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Semi-classical spectrum of the continuous Heisenberg spin chain
ANNALS
OF PHYSICS
120, 107-128 (1979)
Semi-Classical Spectrum
of the Continuous
Heisenberg
Spin Chain*
A. JEVICKI AND N. PAPANICOLAOU~ The Institute
for
Advanced
Study,
Princeton,
New Jersey
08540
Received August 29, 1978
We develop a path integral formalism that allows for semi-classical quantization of systems with spin degrees of freedom. We apply it to study the continuous Heisenberg spin chain, which has been known to possess interesting classical solutions. The calculated semi-classical spectrum turns out to be essentially exact. We also construct a new infinite series of conservation laws that are nonlocal generalizations of the spin.
I. INTRODUCTION
Semi-classical methods have been vigorously investigated in recent years and applied to a number of field theories. They consist of a stationary phase approximation of the functional integral around non-trivial extrema of the action. These are classical solutions of the underlying non-linear field equations. Barring difficuIties in actually solving the classical equations, the method is relatively straightforward for systems with Bose degrees of freedom. Important limitations arise for systems with Fermi degrees of freedom, which are formulated in terms of anti-commuting fields. Semiclassical methods have been successfully used for fermions only in isolated cases, where an effective Bose field may be introduced. Similar problem arise for spin degrees of freedom, since functional integral representations for spin systems have been so far given in terms of anti-commuting fields. It is the purpose of the present article to construct a functional integral representation for spin systems that involves ordinary (commuting) integration variables and, thereby, provides a scheme in which semi-classical quantization follows the standard technique. Interesting physical systems with spin degrees of freedom are wideIy known and there is little need to give here a detailed enumeration. In this paper we shall concentrate on the continuous Heisenberg spin chain. The theory is defined by the Hamiltonian (1.1)
* Research supported by the Department of Energy under Grant EY-76-S-02-2220. +Present address: Department of Physics, University of California, Berkeley, Calif. 94720.
107 0003-49 16/79/070107-22$05.00/O CopyrIght f?‘i 1979 by Aradrmlr Prrss. Inc. All rights of reproduction in any form rescrwd.
108
JEVICKI AND PAPANICOLAOU
which together with the commutation yields the equation of motion
relations [sa(x), @(a!)] = i@sr(~)
6(x - x’)
s2(x) is fixed to be s(s + 1). The exact spectrum of the discrete isotropic chain was obtained long ago by Bethe [l]. His Ansatz for the spin eigenstates (known as Bethe’s hypothesis) was subsequently applied to obtain the exact solution for a class of interesting generalizations of the original Heisenberg chain. The literature in this direction is too extensive to be cited here. At any rate, these generalizations are not directly relevant to our present work. The remarkable structure of Bethe’s solution can be presently understood as deriving from an interesting property of the Heisenberg chain, namely, its complete integrability. This is most directly implied by the existence of higher conservation laws constructed in [2]. More complete results are available for the classical theory of the continuous chain, Eq. (1.2). Besides the spin-wave solution, localized soluton solutions were recently obtained in [3, 4, 51 and subsequently rederived by systematic inverse scattering methods in [6]. However, the relevance of the above classical solutions to the quantum theory has been as yet unclear. This problem is solved in the present article. Semi-classical quantization of the soiiton solution leads, in fact, to the exact magnon spectrum. Aside from providing an interesting illustration of the method for semi-classical quantization of spin, the continuous ferromagnetic chain appears to be an interesting example in which to test inverse scattering methods. Thus we present here some results concerning the inverse problem of the continuous spin chain. We construct a new infinite series of higher conservation laws that are non-local generalizations of the spin. This establishes a close connection with a large class of two-dimensional field theories that have been solved by inverse scattering methods. These are the nonlinear o-models of [7] and the classical Fermi interactions of [8]. The interesting common feature of these theories is that the origin of the Lax representation is understood to