SU(2) Yang-Mills quantum mechanics of spatially constant fields

0.2 MB·English

by  H. -P. Pavel

SU
(2) Yang-Mills quantum mechanics of spatially constant fields
H.-P. Pavel
Institut fu¨r Kernphysik, c/o Theorie, Schlossgartenstr. 9, D-64289 Darmstadt, Germany
and
Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Russia
January 30, 2007
7
0 Abstract and the global spatial rotations Aχ =A R(χ) .
ai aj ji
0 IntheconstrainedHamiltonianformulation(seee.g.[2])
2 As a first step towards a strong coupling expansion of the time dependence of the gauge transformations (2) is

n Yang-Mills theory, the SU(2) Yang-Mills quantum me- exploited to put the Weyl gauge A = 0 , a=1,2,3, and
a0
a chanics of spatially constant gauge fields is investigated the physicalstatesΨ haveto satisfyboththe Schr¨odinger
J
in the symmetric gauge, with the six physical fields rep- equation and the three Gauss law constraints
0
resented in terms of a positive definite symmetric 3 3
3 ×
matrix S. Representing the eigenvalues of S in terms of 1 ∂ 2
1 elementary symmetric polynomials, the eigenstates of the HΨ = 2 ∂A +Ba2i(A) Ψ=EΨ , (3)
v corresponding harmonic oscillator problem can be calcu- Xa,i "(cid:18) ai(cid:19) #
3 lated analytically and used as orthonormal basis of trial ∂
G Ψ = iǫ A Ψ=0 , a=1,2,3 . (4)
8 statesforavariationalcalculationoftheYang-Millsquan- a − abc bi∂Aci
2
tum mechanics. In this way high precision results are ob-
1 The G are the generators of the residual time indepen-
tained in a very effective way for the lowest eigenstates a
0 dent gauge transformations, satisfying [G ,H] = 0 and
7 in the spin-0 sector as well as for higher spin. Further- a
[G ,G ] = iǫ G . Furthermore H commutes with the
0 more I find, that practically all excitation energy of the a b abc c
angularmomentumoperatorsJ = iǫ A ∂/∂A ,i=
h/ eigenstates, independently of whether it is a vibrational i − ijk aj ak
1,2,3. The matrix element of an operator O is given in
t or a rotational excitation, leads to an increase of the ex-
- the Cartesian form
p pectationvalueofthelargesteigenvalue φ3 ,whereasthe
h i
e expectation values of the other two eigenvalues, φ and
h h 1i Ψ′ OΨ dA Ψ′∗(A)OΨ(A) . (5)
: hφ2i, andalso the component hB3i=ghφ1φ2i ofthe mag- h | | i ∝Z
v netic field, remain at their vacuum values.
i Forcarryingoutquantummechanicalcalculationsitisde-
X
sirabletohaveacorrespondingunconstrainedSchr¨odinger
1 Introduction
r equation and to find its eigenstates in an effective way
a
with high accuracy at least for the lowest states. The ba-
Theso-calledYang-MillsmechanicsoriginatesfromYang-
sic ideas and first results of such an gauge fixed approach
Mills field theory under the supposition of the spatial ho-
toYang-Millsquantummechanicswillbepresentedinthe
mogeneity of the gauge fields [1]. For this case the La-
following.
grangian of pure SU(2) Yang-Mills theory reduces to 1
1 2 1
L= A˙ gǫ A A B2 (A) , (1) 2 Unconstrained Hamiltonian
2 ai− abc b0 ci − 2 ai
(cid:16) (cid:17)
with the magnetic field Bai(A) = (1/2)gǫabcǫijkAbjAck. 2.1 The symmetric gauge
The local SU(2) gauge invariance and the rotational in-
The local symmetry transformation (2) of the gauge po-
variance of the original Yang-Mills action reduces to the
tentialsA promtsuswiththesetofcoordinatesinterms
symmetry under the SO(3) transformations local in time ai
ofwhichtheseparationofthegaugedegreesoffreedomoc-
1
Aω (t) = O(ω(t)) A (t) ǫ O(ω(t))O˙(ω(t)) , curs. As in [3] I use the polar decompositionfor arbitrary
a0 ab b0 − 2g abc bc 3 3 quadratic matrices
Aω(t) = O(ω(t)) A (t) , (cid:16) (cid:17)(2) ×
ai ab bi
A (q,S)=O (q)S , (6)
1Everywhere in the paper we put the spatial volume V = 1. ai ak ki
As result the coupling constant g becomes dimensionful with g2/3
with the orthogonal matrix O(q), parametrized by the
having the dimension of energy. The volume dependence can be
restoredinthefinalresultsbyreplacingg2 withg2/V. three angles qi, and the positive definite, symmetric 3 3
×
1
matrix S. The decomposition (6) is unique and corre- The potential term in (19), has a flat valley of degener-
sponds to the symmetric gauge χ (A)=ǫ A =0. The ate absolute minima at zero energy, φ = φ = 0,φ
i ijk jk 1 2 3
−
Jacobian is ∂(A )/∂(q,S) detΩ(q) (φ +φ ), arbitrary, at the edge of (9). Close to the bottom of the
| ai | ∝ i<j i j
where φ ,φ ,φ are the eigenvalues of S and Ω (q) valley the potentialis thatof a harmonicoscillatorwitha
1 2 3 jm
(1/2)ǫ OT(q)∂O(q)/∂q . The vaQriables S an≡d width narrowing down for larger values of φ .
