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A Yang-Mills type of equation for the compatibility conditions
0020-7225/89 $3.00+ 0.00 Pergamon Press plc
Int. J. Engng Sci. Vol. 27, No. 11, pp. 1439-1442.1989
Copyright @I 1989
Printed in Great Britain. All rights reserved
LETTERS
IN APPLIED
AND ENGINEERING
SCIENCES
A YANG-MILLS TYPE OF EQUATION FOR THE COMPATIBILITY CONDITIONS JANUSZ
BADUR
Institute of Fluid-Flow Machinery of the Polish Academy of Sciences, ul. Gen. J. Fiszera 14, 80-952 GdaAsk, Poland Abstract-It is shown that non-classical compatibility gauge-invariant Yang-Mills field strength tensor.
1. THE YANG-MILLS
conditions
PRIMAL
for strains
have a form of
CHAIN
A non-Abelian model of strong interactions, based on the local three-parameter gauge symmetry group SU(2), was first introduced by Yang and Mills [l]. Three massless vector gauge potentials, with values in the algebra of the group SU(2), played the role of intermediate bosons in this model. Since the three-parameter group of rotations O(3) is homomorphic to the unimodular unitary group SU(2), such that to every rotation from O(3) there correspond two matrices u and -u of SU(2) and, conversely, to every element u E SU(2) there corresponds some rotation from O(3), Yang and Mills adopted the three-dimensional natural representation of the group O(3) as the space of the internal degrees of freedom (the so-called isotopic space). The objects in the isotopic space are indexed a, b, c = 1,2, 3, in contradistinction to those from the Minkowski space-time, the arena for spectacle with isotopic vectors and tensors as participants, where the indices are p, Y, p = 0, 1, 2, 3. The primal chain of the gauge field theory depends on the dimension of the extended objects interacting with each other via the gauge potentials and on the dimension of the space-time [3]. In the four-dimensional Minkowski space-time we have different schemes of the primal chain for such extended objects: instantons (O-dimensional in the space-time), solitons (O-dimensional in the space), strings and vortices (one-dimensional), domain walls (twodimensional) or bags (three-dimensional). It means that the type of the gauge potentials is connected with the type of its sources, that is, the singularities of various dimensions of the dual fields [2]. Hence the instanton implies a scalar potential, the monopole-a vector potential, the string-a skew-symmetric tensor potential, etc. The Yang-Mills primal chain can be written as follows [3]: + the pure gauge potential + the gauge potential + the field strength tensor j the Bianchi identities + the “magnetic current”. We shall restrict our presentation of Y-M theory only to the elements belonging to the primal chain. Let us represent the Y-M gauge potential via the isotopic vector b, [l] B, = b;T, = b,T,
b, = {b:, b;, b;}
(1)
where the skew-symmetric matrices T, = E,(~~) are the generators of the group of rotations with the structural constants [T,, Tb] = E,~~T~ being otherwise the three-dimensional Ricci alternators. The action of the structural constants on the isotopic vectors will be denoted by the sign “8” of the cross product in the isotopic space. The pure gauge potentials, the field strength tensor and the Bianchi identities are defined as follows [l, 41 (the coupling constant equal to 1): ,y = -aVgg-l, g(x) l O(3) Fpy = [D,, Dvl= a$,
- a,B, + [B,, B,] = &yT
f,, = a,b, - ii’,b, + b,@b,
DpFv,+ DvFp,+ D&v = kpvp
(2) 1439
1440
Letters in Applied and Engineering Sciences
where the derivative Q = a,, + [B,,, a] = ($1 + b,@)T in (2), transforms covariantly under gauge transformations, whereas the potential and the field strength tensor according to the rules B, -+gB,g-’
- $gg-”
4, + g&g-’
- g[a,,
ag-1 = gtz;,,
- Gag-~
(3)
The Bianchi identities represented via the dual tensor E@”= ~E~“~“F~~and dual “magnetic” current k” = esvpAkppA = &‘T take the form DpPpv = dpEPV + [B,, i-]
= k”
(4)X
and via fP”‘, the form a ccf”‘_tb ccig@‘y=~v
(4)~
Let us express the independent components of the field strength tensor fpv = -fvp through the decomposition into the “electric” part E, = f&, i = 1, 2, 3 and the “magnetic” part Bi = i&ijkf’&in the form of “double” vectors in the space and the isotopic space E, Jj &Z= -a,& + b&3& - grad b,, ~=roth+~@C%)xb
(5)
where 3, = {-C$,, V}, d” = {&, V) when the light velocity c = 1. The Bianchi identities, expressed through &, @, in the rest frame, assume the form of the non-Abelian Maxwell equations with the “magnetic” current D +B =I:div B + bi@B’ = b Dxli:+D,B=rotE+(b~)xE+a,B-b,~B=k
(6)
where k’= {4,, k),
D,, = &,l - (b&3).