a considerable extent. Augmenting the above class by new systems, such as the ferromagnetic chain, should certainly prove useful in trying to identify the origin of the frequent occurrence of integrable systems. The plan of the article is as follows: Section II provides the necessary background from the underlying classical theory and presents the new series of non-local conservation laws. Our path integral representation for a general spin system is given in Section III. It is used in Section IV in order to carry out a DHN [9] semi-classical calculation of the spectrum of the continuous ferromagnetic chain. The problem of small oscillations around the onesoliton solution is solved in an Appendix. II. CLASSICAL THEORY A. The reduction technique. It was already mentioned in the introduction that the purpose of this section is to provide the necessary calculational background for
CONTINUOUS
SPIN
109
CHAIN
our subsequent semi-classical discussion, and to present some new results concerning the mathematical structure of the theory. In particular, we construct a new series of conservation laws, which are non-local generalizations of the spin. This and previous results on the subject suggest a strong analogy with the situation in the non-linear a-models and the classical Fermi interactions analyzed at length in [7, 81. We should add, however, that our present discussion of the classical theory is not meant to be complete. We begin with a brief outline of the reduction technique originally employed in [4] for the construction of the one-soliton solution of the continuous Heisenberg chain. Let Sa, a = 1, 2, 3 denote the continuous spin variable, a function of space and time, such that S2 = SaSa = 1. We form the following set of orthonormal vectors, to be referred to as the moving trihedral:
ela = &
,
EabcSbS,cc
e2a =
t2Hjl,s
,
e3a = S”,
(2.1) H = @‘zaS,a,
eiaeja= aij ,
where cabcis the usual antisymmetric tensor and the summation convention is implied. We have also used an obvious notation for the differentiation symbol. An arbitrary vector formed out of S” and its derivatives may be expanded in the above complete basis with components that are invariant under O(3) rotations. By a simple induction, it can be shown that all such invariants may be expressed in terms of the fundamental set of invariants s2=
1,
H = $S,“S,“,
QSC “bcsas,bs~,
(2.2)
and their derivatives. In this notation, the following expansion will be useful for our calculations:
ela
f
-
(2$/2
e2a
-
2He3a.
We now examine the variation of the trihedral under space displacements. elementary completeness arguments, one finds:
et, = Gikeka, a =
1,2,3,
0.3)
Using
(2.4)
where the matrix C, is a linear superposition of the standard anti-hermitean generators of O(3), with real coefficients that are functions of the fundamental invariants Hand Q: Cl = w1=z=,
[I”, I*] = @ZC,
{cola} = (0, -(2H)‘i2,
-Q/2HJ.
(2.5)
110
JEVICKI
AND
PAPANICOLAOU
Passing to the spinor representation and using the T-matrices defined by ta = (l/2$@, where ua are the Pauli matrices, C, reads (2H)‘/” -i Q/2H
1’
(2.6)
Our statements so far have been independent of the time evolution of the spin variable. The continuous Heisenberg spin chain is described by the evolution equation qa
=
E abSbs:a
=
[~abCSbSZc]Z
(2.7)
)
which is in the form of a conservation law implying the conservation of spin. Using (2.7) and the reduction technique outlined above, it is easy to find that the time displacement of the trihedral is described by e$ = C2,ikeka. The matrix C, is given, in the spinor representation,
c,
by
(WV’“>m ff, - iQ (2H)lj2 ’ (2H)li2 4H2 Hz + iQ Q2 + WW2h. (2H)l12 ’ - 4H2 (2H)‘/”
__-Q2
1
= 3
/ The integrability
G33)
*
(2.9)
1
condition for the system of Eqs. (2.4), (2.8) results into: (2.10)
or, in an equivalent form: Ht + Q, = 0,
Q,=[H2+
HH,,
- Hz2 - Qz II * H
(2.11)
Since all invariants calz be expressed in terms of H, Q and their space derivatives, Eqs. (2.11) provide a complete description of the dynamics of invariants. They constitute the “reduced” system associated with the continuous spin chain. They are found in the form of local conservation laws, implying the conservation of the energy s dx H and revealing a “higher” conserved charge, namely s dx Q. The latter hasbeen incorrectly identified in [4] as the momentum of the system. A correct definition of a momentum integral was given in [5]. We shall return to it in Section III. We emphasize
CONTINUOUS
SPIN
111
CHAIN
here, however, that it does not belong to the set of conservation laws following from Eqs. (2.11). For example, the next conserved charge is given by Q’ = J-r dx[S&S&
+ $(S:S$)‘].