mkl j kl 3
∂/∂S make no contribution to the Gauss law operators
G = iO(cid:2) (q)Ω−1(q)∂/∂q .(cid:3)Hence, assuming the invert-
a − as sj j 2.3 Symmetries
ibility of the matrix Ω, the non-Abelian Gauss laws (4)
are equivalent to the set of Abelian constraints ∂Ψ/∂q =
i As a relic of the rotational invariance of the initial gauge
0 ,i = 1,2,3. and the physical Hamiltonian of SU(2)
field theory the Hamiltonian (53)possesses the symmetry
Yang-Mills quantum mechanics reads
1 ∂ 2 1 [H,Jk]=0 , (13)
H = + γ−2J J +B2 (S)
2Xm,n"−(cid:18)∂Smn(cid:19) 2 mn m n mn with the angular momentum operators Ji = Rijξj satis-
γ−1(S) δ tr(γ−1(S)) ∂ , (7) fying [Ji,Jj] = iǫijkJk and [Ji,ξj] = 0 . Hence the eigen-
− mn − mn ∂Smn # statesofH canbecharacterizedbythequantumnumbers
J andM. FurthermoreH isinvariantunderarbitraryper-
(cid:2) (cid:3)
withγik(S):=Sik−δiktrS andtheangularmomentaJi = mutations σij of any two of the three indices 1,2,3, time
−2iǫijkSaj∂/∂Sak ,i = 1,2,3, in terms of the physical reflections T, and parity reflections P :φi →−φi,
variables (note the factor 2). The matrix element of a
physical operator O is given by [H,σ ]=0 , [H,T]=0 , [H,P]=0 . (14)
ij
Ψ′ OΨ dS (φ +φ ) Ψ′∗(S)OΨ(S) . (8)
i j
h | | i ∝Z hYi<j i 2.4 Boundary conditions
2.2 Unconstrained Hamiltonian in terms of ro- TherequirementofHermiticityofH intheregionbounded
by the three boundary planes φ = 0 ,φ =φ ,φ =φ
tational and scalar degrees of freedom 1 1 2 2 3
and at positive infinity, leads to the conditions at φ
1
In order to achieve a more transparent form for the re-
duced Yang-Mills system (7) I shall limit myself in this
∂ Ψ =0 Ψ =0 , (15)
1
work to the principle orbit configurations φ1=0 ∨ φ1=0
(cid:12) (cid:12)
(cid:12) (cid:12)
0<φ1 <φ2 <φ3 < , (9) corresponding(cid:12)to even or odd parity(cid:12)states respectively. I
∞
for the eigenvalues φ ,φ ,φ > 0 of the positive definite shall consider in the work only the first alternative allow-
1 2 3
symmetricmatrixS (notconsideringsingularorbitswhere ingonlyevenparitystatesinallspinsectors. Furthermore,
twoormoreeigenvaluescoincide)andperformaprincipal- duetothevanishingofthemeasure(12)ontheboundaries
axes transformation φ1 =φ2 and φ2 =φ3, we have to require only that
S =R(α,β,γ) diag (φ ,φ ,φ ) RT(α,β,γ) , (10)
1 2 3
Ψ =finite Ψ =finite . (16)
with the SO(3) matrix R parametrized by the three Eu- φ2=φ1 ∧ φ3=φ2
(cid:12) (cid:12)
ler angles. The Jacobian of (10) is ∂S/∂(α,β,γ,φ) (cid:12) (cid:12)
| | ∝ Finally,no(cid:12)rmalisabilityofthewav(cid:12)efunctionsrequiresthat
sinβ (φ φ ). In terms of the principal-axes vari-
i<j i− j the wave functions vanish sufficiently fast at infinity.