B = v1+ (b@),
The presence of the singular gauge transformations, that is those transformations for which GPY=g-‘[a,,, &]g #O, leads to the occurence of an additional field G,,, in the equations for the field strength tensor (3)* and for the Bianchi identities. This field does not vanish only on the Dirac strings [5]. In this case the Bianchi identities are breaked only in points where “magnetic” current kc”= a,G”” + [B,, G’““] which links the singularity current Gp’(x) = ;
~~“~*d~~ = j- dr j+ dW%F’S4(~ -
y(r, a))T,
does not vanish. In the above equation t and CI are time-like and space-like parameters of the two-dimensional world-sheet trace out by one-dimensionally singularity, oPv =--aYfi?Yv
aTa
(7) invariant string of
aY@aYv
adt
is the normal vector of the world-sheet, y “(r, o)-the position vector of the string and vu, a = 1,2,3 is a unit vector orienting the string in the isotopic space.
2. THE
COMPATIBILITY
CONDITIONS
FOR
STRAINS
The local compatibility conditions of an elastic continuum, based on the metric model of a simple body with translational degrees of freedom, are usually understood as the condition for
1441
Letters in Applied and En~nee~ng Sciences
vanishing of the Riem~n-Ch~stoffel curvature tensor. These conditions should be interpreted as the conditions imposed on the metric created by a given deformation tensor. In the theory of plastic deformation the condition for vanishing of the R-C tensor loses validity, due to the occurrence of dense distribution of singular objects similar to G,, in (7). In [6] it was shown, that the conditions imposed on R-C tensor are equivalent to the conditions imposed on this part of the total deformation which does not occur in the metric description, that is, on its rotational part in the polar decomposition. In a manner similar to that from the semi-metric Cosserat continuum, the tree-dimensional rotation, a function of position, may be identified with the space of internal degrees of freedom described by the gauge group of proper rotations SO(3). Thus, our approach does not consist in gauging of the global Galilean group of rotations, but in admitting the space of internal degrees of freedom (the isotopic space) hidden in the metric formulation. Let us denote, as in [6], by Vi i = 1,2, 3 the connection associated with a curvilinear convected coordinate system parametrizing deformable elastic continuum. Then the matrix of rotation is represented by the isotopic vector of the parameters of rotation w = {w”}, a = 1, 2, 3 according to the formula g*=g-l
g = exp(w”T,) = eKp(u%(bc$
(8)
The so-called tensor of change of curvature (denoted by -Li in [6]) plays the role of the gauge potentials Ai = ajT. The field strength tensor takes the Y-M form &I= ViAi - VjA, + [Ai, Aj] = (Via, - Via1 + a@aj)T = fijT
(9)
Since we are interested in “pure” or “vacuum” continuum, that is free of any topological defects, the condition of vanishing of fii is satisfied by the pure gauge potential in the form A, = -(Vig)g-’
= apTb
(10)
The isovector ai due to (8) and (10) is represented by the derivatives of the isovector of rotation w, 0 = Iw] 1 - cos w a,=-- sm a viw + w w VjW@W+ (E-i)Vi,, (11) , W The condition fij = 0 is nothing else but a condition of absence of singularities transformations (3),. Substituting (IO) to (9) we obtain 4~~ =
vi(-vjgg-')-v~(vigg-')
+
[(-v&T-1>a
in the gauge
(-Vjgg-'>I
g-‘(VjVi - VjV,)g = 0
for nowhere non-singular g(x). The independent components of the tensor fi, (9) may be obtained, form of “double” vector & = (P), B’ = asq*jk
similarly to (5),, in the
B=rote+~(~@)x~
(12)
Since, additionally 3“’ = B”’ [7], the equations (12) contain 6 independent conditions imposed on the rotational part of deformation. On the other hand, in the model considered here, the rotations are related with the gradient of the displacement field by the polar decom~sition of the deformation gradient F = VR (i.e. R = g in our case). Differentiating both sides of F = VR we obtain Ai = V-‘ViV Hence the expression for the pure potential deformation 2Eij = V~& - gii, is [6]
- V-‘ViFg-’ written
in terms of the metric
Atab = (V,)-‘VzV$ - V~(V~)-‘(V~~~~ + VIE, - VkEftr) Additionally
&a = Vai is the root of the expression (2E, + gjj) [8].