(2.12)
The charge density appearing in this equation can be reduced in terms of H and Q. One obtains: Q’ = r’:
dx [9H2 + H’2;
Q2 ]
(3.12)
It is interesting to note that in its reduced form the above density is non-polynomial. The characteristic feature of the present series of charge densities is that they are invariant under O(3) rotations. Roughly speaking, they are generalizations of the energy density. An independent series of conservation laws generalizing the spin will be constructed in the following. For the moment, however, we turn our attention to an interesting identification of the system (2.11) made in [4]. It is shown by direct substitution that given a complex field # = aleiP satisfying the non-linear Schrodinger equation: (2.14) ih + 9L!z + H+*#># = 0 the quantities H E a2/2 and Q = a2pZ satisfy the system of Eqs. (2.11). Hence, the “reduced system” of the continuous Heisenberg chain is the non-linear Schrodinger theory, in a manner analogous to the situation in the non-linear u-models and the Gross-Neveu models, where the “reduced systems” are the sine- and sinh-Gordon equations and their appropriate generalizations. Correspondingly, it is natural to expect that the system of Eqs. (2.4, 9) provide the associated “linear” problem (Lax representation). However, the above line of argument is at this point incomplete, as no principle for the introduction of an eigenvalue parameter has been identified. We shall return to this question in later stages of our discussion. B. Non-local conservation laws. An independent method was proposed by Takhtajan in [6]. For this discussion, it is somewhat more convenient to use matrix notation specified by s = SV, s2 = 1.
(2.15)
The equation of motion (2.7) is transctibed into: 1
1
St = i L%&,I = jj Lx &la!* It is easy to show that the following “linear” R, = -iASR,
Rt =
(2.16)
equations: - G [S, S,] $ 2ih2S/ R
(2.17)
112
JEVICKI
AND
PAPANICOLAOU
are compatible in virtue of (2.16) for arbitrary constant X. The system (2.17) is the Lax representation used in [6] for a systematic solution of the inverse problem. Given the reduction of the present theory to the non-linear Schrodinger theory, a procedure that involves O(3) invariant quantities, it is somewhat surprising that a linear problem such as (2.17) exists-its distinguishing feature being that the Lax matrices are realized in terms of O(3) variant spin variables. However, such phenomenon is not unfamiliar. It systematically occurs in the systems analyzed in [7, 81. An equivalen way of stating Eqs. (2.17) is the following: given a solution S of (2.16), the linear system (2.17) may be solved for R, which can be chosen to be a (space-time dependent) unitary matrix. Furthermore, one can show by direct substitution that [R-‘SRI (x + 4ht, t) is also a solution. It may be said that the above construction provides a non-linear implementation of Galilean boosts. Previous experience suggests that a “linear” system of the above nature generates a new series of conservation laws-the non-local conservation laws. They were first constructed in [lo] for the non-linear u-model and can be easily shown to exist in the classical Fermi interactions of [8]. Their significance for the quantum theory was studied in [II]. We briefiy outline the construction of such conservation laws in the present theory and give a few examples. For a concise derivation of the conservation laws, we introduce dual potentials as follows: Denote by J = (1/2i)[S, S,] so that the equation of motion (2.16) reads S, = J, . The current densities S and J also satisfy the “pure gauge” condition
is,+J[s,J]
=o
(2.18)
following merely from S2 = I. Introduce the dual potential Q = .Qv satisfying the compatible equations fin, = S, f& = J. (2.19) With this variable the current conservation reads trivially on,, = fit, , wheras Eq. (2.18) becomes the equation of motion for Sz: (2.20)
iQ,, + $[!2, ) &Qn,]= 0. Correspondingly,
Eqs. (2.17) are transcribed R, = -iihS,R,
which are compatible in virtue Eqs. (2.21) may be used for a At each stage of this recursion, sequence of conservation laws,
into
R, = [-ihs2,
f 2ih28,]R,
(2.21)
of (2.20). Expanding R as R = 1 + AR, + PR, ,.... recursive determination of the coefficients R, , R, . .. . the integrability condition (2.20) is iterated to yield a the first few of which are:
it&., = $A$+) (a,Q),
= (Q&2 - 2&J,
. .. . .
