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A stable symmetrized Savvidy vacuum
Volume 100B, number 6
PHYSICS LETTERS
23 April 1981
A STABLE I~YMMETRIZED SAVVIDY VACUUM Ch. RAGIADAKOS Department o f Physics, University of Crete, Iraklion, Crete, Greece
Received 7 October 1980 We notice that a space and SU(N) Lie algebra symmetrization of the states, stabilizes the constant chromomagnetic field vacuum (Savvidy vacuum). But the constant chromoelectric field vacuum continues to be unstable after the symmetrization. The hamiltonian is formally quantized around the symmetrized Savvidyvacuum.
It is generally assumed that quantum chromodynamics (QCD) is the background field theory of strong interactions. But it is clear that the naive perturbation theory cannot describe the low energy hadron phenomenology. The physical vacuum of QCD must be different. The works of the Armenian group [1] may be considered a step to this direction. They applied the Schwinger background field technique [2] to study the vacuum structure of the Yang-Mills theories. The Copenhagen group [3 ] proved that the vacuum characterized by a chromomagnetic field H a (the Savvidy vacuum) is unstable and they found explicitly the unstable mode. The real part of the energy up to one loop corrections of the H vacuum is [1 ] Re(ell)
=fdax [½H 2 + (11N/967r2)g2H2 ln(g/la 2 -
½)],
(1)
where g is the coupling constant, N for a general SU(N) gauge group and/a 2 is the normalization point. It has a minimum at gHmi n =/a 2 e x p ( - 4 8 r r 2 / l 1Ng2).
(2)
The imaginary part of the energy which indicates the instability of this vacuum is [3] Im(e H) = - (1/870 g2H2.
(3)
Such a vacuum could never be the physical vacuum, because it is not rotationally invariant. The Copenhagen group [4] built up an approximately stabilized ground state by studying the unstable quantum mode. The space was separated into domains. The chromomagnetic field in each domain is homogeneous, but the field directions change randomly from one domain to the other. This technique permits the creation of a rotationally invariant state. In the present work we stabilize the Savvidy vacuum by postulating a restriction of the Fock space to the sphericaUy and SU(N) invariant states. In'classical quantum mechanics the Pauli principle is an analogeous case. A many (identical) fermion (boson) system occupies only the antisymmetric (symmetric) solutions of the Schrbdinger equation. The other states are excluded by hand from the physical Hilbert space. Hence we actually assume: Postulate. The physical Fock space contains only the rotationally and color-group symmetric states.
After this postulate the proof of the stability of a constant magnetic field vacuum is straightforward. For our purpose it is adequate to use the canonical quantization procedure in order to quantize the Yang-Mills lagrangian around a constant field strength solution, because the quantum field component appears explicitly. This is a sim0 031-9163/81/0000-0000/$ 02.50 © North-Holland Publishing Company
471
Volume 100B, number 6
PHYSICS LETTERS
23 April 1981
pie generalization of the method used by the Copenhagen group to reveal the unstable mode of the Sawidy vacuum. We have the gauge lagrangian lau
1
£?= - -~ Fanv F~ ,
Fan v = OuAav - OvAau + gfabcAbuAcv.
(4)
The non-dynamical component A 0 can be removed [5] from the lagrangian using the constraint equation
(D2)abAbo - D~ib( aoAai) = 0,
(5)
where we denote
D~ib = 6ab ai - gfabe A el"
(6)
From (5) we find
AdO = c ~a + (1/DZ)ab DbC( aoAci),
(7)
where (I)a satisfies the relation (D2)ab *b = 0,
(8)
and c= dc/dt is the zero mode of D 2. Then the lagrangian takes the form [5]
=fd3x [½(E(T))2 _ ½ (Bai)2 + ½ ~2 (D~ib ~b)2 _ ~(D~/bd~b)Eatff.) ],
(9)
where we have
E(aT) = [SadSi] - l~ib (1/D2)bC D~d ] aoA d/,
Bai - ½ eilk Fajk.