ables, the physical Hamiltonian reads
Q
1 3 ∂2 2 ∂ ∂
H = φ φ 2.5 Virial theorem
2cyclic"− ∂φ2i − φ2i −φ2j(cid:18) i∂φi − j∂φj(cid:19)
X Writing H = 1 E2+B2 anddenotingthe eigenstatesof
φ2+φ2 2
+ξ2 j k +g2φ2φ2 . (11) Hby n ,withenergiesEn,oneobtainsthevirialtheorem
i (φ2 φ2)2 j k | i (cid:0) (cid:1)
j − k #
nE2 n =2 nB2 n , (17)
AlltherotationalvariablesintheHamiltonian(53),which h | | i h | | i
agrees with that of [4], are combined into the intrinsic
angular momenta ξ = RTJ obeying [ξ ,ξ ] = iǫ ξ . It proofs to be a very useful tool to judge the quality of
i ij j i j − ijk k theapproximateeigenstatesobtainedusingthevariational
The matrix elements of an operator O are given as
approach. From (17) also follows (G2 :=2 B2 E2 )
cyclic −
Ψ′ OΨ dαsinβdβdγ dφ φ2 φ2 Ψ′∗OΨ. (cid:0) (cid:1)
h | | i∝Z 0<φ1<Zφ2h<Yφ3 i(cid:0) j − k(cid:1)i hn|B2|ni= 32En , hn|E2|ni= 34En, hn|G2|ni=−34En.
(12) (18)
2
3 Low energy spin-0 states
1.00
For J2 =ξ2 =0, the Hamiltonian (11) reduces to
1 3 ∂2 2 ∂ ∂ 0.75
H = φ φ +g2φ2φ2
0 2cyclic"−∂φ2i − φ2i −φ2j(cid:18) i∂φi − j∂φj(cid:19) j k#
X
(19) s30.50
Similar as in the Calogero model [5] one can prove that
the eigenstates of (19) are completely symmetric in the
arguments (φ ,φ ,φ ). We therefore can have only the 0.25
1 2 3
two possible forms, the parity even
Ψ(+)(φ ,φ ,φ )=Ψ(+)(φ2,φ2,φ2) , (20)
1 2 3 1 2 3 0.00
0.00 0.25 0.50 0.75 1.00
and the parity odd
e s2
Ψ(−)(φ1,φ2,φ3)=φ1φ2φ3Ψ(−)(φ21,φ22,φ23) , (21) Figure 1: The (dashed) area of integration in (s2,s3) space.
The dotted line is the limit s3 ≤s32/2 from algebraic inequali-
wherethe functionsΨ(±) arecoempletelysymmetricinthe ties.
arguments (φ2,φ2,φ2). Ψ(+) satisfies the first and Ψ(−)
1 2 3
the second of the boeundary conditions (15). I shall con- ∂2 ∂ ∂
D(0) := 4s 18 2s +1 s , (29)
sider here only the even case (20), lower in energy. The −1 − 3∂s2 − 3∂s 2∂s
2 (cid:18) 3 (cid:19) 3
odd case (21) can be treated completely analogously.
∂ ∂ ∂
D(0) := 12 s +4s +2 . (30)
−2 − 2∂s 3∂s ∂s
3.1 Elementary symmetric polynomials (cid:18) 2 3 (cid:19) 2
NotethatthesingularitiesinH havedisappearedviathe
0
Here it is useful to change to the new coordinates transformation(22)totheelementarsymmetricvariables.