(2.22a)
Expressed in the original variables, they lead to both local and non-local conservation
CONTINUOUS
laws. For instance, the conserved (2.22a) read
SPIN
113
CHAIN
charges corresponding
to the conservation
(2.22b)
plo = i+= d-Y dx’ e(s - s’) S”(x) S”(x’) --a, Q1”
=z c+c
laws
:m rix dY’ e(s - x’) Sb(x) SC(x’) )... . s - rx
19(x - x’) is the step function. As expected the first charge of this series is the integrated spin. We finally note that for ferromagnetic boundary conditions the above integrals diverge in general. They become finite after subtracting their values in the ferromagnetic ground state. This complication will cause no difficulty in our subsequent considerations. The remarkably similar structure of such diverse systems as the non-linear umodels, classical Fermi interactions and the present continuous ferromagnetic chain suggests that there exists a governing fundamental principle, whose full extent remains to be identified. Conserning the present theory, one more step has to be taken, namely, to explain the origin of the eigenvalue parameter in (2.17) and, consequently, introduce an eigenvalue parameter in the linear system of Section A. We shall have additional comments to make in the end of Section 111. C. Calculational details. In the remainder of this section, we turn our attention to more pragmatic considerations, in order to prepare our semi-classical calculation. We summarize here the necessary information concerning the one-soliton solution. Our computations are based on Takhtajan’s inverse problem, Eqs. (2.17). For later purposes, we shall need the explicit form of the kernel of the Gel’fandLevitan-Marchenko (GLM) equation; we therefore, repeat the calculation of [6] in some more detail. The relevant GLM equation reads
0,
(2.23)
for .Y < J’. The matrices Q1 and @, are determined from (2.24) by the relations (2.25)
114
JEVICKI
AND
PAPANICOLAOU
The form (2.24) for F is sufficiently general to cover the one-soliton solution and small fluctuations around it. The time evolution of the scattering data is given by a(h) = const,
5 = const,
b(h, t) = e--4iA2tb(X, O),
m(t) = e-*icetm,
(2.26)
5 is in general complex, Im 5 3 0. In terms of the kernel K, the solution of Eq. (2.16) is given from
S(x, t) = [X(x, x; t) - UJ uJiK(x, x; t) - up,
(2.27)
where o3 is the usual Pauli matrix. The one-soliton solution is obtained by solving the GLM equation with b = 0. The kernel that solves Eq. (2.23) is then
K,(x, Y; t) = Mx; t> ewl&d, (2.28) Here, * P2 = , cm_mi* 12 exp[2i(< - 5*)x - 4i(P - <*3t],
(2.29)
qSl = Re[25x - 412t] + arg m. Inserting these values into (2.27) and taking 5 = v/4 + i(e/2), we obtain the onesoliton solution 1 s, SE cos 6 = 1 2E2 3 + v2/4 cosh2 E(X - vt - x,,) ’
(2.30)
S, = S, + iS2 = sin Bei*, d = ( c2 + f)
t + ; (x - vt) + arctg [c th(x
- vt - x,)] + q.
with appropriate identification of the trivial constants x,, , q,, . Arbitrary value s for the total spin (Sz = s2) is achieved by the substitutions S -+ sS and t -+ st. By an additional resealing sv -+ a, we get S = s{sin 0 cos 4, sin 0 sin #J, cos e}; S2 = s2; cos 8 = 1 r#l = ,(,2
2.52 1 E2+ vy4s2 ' cosh2 ‘(x - vt - x0) ’ + &
) t + % (x - vt) + arctg [+
(2.31) thr(x - vt - %)I + 91 .
CONTINUOUS SPIN CHAIN
115
The physical content of this solution is not difficult to identify. In the rest frame (U = 0), we obtain a quasi-static configuration for which the third component of the spin is time-independent, whereas the total spin undergoes a precession around the third axis with frequency w = SE2. Concerning the space distribution of the spin density, this solution describes a localized excitation from the ferromagnetic ground state. In fact, everywhere but in a small region, the spin points on the (time) average along a fixed direction and possesses the vacuum value S, N 1, which characterizes the ferromagnetic ground state configuration. For a moving soliton (V f 0), this picture remains essentially the same, except for an interesting Doppler, velocity dependent, shift of the frequency: w = s(e2 + v2/4s2). The above intuitive picture will be gradually sharpened, as we will proceed with the semi-classical computation. To conclude this section, we calculate some relevant “observables.” Thus the energy, the total spin along the third direction and the first “higher” charge are given by:
d= I-02+a dx
e = j-r
H(x)
= 4es2,
dx Q(x) = 2Evs2’
The above values refer to the excited spin; the contributions from the ferromagnetic ground state have been subtracted. An important omission in (2.32) is the momentum integral. This should wait for the considerations of the following section.