(10)
Notice that for regular fields the last two terms of (9) vanish and the c mode disappears. Then we find the ordinary lagrangian. Quantization around a constant magnetic field H i solution
A H _- na ~1 ei]k H/x k ,
c = 0,
(11)
gives the hamiltonian ~1~H
=fd3x ½H 2 + ½fd3x((aoAa]) 2 -
Aai [D2(A H)]abAbi +gn a eijkHkfabcAbiAc] } + higher order terms,
(12)
where the background field gauge condition
G =- [Di(AH)]abAbi = 0
(13)
was assumed. In order to quantize canonically around a classical solution A (cl) we simply make the translation Aai(X )-'~ A(cl) +Aai(X ) and we expand the lagrangian relative to the quantum component of the gauge field Aai(X ). By inspection one can show [3] that the quadratic part of (12) is not positive definite, which means that the vacuum is not stable There are configurations of the field Aai , found by the Copenhagen group [3], which give negative values to the quadratic part of (12) while at the same time they satisfy the background field gauge condition (13). Let S be the projection operator which symmetrizes over space and group directions i.e.
S [Hk, n a) =
[r((N2 -
1)/2)/8 rr(N2 + 1)/2 ] fd~ dw IHk, n a),
(14)
where IHk, n a) is any mode created around the unstable Sawidy vacuum, ~2 is the space spherical angle and 6o the SU(N) Lie algebra spherical angle. In this space the hamiltonian becomes 472
Volume 100B, number 6 ~' =S~S
(S =S t ,
PHYSICS LETTERS
23 April 1981
S 2 = 1).
(15)
Using the expansion (12) we find
~ =fd3x~ H 2 +½fd3x((aoA,,i)2 - AaiV2Aai + [g2H2N/6(N2 - 1 ) ] X2ZaiAai)
+ h.o.t.,
(16)
Cz(SU(N)) ----fabcfabd = N6ca.
(17)
where we used the relations mean (tljHt) = ~HZ6fl,
mean (nanb) = [N2 - 1 ] -16ab,
We see that the quadratic part of the projected hamiltonian ~ 1 is positive definite. This means that it has a stable vacuum and a stable Fock space can be constructed on this vacuum by quantizing Aai. It is easy to see that the projected background field gauge condition (13) becomes the ordinary transverse gauge condition
G' = SGS = ~iAai = 0,
(18)
because it is symmetrized relative to the space and group directions. The vacuum expectation value of the magnetic field minimizes the vacuum energy with quantum corrections (and properly renormalized) in analogy to the Savvidy method. From the value (2) of the magnetic field in the Savvidy vacuum 10) we expect (svl H 2 Isv) = (01SH2SIO) ~- (kt4/g 2) e x p ( - 967r2/11Ng2),
(19)
where Isv) is the symmetrized vacuum. The naive expectation would be this value to minimize the energy of the vacuum of the symmetrized hamiltonian (16). But we cannot prove it. The quantization around the symmetrized Savvidy vacuum, which we will make below, is too formal to give the vacuum energy. A field translation around a constant electric field solution
A E = O,
c=~
(~ba = n a Eix i/~),
(20)
gives the hamiltonian
C~E = _ f +
d3x 1E2 +½f d3x [(E2/~2)
~fd
~2 + (~oAa])2 + (3jAak)2 _g2(Eixi)2fabcfadenbndaei]
EiE~i
3x d3x' Ix - x ' l fabcfadenbndAci(x)Aej(X ) + h.o.t.,
(21)
where the gauge condition DiAai = 0 was assumed. Then we can find the symmetrized hamiltonian
~ E -- S ~ S = - f d3x ½ E 2 + l f d 3 x +
g2NE2
f d 3 x d3x '
rr(N2-1) d
((E2/~ 2) ~2 + (aoAa])Z + (a]Aak)2 _ [g2NE2/3(NZ_I)] xZAaiAai)
Aai(x) Aai(x') Ix - x'[
+ h.o.t.