measure
of
(13)
1442
Letters in Applied and Engineering Sciences
In other words, to guarantee existence of a unique displacement field for a given strain field Eii we may demand that the six compatibility conditions in the form of Yang-Mills equations (12) be satisfied. The advantage of such formulation of the compatibility conditions follows from the fact that they have the structure of a gauge theory and may be regarded as a particular case of the gauge theory of dislocations and disclinations [9].
REFERENCES [l] C. N. YANG and R. L. MILLS, Phys. Rev. 96, 191 (1954). (21 Y. NAMBU, Leer. Appl. Math. 21, 3 (1985). [3] E. TONTI, On rhe Fonnul Structure of Physical Theories. Inst. di Matematia Politecnico di Milan0 (1976). (41 M. CHAICHIAN and N. F. NELIPA, Introduction to Gauge Field Theories. Springer, Berlin (1984). [5] P. A. M. DIRAC, Phys. Rev. 74, 817 (1948). [6] W. PIETRASZKIEWICZ and J. BADUR, ht. J. Engng Sci. 21, 1097 (1983). [7] W. PIETRASZKIEWICZ and J. BADUR, Trends in Applications of Pure Mathematics to Mechanics (Edited by J. BRILLA). Pitman (1984). [8] A. HOGER and D. E. CARLSON, (2. Appl. Math. 42, 113 (1983). [9] A. KADIC and D. G. B. EDELEN, ht. J. Engng Sci. 20,433 (1982). (Received
1 February 1989)
Int. J. Engng Sci. Vol. 27, No. 11, pp. 1439-1442.1989
Copyright @I 1989
Printed in Great Britain. All rights reserved
LETTERS
IN APPLIED
AND ENGINEERING
SCIENCES
A YANG-MILLS TYPE OF EQUATION FOR THE COMPATIBILITY CONDITIONS JANUSZ
BADUR
Institute of Fluid-Flow Machinery of the Polish Academy of Sciences, ul. Gen. J. Fiszera 14, 80-952 GdaAsk, Poland Abstract-It is shown that non-classical compatibility gauge-invariant Yang-Mills field strength tensor.
1. THE YANG-MILLS
conditions
PRIMAL
for strains
have a form of
CHAIN
A non-Abelian model of strong interactions, based on the local three-parameter gauge symmetry group SU(2), was first introduced by Yang and Mills [l]. Three massless vector gauge potentials, with values in the algebra of the group SU(2), played the role of intermediate bosons in this model. Since the three-parameter group of rotations O(3) is homomorphic to the unimodular unitary group SU(2), such that to every rotation from O(3) there correspond two matrices u and -u of SU(2) and, conversely, to every element u E SU(2) there corresponds some rotation from O(3), Yang and Mills adopted the three-dimensional natural representation of the group O(3) as the space of the internal degrees of freedom (the so-called isotopic space). The objects in the isotopic space are indexed a, b, c = 1,2, 3, in contradistinction to those from the Minkowski space-time, the arena for spectacle with isotopic vectors and tensors as participants, where the indices are p, Y, p = 0, 1, 2, 3. The primal chain of the gauge field theory depends on the dimension of the extended objects interacting with each other via the gauge potentials and on the dimension of the space-time [3]. In the four-dimensional Minkowski space-time we have different schemes of the primal chain for such extended objects: instantons (O-dimensional in the space-time), solitons (O-dimensional in the space), strings and vortices (one-dimensional), domain walls (twodimensional) or bags (three-dimensional). It means that the type of the gauge potentials is connected