III.
PATH INTEGRAL
FOR SPIN
It is well known that the path integral method provides the most appropriate framework for semi-classical quantization, [12-141. In practice, however, important limitations of the applicability of the method may occur: whereas for Bose fields classical solutions can be used directly to dominate the path integral, in the case of Fermi or spin degrees of freedom additional effort is required. This is mostly due to the fact that the corresponding path integral is usually defined over anti-commuting numbers. Consequently, semi-classical calculations involving fermions have been performed only in a few isolated cases, where it was possible to introduce an effective Bose field (see [ 151). We shall be interested in quantum systems that contain spin degrees of freedom. At the operator level, the procedure is standard. Let 9, CI= 1, 2, 3 be the spin operators satisfying the commutation relations: [9, sib] = i@c!3, 32 = s(s + l), s = 4, l... .
(3.1)
116
JEVICKI
Given a Hamiltonian equation of motion
AND
H = H(s),
PAPANICOLAOU
the time evolution
is dictated by the Heisenberg
$9(r) = f [9. H(S)].
(3.2)
The standard functional integral representation for this system is given in terms of anti-commuting numbers. For example, for spin one-half one has, [16] (3.3) where Q , Q and Q represent generators of a Grassman algebra. Clearly, this representation is not very useful for semi-classical quantization, especially in case of many degrees of freedom. It is difficult to find classical solutions in terms of anti-commuting numbers [28]. In what follows, we describe an alternative path integral for spin that involves ordinary c-numbers and, consequently, is more appropriate for our purpose. We start from the well known harmonic oscillator representation of angular momentum [7],
Here, era are the standard Pauli matrices and we have used the notation
A,+]. [ 21 A+=[Al+, n
d=
Ai and &
2,
are Bose creation-annihilation [Ai
) Al,+]
=
sij
operators: )
1,2.
i,j=
Introducing the total occupation number operator fl = A+& relation 92 =f(-$+
1).
(3.6) one obtains the
(3.7)
Consequently, this description allows for arbitrary value of the total spin. In order to account for a fixed value of the angular momentum, denoted by s, one is forced to work in the subspacecharacterized by the constraint: [IQ - 2S]l) = 0.
(3.8)
A functional integral on this restricted subspaceis obtained by using the modified Hamiltonian H, = H(A+iA)
+ X(AfA - 2S),
(3.9)
CONTINUOUS
SPIN
117
CHAIN
where the Lagrange multiplier X is introduced to enforce the constraint. The presence of a constraint necessitates a gauge condition #(A). In choosing #J(A), the only restriction is that the Poisson bracket
{4(4 A+&.B.
(3.10)
be different from zero. Following the general formalism integral representation of the transition amplitude reads 2 =
s
fi BA:9Ai
of Faddeev [18], the path
n 6[A+(t) A(t) - 24 ’ &#(A)]
i=l
t
x W4(4, WP.~.exp[i/dtL], where the Lagrangian
(3.11)
is given by 2 L = ; A+ z A - H(A+?A).
Next, we introduce the spin integration
(3.12)
variables through the identity
IS%? g S[P(t)
- A++A]
= 1,
(3.13)
and totally eliminate the harmonic oscillator variables A and A*. This is possible, because the &conditions (3.13) together with the gauge condition present in (3.11) provide four equations, which allow for the inversion. We use the following simple gauge condition: $(A) = A, + A; = 0.
(3.14)
It then follows that A, = i(s + S3)1/2,
(3.15) A
2
The Lagrangian (3.12) now becomes L(S, S) =
s,s,
- s,s, s + s,
- H($,
(3.16)
whereas the functional integral reduces to 2 = I 99 v S[Sz(t) - s2] exp [i I dt L(f?, !?)I,
(3.17)
118
JEVICKI
AND
PAPANICOLAOU
which is our final form. Here, S(t) is a c-number integration variable and, consequently, such a representation allows for semi-classical approximations. The classical equations of motion obtained by varying our Lagrangian (3.16), with the constraint sz = s2, are identical in form to the Heisenberg equations satisfied by the spin operators. Conserning the form of this Lagrangian, we mention the following. Classical spin Lagrangians have been proposed by solid state physicists, who have extensively studied the classical theory of rigid magnetic continuum. They all differ from (3.16). It now becomes clear, however, that the form of the Lagrangian depends on the gauge condition. If, instead of (3.14), we choose Re A, = 0 one gets
L’ = _ s2s, - s,s, s - s, The well-known Doering-Gilbert L DG
Lagrangian =
&(&~2 s2 -
- H(s).