(22)
We see that the quadratic part of this symmetrized hamfltonian is not positive definite. For one may find configurations ofAai(X ) satisfying the gauge condition where the negative quadratic term of (21) is dominant. Hence only the constant chromomagnetic field vacuum is stable while the constant chromoelectric field solution does not correspond to a stable vacuum. The gauge condition (18) complicates considerably the quantization of the hamiltonian (16). In order to quantize the system we define the variables
bai(n ) = f d3x Aai(X , t) *n(X),
(23) 473
Volume 100B, number 6
PHYSICS LETTERS
23 April 1981
where n = (n 1, n2, n 3 ) ; n l , n2, n3 = 0, 1,2,3 .... are the three integers which characterize the orthonormal set of eigenfunctions @n(x) of the harmonic oscillator operator which appears in (16). Then this hamiltonian takes the form ~ H,
=
f d3x ~" 1 H2 + ~1 ~
[i~ai(n)i~ai(n)+Enbai(n)bai(n)]
+ h.o.t.
(24)
n
But we cannot consider
dai(n) =--(2x/~n)-1/2 [i~ai(n) _ i v a n bai(n) ]
(25)
and their conjugate dtai(n) as annihilation and creation operators because they are not independent variables. The gauge condition is equivalent to the following relations
f d3x3iAai(X , t ) Xltn(X) = O,
Vn - (nl, n2, n3).
(26)
Using the recursion relations of the Hermite polynomials, we find the following recursion relation for the variables
b ai(n) VCff-1bal ( n l - 1 , n2, n3) - (nl +l )l/2 bal (nl +1, n2, n3) +x/~2 ba2(nl , n 2 - 1 , n3) - (n 2 +l)l12ba2(n1, n 2 +1, n3) + x/~3 ba3 (nl, n2, n 3 - 1 ) - (n 3 +1) 1/2 ba3 (nl, n2, n 3 +1) = O,
(27)
and exactly the same relations for their time derivatives bai(n). Then we can write the variables ba3(n ) as functions of bal (n), ba2(n) and ba3(nl, n2,0). After their substitution in (24), the hamiltonian takes the general form
--fd3x } ,2 + E
+G
G) + h.o.t.,
(28)
where/1 (v) is the general index of the independent variables and Muv , Y~uvare symmetric matrices. We diagonalize them and we redefine the new variables such that the hamiltonian takes the general form ,
1
~ H = f d3x ~-H2 +~ v~ (~av~av + m 2 cavCav) +h.o.t.,
(29)
car and their conjugate momenta ~av are independent variables. Then the system can be ordinarily quantized
[ca., ~b~1 = Gb Gv.
(30)
The constants m v become the energies (masses) of the states. Of course the whole preceding quantization procedure is formal. We think that any practical effort to diagonalize Muv and Euv will reveal infinities which we cannot actually discuss. But we presented this formalism in order to show that we can construct the Fock space of the hamiltonian at least formally.
Conclusion. The essential result of the present work is the observation that a space and Lie algebra symmetrization stabilizes the unstable constant chromomagnetic field vacuum while the constant chromoelectric field vacuum continues to be unstable after the symmetrization. We cannot actually argue on the symmetrization postulate. But we only want to point out that there is no particular space or Lie algebra direction in the observed physical vacuum. Hence, any considered vacuum must be somehow symmetrized. These considerations do not affect the short distance calculations of the perturbative quantum chromodynamics, because the hadrons are believed [6] to be bubbles of the perturbative vacuum in the real chromodynamic vacuum.
474
Volume 100B, number 6
PHYSICS LETTERS
23 April 1981
References [1] G.K. Sawidy, Phys. Lett. 71B (1977) 133; S.G. Matinian and G.K. Sawidy, Nucl. Phys. B134 (1978) 539; M.J. Duff and R. Ramon-Medrano, Phys. Rev. D12 (1976) 3357; H. Pagels and E. Tomhoulis, Nucl. Phys. B143 (1978) 485. [2] J. Schwinger, Phys. Rev. 82 (1951) 664. [3] N.K. Nielsen and P. Olesen, Nucl. Phys. B144 (1978) 376. [4] H.B. Nielsen and P. Olesen, NucL Phys. B160 (1979) 380; H.B. Nielsen and M. Ninomiya, Nucl. Phys. B156 (1979) 1; J. Ambjorn and P. Olesen, Nucl. Phys. B170 (1980) 60. [5] I.V. Polubarinov, JINR R-2421, Dubna (1965) (in Russian); V.N. Pervushin, JINR R2-12225 Dubna (1979) (in Russian); V.N. Pervushin, JINR E2-12514 Dubna (1979). [6] H.B. Nielsen, Phys. Lett. 80B (1978) 133.