with the type of its sources, that is, the singularities of various dimensions of the dual fields [2]. Hence the instanton implies a scalar potential, the monopole-a vector potential, the string-a skew-symmetric tensor potential, etc. The Yang-Mills primal chain can be written as follows [3]: + the pure gauge potential + the gauge potential + the field strength tensor j the Bianchi identities + the “magnetic current”. We shall restrict our presentation of Y-M theory only to the elements belonging to the primal chain. Let us represent the Y-M gauge potential via the isotopic vector b, [l] B, = b;T, = b,T,
b, = {b:, b;, b;}
(1)
where the skew-symmetric matrices T, = E,(~~) are the generators of the group of rotations with the structural constants [T,, Tb] = E,~~T~ being otherwise the three-dimensional Ricci alternators. The action of the structural constants on the isotopic vectors will be denoted by the sign “8” of the cross product in the isotopic space. The pure gauge potentials, the field strength tensor and the Bianchi identities are defined as follows [l, 41 (the coupling constant equal to 1): ,y = -aVgg-l, g(x) l O(3) Fpy = [D,, Dvl= a$,
- a,B, + [B,, B,] = &yT
f,, = a,b, - ii’,b, + b,@b,
DpFv,+ DvFp,+ D&v = kpvp
(2) 1439
1440
Letters in Applied and Engineering Sciences
where the derivative Q = a,, + [B,,, a] = ($1 + b,@)T in (2), transforms covariantly under gauge transformations, whereas the potential and the field strength tensor according to the rules B, -+gB,g-’
- $gg-”
4, + g&g-’
- g[a,,
ag-1 = gtz;,,
- Gag-~
(3)
The Bianchi identities represented via the dual tensor E@”= ~E~“~“F~~and dual “magnetic” current k” = esvpAkppA = &‘T take the form DpPpv = dpEPV + [B,, i-]
= k”
(4)X
and via fP”‘, the form a ccf”‘_tb ccig@‘y=~v
(4)~
Let us express the independent components of the field strength tensor fpv = -fvp through the decomposition into the “electric” part E, = f&, i = 1, 2, 3 and the “magnetic” part Bi = i&ijkf’&in the form of “double” vectors in the space and the isotopic space E, Jj &Z= -a,& + b&3& - grad b,, ~=roth+~@C%)xb
(5)
where 3, = {-C$,, V}, d” = {&, V) when the light velocity c = 1. The Bianchi identities, expressed through &, @, in the rest frame, assume the form of the non-Abelian Maxwell equations with the “magnetic” current D +B =I:div B + bi@B’ = b Dxli:+D,B=rotE+(b~)xE+a,B-b,~B=k
(6)
where k’= {4,, k),
D,, = &,l - (b&3).
B = v1+ (b@),
The presence of the singular gauge transformations, that is those transformations for which GPY=g-‘[a,,, &]g #O, leads to the occurence of an additional field G,,, in the equations for the field strength tensor (3)* and for the Bianchi identities. This field does not vanish only on the Dirac strings [5]. In this case the Bianchi identities are breaked only in points where “magnetic” current kc”= a,G”” + [B,, G’““] which links the singularity current Gp’(x) = ;
~~“~*d~~ = j- dr j+ dW%F’S4(~ -
y(r, a))T,
does not vanish. In the above equation t and CI are time-like and space-like parameters of the two-dimensional world-sheet trace out by one-dimensionally singularity, oPv =--aYfi?Yv
aTa
(7) invariant string of
aY@aYv
adt
is the normal vector of the world-sheet, y “(r, o)-the position vector of the string and vu, a = 1,2,3 is a unit vector orienting the string in the isotopic space.