(e1,e2,e3) defined as the elementary symmetric combina- The matrix elements become
tions
∞ 3/4 s(3up)(s2)
ds
e =φ2+φ2+φ2, e =φ2φ2+φ2φ2+φ2φ2, e =φ2φ2φ2. Ψ′ OΨ ds s7/2 ds 3 Ψ′∗OΨ
1 1 2 3 2 1 2 2 3 3 1 3 1 2(232) h | | i ∝Z 1 1 "Z 2Z √s3
0 0 0
The corresponding Jacobian
1 s(3up)(s2)
ds
∂(φ ,φ ,φ ) 1 + ds 3 Ψ′∗OΨ . (31)
(cid:12)(cid:12) ∂(e11,e22,e33) (cid:12)(cid:12)= √e3√∆ , (23) 3Z/4 s(32lowZ)(s2√) s3 #
(cid:12) (cid:12)
with the square(cid:12)(cid:12)root of the d(cid:12)(cid:12)iscriminant Here the region of integration is given by the (s2,s3) sat-
isfying the inequality, obtained from 24,
∆ (φ2 φ2)2= 27e2+18e e e 4e3e 4e3+e2e2,
≡ i − j − 3 1 2 3− 1 3− 2 2 1 ∆˜ s2+6s s 4s 4s3+3s2 >0 . (32)
i<j ≡ − 3 2 3− 3− 2 2
Y
(24) Denoting t(cid:0)he two roots of the equation(cid:1)∆˜ (s ,s ) = 0,
3 2
cancels the original measure (φ2 φ2) in (12). Fur-
i<j i − j quadratic in s3, by
thermore let us consider the scaling transformation
Q s(up)(s ):= 1 √1 s 2 1+2√1 s , (33)
s =e , s =3e /e2 , s =27e /e3 , (25) 3 2 − − 2 − 2
1 1 2 2 1 3 3 1
well defined and(cid:0)positive for a(cid:1)ll(cid:0)0 s2 1/3,(cid:1)and
≤ ≤
with the Jacobian ∂(e ,e ,e )/∂(s ,s ,s ) s5. Then
| 1 2 3 1 2 3 | ∝ 1 s(low)(s ):= 1+√1 s 2 1 2√1 s , (34)
the Schr¨odinger equation of (19) becomes 3 2 − 2 − − 2
welldefinedfor a(cid:0)ll0 s2 1,(cid:1)bu(cid:0)t positiveonly(cid:1)for 3/4
1 g2s2s 2s ∂2 9 ∂ + 1 D(0) 49 Ψ=EΨ, s2 1. The region o≤f pos≤itive ∆˜ is the shaded region i≤n
"6 1 2− 1∂s21 − ∂s1 2s1 − 4 !# Fig≤. 1.
(26)
with 3.2 Exact solution of the corresponding har-
D(0) := D(0)+ D(0)+ D(0) , (27) monic oscillator problem
0 −1 −2
where Note that for the case when the Yang-Mills potential in
∂ ∂ 7 2 (19) is replaced by the harmonic oscillator potential
D(0) := 2 2s +3s + , (28)
0 2∂s 3∂s 2 g2 φ2φ2+φ2φ2+φ2φ2 ω2 φ2+φ2+φ2 , (35)
(cid:18) (cid:18) 2 3(cid:19) (cid:19) 1 2 2 3 3 1 −→ 1 2 3
(cid:0) (cid:1) (cid:0) (cid:1)
3
thatisg2s s2/6replacedbyωs /2,theaboveSchr¨odinger We have choosen the overall normalisation constant such
2 1 1
eigenvalueproblem(26)separatesintoadensityandade- that the constant solution, P , corresponding to the
(0,0)
formation problem2 (k separation const., µ multiplicity), lowest eigenvalue k = 7/2, is equal to one. Note that
there is no eigenstate with k = 11/2. To the eigenvalue
Φ (s)=R (s )P (s ,s ) . (36)
nkµ nk 1 kµ 2 3 k = 31/2 correspond two states P and P . One
(3,0) (0,2)
The solutions of the density equations are given by can easily check that states with different values of k are
orthonormal to each other as it should be. To orthonor-
Rnk(s1)=sΓ(n+nk! +1) ω49 (ωs1)12(k−27)e−ωs1/2Lkn(ωs1), musaeliGseratmhe-Sscthamteisdotfoerqthuoanlokrmwaitlhisarteisopne,cPt(tpo,q)(4→5)P, o(′pn,qe).caInn
(37) summary we obtain the eigenstates
satisfying the orthonormality relations
Φ (s)=R (s )P′ (s ,s ) , (45)
∞ npq nk 1 (p,q) 2 3
ds1 s71/2 Rnk(s1)Rn′k(s1)=δnn′ , (38) with the energy eigenvalues
Z0
with the energy eigenvalues 9
E = 2(n+2p+3q)+ ω , (46)
n,p,q
2
Enk =(2n+1+k)ω . (39) (cid:18) (cid:19)
The values of k are determined by the corresponding de- which are equidistant and depend only on the total num-
formation problem ber 2(n+2p+3q) of nodes. The degeneracy is therefore
rapidly increasing with energy.