475
PHYSICS LETTERS
23 April 1981
A STABLE I~YMMETRIZED SAVVIDY VACUUM Ch. RAGIADAKOS Department o f Physics, University of Crete, Iraklion, Crete, Greece
Received 7 October 1980 We notice that a space and SU(N) Lie algebra symmetrization of the states, stabilizes the constant chromomagnetic field vacuum (Savvidy vacuum). But the constant chromoelectric field vacuum continues to be unstable after the symmetrization. The hamiltonian is formally quantized around the symmetrized Savvidyvacuum.
It is generally assumed that quantum chromodynamics (QCD) is the background field theory of strong interactions. But it is clear that the naive perturbation theory cannot describe the low energy hadron phenomenology. The physical vacuum of QCD must be different. The works of the Armenian group [1] may be considered a step to this direction. They applied the Schwinger background field technique [2] to study the vacuum structure of the Yang-Mills theories. The Copenhagen group [3 ] proved that the vacuum characterized by a chromomagnetic field H a (the Savvidy vacuum) is unstable and they found explicitly the unstable mode. The real part of the energy up to one loop corrections of the H vacuum is [1 ] Re(ell)
=fdax [½H 2 + (11N/967r2)g2H2 ln(g/la 2 -
½)],
(1)
where g is the coupling constant, N for a general SU(N) gauge group and/a 2 is the normalization point. It has a minimum at gHmi n =/a 2 e x p ( - 4 8 r r 2 / l 1Ng2).
(2)
The imaginary part of the energy which indicates the instability of this vacuum is [3] Im(e H) = - (1/870 g2H2.
(3)
Such a vacuum could never be the physical vacuum, because it is not rotationally invariant. The Copenhagen group [4] built up an approximately stabilized ground state by studying the unstable quantum mode. The space was separated into domains. The chromomagnetic field in each domain is homogeneous, but the field directions change randomly from one domain to the other. This technique permits the creation of a rotationally invariant state. In the present work we stabilize the Savvidy vacuum by postulating a restriction of the Fock space to the sphericaUy and SU(N) invariant states. In'classical quantum mechanics the Pauli principle is an analogeous case. A many (identical) fermion (boson) system occupies only the antisymmetric (symmetric) solutions of the Schrbdinger equation. The other states are excluded by hand from the physical Hilbert space. Hence we actually assume: Postulate. The physical Fock space contains only the rotationally and color-group symmetric states.
After this postulate the proof of the stability of a constant magnetic field vacuum is straightforward. For our purpose it is adequate to use the canonical quantization procedure in order to quantize the Yang-Mills lagrangian around a constant field strength solution, because the quantum field component appears explicitly. This is a sim0 031-9163/81/0000-0000/$ 02.50 © North-Holland Publishing Company
471
Volume 100B, number 6
PHYSICS LETTERS
23 April 1981
pie generalization of the method used by the Copenhagen group to reveal the unstable mode of the Sawidy vacuum. We have the gauge lagrangian lau
1
£?= - -~ Fanv F~ ,
Fan v = OuAav - OvAau + gfabcAbuAcv.
(4)
The non-dynamical component A 0 can be removed [5] from the lagrangian using the constraint equation
(D2)abAbo - D~ib( aoAai) = 0,
(5)
where we denote
D~ib = 6ab ai - gfabe A el"
(6)
From (5) we find
AdO = c ~a + (1/DZ)ab DbC( aoAci),
(7)
where (I)a satisfies the relation (D2)ab *b = 0,
(8)
and c= dc/dt is the zero mode of D 2. Then the lagrangian takes the form [5]
=fd3x [½(E(T))2 _ ½ (Bai)2 + ½ ~2 (D~ib ~b)2 _ ~(D~/bd~b)Eatff.) ],
(9)
where we have
E(aT) = [SadSi] - l~ib (1/D2)bC D~d ] aoA d/,
Bai - ½ eilk Fajk.