2. THE
COMPATIBILITY
CONDITIONS
FOR
STRAINS
The local compatibility conditions of an elastic continuum, based on the metric model of a simple body with translational degrees of freedom, are usually understood as the condition for
1441
Letters in Applied and En~nee~ng Sciences
vanishing of the Riem~n-Ch~stoffel curvature tensor. These conditions should be interpreted as the conditions imposed on the metric created by a given deformation tensor. In the theory of plastic deformation the condition for vanishing of the R-C tensor loses validity, due to the occurrence of dense distribution of singular objects similar to G,, in (7). In [6] it was shown, that the conditions imposed on R-C tensor are equivalent to the conditions imposed on this part of the total deformation which does not occur in the metric description, that is, on its rotational part in the polar decomposition. In a manner similar to that from the semi-metric Cosserat continuum, the tree-dimensional rotation, a function of position, may be identified with the space of internal degrees of freedom described by the gauge group of proper rotations SO(3). Thus, our approach does not consist in gauging of the global Galilean group of rotations, but in admitting the space of internal degrees of freedom (the isotopic space) hidden in the metric formulation. Let us denote, as in [6], by Vi i = 1,2, 3 the connection associated with a curvilinear convected coordinate system parametrizing deformable elastic continuum. Then the matrix of rotation is represented by the isotopic vector of the parameters of rotation w = {w”}, a = 1, 2, 3 according to the formula g*=g-l
g = exp(w”T,) = eKp(u%(bc$
(8)
The so-called tensor of change of curvature (denoted by -Li in [6]) plays the role of the gauge potentials Ai = ajT. The field strength tensor takes the Y-M form &I= ViAi - VjA, + [Ai, Aj] = (Via, - Via1 + a@aj)T = fijT
(9)
Since we are interested in “pure” or “vacuum” continuum, that is free of any topological defects, the condition of vanishing of fii is satisfied by the pure gauge potential in the form A, = -(Vig)g-’
= apTb
(10)
The isovector ai due to (8) and (10) is represented by the derivatives of the isovector of rotation w, 0 = Iw] 1 - cos w a,=-- sm a viw + w w VjW@W+ (E-i)Vi,, (11) , W The condition fij = 0 is nothing else but a condition of absence of singularities transformations (3),. Substituting (IO) to (9) we obtain 4~~ =
vi(-vjgg-')-v~(vigg-')
+
[(-v&T-1>a
in the gauge
(-Vjgg-'>I
g-‘(VjVi - VjV,)g = 0
for nowhere non-singular g(x). The independent components of the tensor fi, (9) may be obtained, form of “double” vector & = (P), B’ = asq*jk
similarly to (5),, in the
B=rote+~(~@)x~
(12)
Since, additionally 3“’ = B”’ [7], the equations (12) contain 6 independent conditions imposed on the rotational part of deformation. On the other hand, in the model considered here, the rotations are related with the gradient of the displacement field by the polar decom~sition of the deformation gradient F = VR (i.e. R = g in our case). Differentiating both sides of F = VR we obtain Ai = V-‘ViV Hence the expression for the pure potential deformation 2Eij = V~& - gii, is [6]
- V-‘ViFg-’ written
in terms of the metric
Atab = (V,)-‘VzV$ - V~(V~)-‘(V~~~~ + VIE, - VkEftr) Additionally
&a = Vai is the root of the expression (2E, + gjj) [8].
measure
of
(13)
1442
Letters in Applied and Engineering Sciences
In other words, to guarantee existence of a unique displacement field for a given strain field Eii we may demand that the six compatibility conditions in the form of Yang-Mills equations (12) be satisfied. The advantage of such formulation of the compatibility conditions follows from the fact that they have the structure of a gauge theory and may be regarded as a particular case of the gauge theory of dislocations and disclinations [9].
REFERENCES [l] C. N. YANG and R. L. MILLS, Phys. Rev. 96, 191 (1954). (21 Y. NAMBU, Leer. Appl. Math. 21, 3 (1985). [3] E. TONTI, On rhe Fonnul Structure of Physical Theories. Inst. di Matematia Politecnico di Milan0 (1976). (41 M. CHAICHIAN and N. F. NELIPA, Introduction to Gauge Field Theories. Springer, Berlin (1984). [5] P. A. M. DIRAC, Phys. Rev. 74, 817 (1948). [6] W. PIETRASZKIEWICZ and J. BADUR, ht. J. Engng Sci. 21, 1097 (1983). [7] W. PIETRASZKIEWICZ and J. BADUR, Trends in Applications of Pure Mathematics to Mechanics (Edited by J. BRILLA). Pitman (1984). [8] A. HOGER and D. E. CARLSON, (2. Appl. Math. 42, 113 (1983). [9] A. KADIC and D. G. B. EDELEN, ht. J. Engng Sci. 20,433 (1982). (Received
1 February 1989)