(3.18)
[9]
- S2Sl) s,2
-
H(S)
is now given by &(L + L’). All these Lagrangians lead to the same equations of motion, as they differ by a total divergence. Nevertheless, surface terms are important in semiclassical quantization and, for such a purpose, (3.16) will be more convenient. We finally mention that the lack of manifest O(3) invariance in the above expressions is quite analogous to the familiar situation in Quantum Electrodynamics, formulated in a non-covariant gauge. The formalism described above obviously generalizes to systems with many degrees of freedom, like for example, the continuous Heisenberg ferromagnet. Having a Lagrangian, we may now deduce the generator of space translations, namely, the momentum integral. Under the variation $x, t) --+ 3(x + 8X(t), t), the action over a finite time interval [ti , tf] varies according to (3.20)
Hence, the momentum
is given by p = J-+” dx[s + SJ’[s, -m
&s,],
(3.21)
which agrees with the form derived previously in Ref. [5]. It is of particular importance to observe that in contrast to the energy and higher conserved charges, the momentum density is IocalIy gauge dependent. We shall return to this point in the end of this section. We first comment on the validity of our canonical procedure. In fact, in intermediate steps of our derivation, canonical transformations were made, which are usually beset by ordering problems. For a more precise specification of the functional integral, one would need to consider the short-time definition of (3.17). Alternatively, it is more efficient to start from the formalism of geometric quantization [20-221 and
119
CONTINUOUS SPIN CHAIN
derive the precise short-time expression for (3.17). These finer details are not relevant for our subsequent use of (3.17) as we will not be interested in higher order computations. As promised earlier, we return to the classical theory, in view of the canonical procedure developed in this section. At a purely classical level, our preceding discussion may be summarized, or rephrased, asfollows: given a solution A of the classical field equations iAt + [A+T~A]~~ +A + hA = 0, (3.22)
AfA = 2s
the spin density Sa = A++A satisfies the conservation equation (Noether’s equation associated with O(3) invariance)
sta = [@%ss*~]z.
(3.23)
In fact, this equation can be derived from (3.22) for ;\ chosen to be an arbitrary real function X(x, t), since the constraint AI-A = 2s may be enforced by appropriately choosing the initial data. Therefore, the equations of motion are form invariant under the Abelian local gauge transformation: A -+ Aeif(z*t), h + h +ft
,
(3.24)
with f(x, t) arbitrary. The spin density and, consequently, the energy and higher conservation laws are locally gauge invariant, whereas the momentum density P = (i/2) A+a”,A transforms as P ---f P - fz . This explains the absence of the momentum density in the reduction of Sec. IIA. A reduction in terms of the field A, however, necessitates three fundamental invariants, namely P, Hand (2. This and the fact that P is locally gauge dependent is, we believe, the origin of the eigenvalue parameter in the inverse problem. In practice the eigenvalue parameter is introduced by the Galilean transformation described in section 1I.B.
IV. SEMI-CLASSICAL
QUANTIZATION
In what follows, we will quantize the classical soliton solution described in Section II and derive the associated energy spectrum. This semi-classical quantization is based on the path integral representation introduced in the preceding section and the general method of Dashen, Hasslacher and Neveu [9]. Consider G(E) = i I” dT eiETTr[e-iA7]. 0
(4.1)
120
JEVICKI
AND
PAPANICOLAOU
The trace involved here has the path integral representation given by
(3.17), with the action
(4.2)
Summation over all periodic orbits is implied. The general features of the following calculation are similar to those of [23], where the semi-classical spectrum of the non-linear Schrodinger theory was studied. In fact, as was explained in Set 11 at the classical level our present theory is related to the Schrodinger theory. However, the spectrum is totally different. The classical solution (2.31) provides infinitely many periodic orbits, if the following requirements are imposed on the parameters: 2rr
T
7t S(E2+ a214~2) - - 1 Lz -.