(10)
Notice that for regular fields the last two terms of (9) vanish and the c mode disappears. Then we find the ordinary lagrangian. Quantization around a constant magnetic field H i solution
A H _- na ~1 ei]k H/x k ,
c = 0,
(11)
gives the hamiltonian ~1~H
=fd3x ½H 2 + ½fd3x((aoAa]) 2 -
Aai [D2(A H)]abAbi +gn a eijkHkfabcAbiAc] } + higher order terms,
(12)
where the background field gauge condition
G =- [Di(AH)]abAbi = 0
(13)
was assumed. In order to quantize canonically around a classical solution A (cl) we simply make the translation Aai(X )-'~ A(cl) +Aai(X ) and we expand the lagrangian relative to the quantum component of the gauge field Aai(X ). By inspection one can show [3] that the quadratic part of (12) is not positive definite, which means that the vacuum is not stable There are configurations of the field Aai , found by the Copenhagen group [3], which give negative values to the quadratic part of (12) while at the same time they satisfy the background field gauge condition (13). Let S be the projection operator which symmetrizes over space and group directions i.e.
S [Hk, n a) =
[r((N2 -
1)/2)/8 rr(N2 + 1)/2 ] fd~ dw IHk, n a),
(14)
where IHk, n a) is any mode created around the unstable Sawidy vacuum, ~2 is the space spherical angle and 6o the SU(N) Lie algebra spherical angle. In this space the hamiltonian becomes 472
Volume 100B, number 6 ~' =S~S
(S =S t ,
PHYSICS LETTERS
23 April 1981
S 2 = 1).
(15)
Using the expansion (12) we find
~ =fd3x~ H 2 +½fd3x((aoA,,i)2 - AaiV2Aai + [g2H2N/6(N2 - 1 ) ] X2ZaiAai)
+ h.o.t.,
(16)
Cz(SU(N)) ----fabcfabd = N6ca.
(17)
where we used the relations mean (tljHt) = ~HZ6fl,
mean (nanb) = [N2 - 1 ] -16ab,
We see that the quadratic part of the projected hamiltonian ~ 1 is positive definite. This means that it has a stable vacuum and a stable Fock space can be constructed on this vacuum by quantizing Aai. It is easy to see that the projected background field gauge condition (13) becomes the ordinary transverse gauge condition
G' = SGS = ~iAai = 0,
(18)
because it is symmetrized relative to the space and group directions. The vacuum expectation value of the magnetic field minimizes the vacuum energy with quantum corrections (and properly renormalized) in analogy to the Savvidy method. From the value (2) of the magnetic field in the Savvidy vacuum 10) we expect (svl H 2 Isv) = (01SH2SIO) ~- (kt4/g 2) e x p ( - 967r2/11Ng2),
(19)
where Isv) is the symmetrized vacuum. The naive expectation would be this value to minimize the energy of the vacuum of the symmetrized hamiltonian (16). But we cannot prove it. The quantization around the symmetrized Savvidy vacuum, which we will make below, is too formal to give the vacuum energy. A field translation around a constant electric field solution
A E = O,
c=~
(~ba = n a Eix i/~),
(20)
gives the hamiltonian
C~E = _ f +
d3x 1E2 +½f d3x [(E2/~2)
~fd
~2 + (~oAa])2 + (3jAak)2 _g2(Eixi)2fabcfadenbndaei]
EiE~i
3x d3x' Ix - x ' l fabcfadenbndAci(x)Aej(X ) + h.o.t.,
(21)
where the gauge condition DiAai = 0 was assumed. Then we can find the symmetrized hamiltonian
~ E -- S ~ S = - f d3x ½ E 2 + l f d 3 x +
g2NE2
f d 3 x d3x '
rr(N2-1) d
((E2/~ 2) ~2 + (aoAa])Z + (a]Aak)2 _ [g2NE2/3(NZ_I)] xZAaiAai)
Aai(x) Aai(x') Ix - x'[
+ h.o.t.
(22)
We see that the quadratic part of this symmetrized hamfltonian is not positive definite. For one may find configurations ofAai(X ) satisfying the gauge condition where the negative quadratic term of (21) is dominant. Hence only the constant chromomagnetic field vacuum is stable while the constant chromoelectric field solution does not correspond to a stable vacuum. The gauge condition (18) complicates considerably the quantization of the hamiltonian (16). In order to quantize the system we define the variables
bai(n ) = f d3x Aai(X , t) *n(X),
(23) 473
Volume 100B, number 6
PHYSICS LETTERS
23 April 1981
where n = (n 1, n2, n 3 ) ; n l , n2, n3 = 0, 1,2,3 .... are the three integers which characterize the orthonormal set of eigenfunctions @n(x) of the harmonic oscillator operator which appears in (16). Then this hamiltonian takes the form ~ H,
=
f d3x ~" 1 H2 + ~1 ~
[i~ai(n)i~ai(n)+Enbai(n)bai(n)]
+ h.o.t.
