Ab ? =xc 苷 G0 2 =2 xa 1 gcabc fb 2 ≠t ≠t Using these equations, one may address the of question whether the classical solutions of Yang-Mills field theory exhibit a signature of confinement. For that purpose, one should discretize Eqs. (31) and (32), as was done for the electromagnetic fields.
*Electronic address:
[email protected] [1] B. Lautrup, Mat. Fys. Medd. K. Dan. Vidensk. Selsk. 35, No. 11 (1967). [2] Nakanishi, Prog. Theor. Phys. 35, 1111 (1966).
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1 n ≠ x 苷 Jn, G0
(27)
Dn ≠n x 1 ≠n An 苷 2G0 Dn J n .
(28)
Using the conventions of g00 苷 1, gii 苷 21, and 关Li , Lj 兴 苷 icijk Lk , the Maxwell-Yang-Mills field strengths are Ea 苷 2
≠Aa 2 =fa 2 gcabc fb Ac , ≠t
(29)
1
Ba 苷 = 3 Aa 2 2 gcabc Ab 3 Ac .
(30)
The generalized Gauss’ law is = ? Ea 2 gcabc Ab ? Ec 2
1 ≠x 苷 ra , G0 ≠t
(31)
whereas the Maxwell-Ampere equation is ≠Ea 2 gcabc fb Ec 2 gcabc Ab 3 Bc ≠t 1 1 1 =xa 苷 Ja 2 gcabc Ab 3 Bc 1 =xa 苷 Ja . G0 G (32)
= 3 Ba 2
∑
The generalized Lorentz gauge takes the following form: ∏ ≠ra 1 gcabc fb rc 1 = ? Jc 1 gcabc Ab ? Jc . (33) ≠t
M. Veltman, Nucl. Phys. B7, 637 (1968). K. G. Wilson, Phys. Rev. D 10, 2445 (1974). W. D. Gropp, J. Comput. Phys. 123, 254 (1996). K. S. Yee, IEEE Trans. Antennas Propag. 14, 302 (1966). M. Clemens and T. Weiland, IEEE Trans. Magn. 35, 1163 (1999). [8] T. Weiland, Part. Accel. 17, 227 (1985). [9] M. Clemens and T. Weiland, in Proceedings of the 13th Conference on Computation of Electromagnetic Fields, Lyon-Evian, Compumag, 2000 [IEEE Trans. Magn. (to be published)]. [10] P. Meuris, W. Schoenmaker, and W. Magnus, IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 20, 753 (2001). [3] [4] [5] [6] [7]
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