D(0)P (s ,s )=k2 P (s ,s ) . (40)
kµ 2 3 kµ 2 3
In the space of monomials sp sq , p,q N ordered 3.3 Low energy spin-0 spectrum from varia-
by increasing 2p+3q, the op2era3tor D(0) h∈as tr0idiagonal tional calculation
form, since the part D0(0) is diagonal, and the parts D−(01) To obtain the low energy spectrum of the Hamiltonian I
and D(0) act as lowering operators, as can be easily seen shall use the Rayleigh-Ritz variational method. Lead by
−2
from(28)-(30). HenceD(0)caneasilybediagonalisedwith theparabolicformoftheYang-Millspotentialclosetothe
eigenvectors P of the form bottom of the classical zero-energy valley, I shall use the
(p,q)
orthonormalbasisΦ in(45)astrialfunctions. Inorder
npq
to haverapidconvergencethe frequency is fixed using the
P(p,q) =N(p,q) sp2sq3+ a(p,q;p′,q′) sp2′sq3′, (41) lowest eigenstate
2p′+3q′≤2p+3q
X
and eigenvalues  Φ000 = Γ(9/2) ω9/4exp[−ωs1/2] . (47)
p
7 The stationarity conditions for the energy functional of
k =2(2p+3q)+ . (42)
2 this state,
The first 3 eigenstates P(p,q), orderedby increasing eigen- 9 9 g2
value k, are then E[Φ000]=hΦ000|H0|Φ000i=4 ω+ 4 ω2 ,
k = 7/2 : P =1 , (43)
(0,0) leads to the optimal choice
11 13 6
k =15/2 : P(1,0) = 2 r15(cid:18)s2− 11(cid:19) , ω = √32 g2/3 ≃1.259921g2/3 . (48)
221 209 9 36 As a first upper bound for the groundstate energy of the
k =19/2 : P = s s + ,
(0,1) 126 10 3− 17 2 221 Hamiltonian one therefore finds
r (cid:18) (cid:19)
and so on. Orthonormality of the states Φ with respect E E[Φ ] 4.25223g2/3 . (49)
0 000
to the scalar product (31), leads to the correponding or- ≤ ≃
thonormality relations for the Pkµ, In order to improve this upper bound, one truncates the
Fockspaceatacertainnumberofnodesofthe wavefunc-
3/4 s(3up)(s2) ds3 tionsanddiagonalizesthecorrespondingtruncatedHamil-
ds2 Pk′µ′ Pkµ (44)
Z0 Z0 √s3 tonian Htrunk to find its eigenvalues and eigenstates. Ex-
+ 1 ds2 s(3up)(s2) ds3 Pk′µ′ Pkµ = 6√3 δk′kδµ′µ . ttehnedlionwg etnoehrgigyhsepreacntrdumhigihnetrhneusmpibne-r0ssoefctnoordteoshIigohbtnauin-
Z3/4 Zs(3low)(s2) √s3 35 merical accuracy. Including all 174(1041) trial states up
2Notethat thevariables s1,s2 ands3 arerelatedtothedensity to 30(60) nodes, I obtain for the lowest state S0,
and deformation variables ρ,β andγ used in [6] via s1 = ρ2, s2 =
1−β2 ands3=1−3β2+2β3cosγ. E[S0] = 4.116719740(35)g2/3 , (50)
4
(i.e. the lasttwo digits have to be replacedby the ones in 4.1 Spin 1
brackets when going from 30 to 60 nodes). The state S ,
0 The spin-1 Schr¨odinger equation decays into three
given explicitly as (up to coefficients <0.01)
equations, one for each member of the cyclic triplet
S = 0.98 Φ 0.07 Φ 0.04 Φ (Ψ(1),Ψ(1),Ψ(1))=(Ψ(1) ,Ψ(1) ,Ψ(1)),
0 000− 100− 200 1 2 3 M1− M1+ M0
0.16 Φ +0.02 Φ +0.03 Φ , (51)
− 010 001 020 1
H E+ (V +V ) Ψ(1) =0 , and cycl. perm.
nearly coincides with the state Ψ , The energies of the 0− 2 2 3 1
000 (cid:20) (cid:21)
groundstateandthefirstexcitationstogetherwiththecor-
One can easily show that no solutions exist which satisfy
responding deviation from the virial theorem (17)
the boundary conditions (16).
∆E :=2 B2 E2 , (52)
virial
h i−h i
4.2 Spin 2
are shown in Tab. 1.