L’,,L
T’
1 = 1, 2,..., I1 = 0, kl,...
L represents the length of the “space box.” Calculating
.
(4.4)
the action for the configuration
(2.31) we find AC1 = T[--8~s~ + 4sv sin-lfl],
(4.5)
where /3=
(2 + U~,4S31/2 .
In terms of the parameter & the relation (4.3) reads nL 1 1 (1 - P)l/2
This will be enforced by an appropriate form for the trace (4.1) reads:
nL
x6 [ I
- 2(2%-s7)1/2= 0. A-function identity. Hence, the approximate
1 - 2(27r.~r)~~~] AIeiTzE (1 - /32)1/2
x exp I i4snL [ -
B (1 _ 82J1,2 + sin-l /3] 1.
(4.8)
The S-function has been introduced in a manner that will be convenient for subsequent
CONTINUOUS
SPIN
121
CHAIN
calculations. d, summarizes the contribution of symmetry modes and will be evaluated explicitly in later stages. We have, finally, ignored quantum fluctuations in writing (4.8).
In order to perform the Q--and p-integrations, tion of the a-function occurring in (4.8):
we introduce the following representa-
The effective action separates into T- and P-dependent parts. We first perform the T-integration by stationary phase approximation. The stationary point of the corresponding action A, = /[-88~~(27i.m)~/~ + ET]
(4. IO)
occurs at Tl,z _ 4~~(2+~ 0 E
’
and then
Furthermore,
the Gaussian integration
yields the contribution (4.13)
Next, we perform the p-integration occurs at
again by stationary phase. The stationary point (4.14)
PO = P and the corresponding value of the action is A, = AC0 = 4nsL sin-lp..
The corresponding
Gaussian integration p(1 [
(4.15)
provides the contribution -
p3)3/3
2tlsLI.L
112
1.
(4.16)
Putting everything together, (4.8) simplifies to G(E)
= i const * 1 1 dp (1 -Piz)l,z
n*l
. -$ . Id, exp[i(A,O +.
&‘)I.
(4.17)
122
JEVICKI
AND
PAPANICOLAOU
We have omitted the explicit specification of the overall constant in the above expression, as it is irrelevant for the evaluation of the spectrum. We now elaborate on the computation of d, , which comes from the symmetry modes. Since there is a preferred macroscopic direction, namely, the direction of the spin in the ferromagnetic ground state, implied here by the boundary condition (4.18)
/pm &(x, t) = s,
there exist only two collective coordinates: the phase yO and the position parameter x0. In computing Ll, , we have to consider a slightly more general situation described by the boundary conditions d(O) = y’, f)(T) = cpv+ 2?Tz
(4.19)
x(0) = x’, x(T) = xn + nL.
(4.20)
and
The relations (4.3) and (4.4) are then replaced by 277 S(E2 + v2/4s”) =
‘p” - cpj + 2rrE T ’ (4.21)
u =
X” -
$
+
HL
which in turn imply an explicit dependence of the classical action (4.5) on the boundary values {c$‘, y’} and {x”, x’j. d, is then defined by a2AcL a$ Fx”
2ri A, = 271-L det
a2Ac2 ax’
ax”
’
(4.22)
I
After some algebra, one finds: A&.
(4.23)
Inserting this into (4.17), we obtain G(E) = i const L f f s d,u (1 -.fL2)1/2 & n=-cr. I=1 X exp Ii [-2x1(-)
+ 4nsL sin-l p]l.
(4.24)
CONTINUOUS
SPIN
123
CHAIN
The sum over 1 is a geometric series and for the summation summation formula
over n we use the Poisson
(4.25)
n=--m Consequently, (4.24) becomes
2~n x 6 4s sin-l p - 7j.
(
(4.26)
This is our final formula for G(E). The spectrum is now easily obtained by looking at the poles of G(E). They are given by 16s3,u2 E = my
m = 1, 2,...,
(4.27)
with 2rrn 4s sin-l p = L
= pn ,
n = 0, *I,...
.