(24)
n
But we cannot consider
dai(n) =--(2x/~n)-1/2 [i~ai(n) _ i v a n bai(n) ]
(25)
and their conjugate dtai(n) as annihilation and creation operators because they are not independent variables. The gauge condition is equivalent to the following relations
f d3x3iAai(X , t ) Xltn(X) = O,
Vn - (nl, n2, n3).
(26)
Using the recursion relations of the Hermite polynomials, we find the following recursion relation for the variables
b ai(n) VCff-1bal ( n l - 1 , n2, n3) - (nl +l )l/2 bal (nl +1, n2, n3) +x/~2 ba2(nl , n 2 - 1 , n3) - (n 2 +l)l12ba2(n1, n 2 +1, n3) + x/~3 ba3 (nl, n2, n 3 - 1 ) - (n 3 +1) 1/2 ba3 (nl, n2, n 3 +1) = O,
(27)
and exactly the same relations for their time derivatives bai(n). Then we can write the variables ba3(n ) as functions of bal (n), ba2(n) and ba3(nl, n2,0). After their substitution in (24), the hamiltonian takes the general form
--fd3x } ,2 + E
+G
G) + h.o.t.,
(28)
where/1 (v) is the general index of the independent variables and Muv , Y~uvare symmetric matrices. We diagonalize them and we redefine the new variables such that the hamiltonian takes the general form ,
1
~ H = f d3x ~-H2 +~ v~ (~av~av + m 2 cavCav) +h.o.t.,
(29)
car and their conjugate momenta ~av are independent variables. Then the system can be ordinarily quantized
[ca., ~b~1 = Gb Gv.
(30)
The constants m v become the energies (masses) of the states. Of course the whole preceding quantization procedure is formal. We think that any practical effort to diagonalize Muv and Euv will reveal infinities which we cannot actually discuss. But we presented this formalism in order to show that we can construct the Fock space of the hamiltonian at least formally.
Conclusion. The essential result of the present work is the observation that a space and Lie algebra symmetrization stabilizes the unstable constant chromomagnetic field vacuum while the constant chromoelectric field vacuum continues to be unstable after the symmetrization. We cannot actually argue on the symmetrization postulate. But we only want to point out that there is no particular space or Lie algebra direction in the observed physical vacuum. Hence, any considered vacuum must be somehow symmetrized. These considerations do not affect the short distance calculations of the perturbative quantum chromodynamics, because the hadrons are believed [6] to be bubbles of the perturbative vacuum in the real chromodynamic vacuum.
474
Volume 100B, number 6
PHYSICS LETTERS
23 April 1981
References [1] G.K. Sawidy, Phys. Lett. 71B (1977) 133; S.G. Matinian and G.K. Sawidy, Nucl. Phys. B134 (1978) 539; M.J. Duff and R. Ramon-Medrano, Phys. Rev. D12 (1976) 3357; H. Pagels and E. Tomhoulis, Nucl. Phys. B143 (1978) 485. [2] J. Schwinger, Phys. Rev. 82 (1951) 664. [3] N.K. Nielsen and P. Olesen, Nucl. Phys. B144 (1978) 376. [4] H.B. Nielsen and P. Olesen, NucL Phys. B160 (1979) 380; H.B. Nielsen and M. Ninomiya, Nucl. Phys. B156 (1979) 1; J. Ambjorn and P. Olesen, Nucl. Phys. B170 (1980) 60. [5] I.V. Polubarinov, JINR R-2421, Dubna (1965) (in Russian); V.N. Pervushin, JINR R2-12225 Dubna (1979) (in Russian); V.N. Pervushin, JINR E2-12514 Dubna (1979). [6] H.B. Nielsen, Phys. Lett. 80B (1978) 133.
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