Theenergyspectrumofthefirstspin-0eigenstates(with The spin-2 Schr¨odinger equation decays into and into
trial states up to 30 nodes) are shown in Fig. 2. The three equations, one for each member of the cyclic triplet
errorbars are inside the lines. The energies obtained here (Ψ(2),Ψ(2),Ψ(2))=(Ψ(2) ,Ψ(2) ,Ψ(2) ),
1 2 3 M1+ M1− M2−
areingoodagreementwiththoseobtainedbyLu¨scherand
Mu¨nster [7] and by Koller and van Baal [8], using gauge H E+ 1(V +V )+2V Ψ(2) =0 , and cycl. perm.
invariant wave functions as trial states. 0− 2 2 3 1 1
(cid:20) (cid:21)
for which no solutions exist which satisfy the boundary
4 Higher spin states
conditions (16), and the coupled system
Fornon-vanishingspinwe canwritethe Hamiltonian(11) 3 √3
H E+ (V +V ) Ψ(2) + (V V )Ψ(2) = 0
1 3 φ2+φ2 (cid:20) 0− 2 1 2(cid:21) M0 2 1− 2 M2+
H =H0+ 2Xi=1ξi2Vi , Vi = (φ2jj−φ2kk)2, i,j,k cyclic H0−E+2V3+21(V1+V2) Ψ(M2)2++√23(V1−V2)Ψ(M2)0= 0
(cid:20) (cid:21)
with the spin-0 Hamiltonian (19). The eigenstates can be
classified according to their J2 and J quantum numbers for Ψ(2) and Ψ(2) . The solution of this spin-2 singlet
3 M0 M2+
andaresuperpositionsofsimultaneouseigenstatesofJ2,J system (56) can be written in the form
3
and ξ
3 2M =Ψ (s ,s ,s )s−1Y (φ2,φ2,φ2;α,β,γ)
| i 1 1 2 3 1 M 1 2 3
|JMi= ΨM(J)M′(φ1,φ2,φ3)|JMM′i , (53) +Ψ2(s1,s2,s3)s−12YM(φ21,φ22,φ23;α,β,γ) , (56)
M′=0,1±,..,J±
X
with the cyclic symmetric functions Ψ (s ,s ,s ) and
with the combinations (for M′ >0) e 1,2 1 2 3
elementary spin-2 fields Y (φ ,φ ,φ ;α,β,γ)
M 1 2 3
1
JMM′ := (JMM′ 2M M′ ) , (54) 2 1 √3
| ±i √2 | i±| − i Y = φ (φ +φ ) 2M0 + (φ φ )2M2+
M 3 1 2 1 2
of r3"(cid:18) −2 (cid:19)| i 2 − | i#
(57)
2J +1
|JMM′i:=iJr 8π2 DM(JM) ′(α,β,γ) . (55) and its dual YM := YM φ1→φ2φ3. The matrix element of
an operator O(s) can be written as
(cid:12)
e (cid:12)
1
Table 1: The energies and the deviation from the virial h2M′|O|2Mi=hΨ′1|(1−s2)O|Ψ1i+ 6hΨ′1|(s3−s2)O|Ψ2i
theoremforthelowestspin-0eigenstates(usingtrialstates 1 h 1
up to 30 (60) nodes). +6hΨ′2|(s3−s2)O|Ψ1i+ 9hΨ′2|(s22−s3)O|Ψ2i .
i
0+ E[g2/3] ∆E [g2/3] intermsofthecorrespondingspin-0matrixelements(31).
virial
0 4.116719740(35) 7.6 10−8 (1.2 10−12) Specifying again to parity even states, the vector
· ·
1 6.386361(58) 2.4 10−7 (1.3 10−10) Ψ = (Ψ ,Ψ ) satisfies the Schr¨odinger equation of the
1 2
· ·
2 7.97348(34) 4.8 10−4 (1.3 10−8) same formas(26)only with the scalarD(0) replacedwith
· ·
3 9.2039(23) 1.8 10−3 (3.1 10−7) the new matrix operators D(2) with
· ·
4 10.092(86) 7.5 10−4 (1.9 10−6)
5 10.966(37) 1.8· 10−2 (1.8· 10−5) 2 2s ∂ +3s ∂ +1 + 7 2 0
6 12.17(05) − 0·.22 (1.5 10·−4) D(2)= 2∂s2 3∂s3 2 ,
− · 0 (cid:16) (cid:16) 0 2 2s (cid:17)∂ +3(cid:17)s ∂ +2 + 7 2
 2∂s2 3∂s3 2 
 (cid:16) (cid:16) (cid:17) (cid:17)
5
D(0) 8s ∂ +4
D−(21)= 24−∂∂s12 3D∂s−(301) ! , 1122 0+ 2+
D(0) 0
D(2)= −2 .
−2 0 D−(02)−24∂∂s2 ! 3 ] 1100
2/
g
Again the corresponding harmonic oscillator Schr¨odinger y [
Equation separates g 88
r
e
n
Φ =R (s )P (s ,s ) . (58) E
nkµ nk 1 kµ 2 3
66
ThedensityequationsforR arethesameasinthespin-
nk
0 problem The values of k are determined by the corre-
44
sponding deformation problem
Figure 2: The energy spectrum of the first spin-0 and spin-2
D(2)P (s ,s )=k2P (s ,s ) , (59)
kµ 2 3 kµ 2 3 eigenstates.
using the basis (spsq,0) and (0,sp˜sq˜) The first 3 eigen-
2 3 2 3
states P , ordered by increasing eigenvalue k, are then
(p,q)
2 1 3 < >
k =11/2 : P = , 3
(0,0) √195 0
(cid:18) (cid:19)
k =15/2 : P(˜0,˜0) =r172365(cid:18)192/11s3(cid:19) ,6/17 -1/3 [g]2
k =19/2 : P(1,0) = 7 r15(cid:18) 21−2/17 (cid:19) , 1 < 2>
and so on. < >
1
Asforthespin-0caseonecanusethisorthonormalbasis 0
4 6 8 10 12
ofspin-2statesofthe harmonicoscillatorproblemastrial
2/3
states for a variational calculation of the corresponding Energy [g ]
eigenstatesofthespin-2Yang-MillsHamiltonian. Theen- Figure 3: The expectation values hφ1i, hφ2i and hφ3i of the
ergiesofthegroundstateandthefirstexcitationstogether scalar fields along the principal axes as function of the energy
with the corresponding deviation from the virial theorem for the lowest spin-0 (black boxes) and spin-2 (open circles)
(17) are shown in Tab. 2. states.
The energy spectrum of the first spin-2 eigenstates (in-
cluding the 270 trial states with up to 30 nodes) is shown 5 Expectation values of permutation
in Fig. 2. The errorbars are inside the lines. Also for
non-invariant operators
spin-2 the energies obtained here are in good agreement
with those obtained by Lu¨scher and Mu¨nster [7] and by
Expandingtheobtainedenergyeigenstatesintermsofthe
Koller and van Baal [8]. In particular I confirm that the
original φ ,φ ,φ and analytically calculating the Gaus-
first spin-2 state is lower in energy than the first spin-0 1 2 3
sian integrals in the region(9), we obtain the expectation
excitation.
values φ , φ and φ ofthethreefieldsalongtheprin-
1 2 3
h ih i h i
cipal axes. For the spin-0 grounstate e.g. we obtain
Table 2: The energies and the deviation from the virial
0φ 0 = 0.242 g−1/3,
theoremforthelowestspin-2eigenstates(usingtrialstates h | 1| i+
up to 30 nodes). 0φ2 0 + = 0.806 g−1/3,
h | | i
0φ 0 = 1.699 g−1/3.
3 +
2+ E[g2/3] ∆E [g2/3] h | | i
virial
1 6.014500 6.3 10−6 Fig. 3 shows the results for the lowest spin-0 (dark sym-
·
2 7.82062 4.8 10−4 bols)andspin-2states(openwhitesymbols)asfunctionof
·
3 9.3335 4.2 10−3 the energyofthe states. Ifindthat φ raisesuniversally
4 9.9285 6.0·10−4 with increasing energy, independenthw3hiether oscillator or
·
5 10.812 3.0 10−2 spinexcitation,whereas φ and φ staypracticallycon-
− · h 1i h 2i
6 11.91 0.10 stant.
−
The corresponding expectation values of the three in-
6
Several techniques in my treatment, such as the sepa-
<B > ration of the harmonic oscillator problem into a density
1
andadeformationproblemandthe diagonalizationofthe
2
deformation operator in terms of symmetric polynomials,
haveforthecaseofspin-0tosomeextentalreadybeende-
1/3 ] velopedin the contextofthe Calogeromodel [5]andthen
g
[1 liacsteirna[l8s]o. iInntdhiffeecroenntceexttoo[f8Y],atnhge-Mrainllgseqoufainnttuemgramtieocnha(9n)-
here is positive definite, and the corresponding range and
<B >
2
measure (31) in terms of the elementary symmetric vari-
ables is derived explicitly from the original measure (12).
<B >
3
0 Inparticulartherelativenormalizationofthedeformation
4 6 8 10 12
states (43) is different to that given in [8]. Furthermore
2/3
Energy [g ] the generalizationof the use of the elementary symmetric
Figure 4: The expectation values hB1i, hB2i and hB3i of the variables to spin-2 and the consideration of permutation
magnetic fields along the principal axes as function of the en- non-invariant expectation values here is new to the best
ergy for the lowest spin-0 (black boxes) and spin-2 (open cir- of my knowledge.
cles) states.
Acknowledgments
trinsic magnetic fields B =gφ φ etc.
1 2 3 I would like to thank A. Dorokhov, J. Wambach, and W.
Weisefortheirinterestandsupport. Financialsupportby
0B 0 = 1.391 g1/3 ,
h | 1| i+ the GSI-Darmstadt is gratefully acknowledged.
0B 0 = 0.415 g1/3 ,
2 +
h | | i
0B 0 = 0.211 g1/3 ,
h | 3| i+ References
in accordance with the opposite ordering B > B > B ,
1 2 3
[1] G.Z.Basean,S.G.MatinyanandG.K.Savvidi,JETP
following from (9). The results for B , B and B
1 2 3
h ih i h i Let. 29 (1979) 587.
as function of the energy for the lowest spin-0 and spin-2
statesareshowninFig. 4. B stayspracticallyconstant
3 [2] N.H. Christ and T.D. Lee, Phys. Rev. D 22 (1980)
h i
withincreasingenergy,whereas B and B areraising.
1 2 939.
h i h i
Up tothe considerednumber of30nodes the numerical
errorsaremuchsmallerinbothcasesthanthesystematical [3] S.A. Gogilidze, A.M. Khvedelidze, D. M. Mladenov
error of the obtained states ( 1%). and H.-P. Pavel, Phys. Rev. D 57 (1998) 7488; A.M.
≤
KhvedelidzeandH.-P.Pavel,Phys.Rev.D 59(1999)
105017; A. Khvedelidze and H.-P. Pavel, Phys. Lett.
6 Summary and discussion A 267 (2000) 96; A. Khvedelidze, H.-P. Pavel, and
G. Ro¨pke, Phys. Rev. D 61 (2000) 025017; A.M.
It has been shown that SU(2) Yang-Mills quantum me- Khvedelidze, D. M. Mladenov, H.-P. Pavel, and G.
chanics can be studied very effectively and with high ac- Ro¨pke, Phys. Rev. D 67 (2003) 105013 .
curacy, by choosing the symmetric gauge and perform-
ing a variational calculation using the orthonormal basis [4] Yu. Simonov, Sov. J. Nucl. Phys. 41 (1985) 835. G.
ofexacteigenstatesthecorrespondingharmonicoscillator K. Savvidy, Phys. Lett. 159B (1985) 325.
problemas trialstates inthe spin-0andthe spin-2 sector.
[5] F. Calogero, J. Math. Phys. 10 (1969) 2191, 2197; J.
The virial theorem is shown to be fulfilled very well for
Math. Phys. 12 (1971) 419.
the eigenstates obtained. The energies obtained here are
in good agreement with those obtained by Lu¨scher and [6] C. Martin and D. Vautherin, Nucl. Phys. B (Proc.
Mu¨nster [7] and by Koller and van Baal [8], using gauge Suppl.) 39 B,C (1995) 231.
invariant wave functions as trial states. In particular I
[7] M.Lu¨scherandG.Mu¨nster,Nucl.Phys.B232(1984)
confirm that the first spin-2 state is lower in energy than
the first spin-0 excitation. Furthermore I have found that 445.
practicallyallexcitationenergy,independently whetherit
[8] J. Koller and P. van Baal, Nucl. Phys. B273 (1986)
isduetoavibrationalorarotationalexcitation,goesinto
387; Nucl. Phys. B302 (1988) 1; P. van Baal and J.
the increase of the field strength along one of the three
Koller, Ann. of Phys. (N.Y.) 174 (1987) 299.
principalaxisfields, φ , whereas φ and φ ,andalso
3 1 2
h i h i h i
thecomponent B ofthemagneticfieldalongtheintrin-
3
h i
sic 3-direction,remainunchangedattheir vacuum values.
7