(4.28)
Thus we obtain the following dispersion curve (p,, --+ p) MP)
=
1 - cos(p/2s)
8s3
no
3
m = 1,2,...,
(4.29)
which provides the complete excitation spectrum for the continuous Heisenberg ch)in. The classical version of this result is the dispersion relating the basic observables of the theory, the total energy momentum and spin [5]. Calculating the momentum (3.21) over the one-soliton solution (2.31), the table (2.32) is completed to be Q3 =
d = 4a2,
4ES 3 + v2/4P
’
(4.30)
For fixed s, the three observables E, Q3 andp are expressed in terms of two independent parameters, E and a. Therefore, there exists a relation among them, which is easily found to be B =
8~3
1 - cos(p/2s) Q3
Although this procedure does not explain reproduces Eq. (4.29).
’
the quantization
(4.31) of Q3, it essentially
124
JEVICKI
AND
PAPANICOLAOU
As will be shown below, the leasing semi-classical approximation given by (4.29) appears to yield essentially the exact spectrum of the theory. For this reason, any further considerations concerning the quantum corrections will be omitted from our discussion. However, the reader, who might be interested in this issue, will find the solution of the problem of small oscillations in our appendix. It was mentioned already that this semi-classical spectrum agrees with Bethe’s exact solution, which for a discrete spin one-half chain reads 4KJ
= 2J-
1 - cos K, ,11 .
(4.32)
Here, 1”(, = 2rr17/N, rr = 0, f I,... N, N is the total number of sites taken to be large, and m = 1, 2... . To compare with our result for the continuous chain, we first write the classical Lagrangian for the discrete case
(4.33) where the constraint $” = f is assumed and J is a coupling constant. continuum limit we define
where a is the lattice spacing. With the additional J’S. The Lagrangian
To take the
specification
I
(4.35)
(4.33) reads (4.36)
In the continuum limit, (4.36) leads to our previous Lagrangian (4.2). For the special value of the coupling constant (4.35) and writing K, = 27rn/N = (2nn/L) n = pa, Bethe’s formula (4.32) reads 1 I - cos(pa) %,L(P) = 2 m
>
m --: 1. 2,...,
whereas the semi-classical spectrum for spin one-half is obtained from (4.29) by setting s = 1/2a. It is thus found to coincide with the exact spectrum.
CONTINUOUS SPIN CHAIN
125
V. CONCLUSION The successful treatment of the Heisenberg spin model, with the present semiclassical method, encourages one to study other field theories, with internal degrees of freedom, in a similar fashion. Of current interest is the system of static colored quarks coupled to the Yang-Mills field. The color degreesof freedom may be treated using the first functional integral representation (3.3) which then leads to a classical mechanicsover a Grassman algebra. In some senseAdler’s recent proposal [24] may have someconnection with such an approach. However, it appears much simpler to work with the alternative c-number representation (3.17). A semi-classicalapproach would then require the solution of ordinary classical equations, in fact those constructed in [25]. Khriplovich [26] has recently discusseda particular Ansatz for the solution of this complex system of equations. An attempt to understand the quantum meaning of his Ansatz was made in [27]. In our opinion, the interesting question is to search for more general, non-perturbative solutions of the classical equations including static quark sources. The quantization can then be done in the manner discussedin the present paper. Such solutions do not exist at present. However, we have analyzed in this way the old static strong coupling theories and in that case one easily rederives the well-known isobar energy levels. APPENDIX:
SMALL
FLUCTUATIONS
The Gaussian integration requires the solution of the linearized equations, around the one-soliton solution. The construction of the eigenfunctions and, thereby, the identification of the stability angles and phase shifts [9] is performed systematically through the inverse scattering formalism. The method used in the following was implied by the considerations of [29] and later amplified in [30, 311. Specifically, a sohrtion of the non-linear equations is parametrized by p(X) == b(h, 0)/a(h) and the discrete parameters m and 5 (the notation is explained in Section TIC). We denote it by S”(; p(h), m, 5). The one-soliton solution isgiven by S” ; p(h) = 0, m, 5). The linearized equations are solved by the eigenfunctions S,S”, &,Ja and aJ” calculated at p(X) = 0. The variation with respect to the discrete parameters may be performed directly by using the explicit expressions for the one-soliton solution. A less trivial calculation is required for 6,s” /0=0and is described below. Consider the GLM equation (2.23) for the more general kernel (2.24) which is written as
(A.11 Varying (2.23) with respect to p(h) and setting p(X) = 0 (this is implied in the following) we obtain the equation .’
3
(A.3
126
JEVICKI
AND
PAPANICOLAOU
Here, Kr is the one-soliton kernel of Eq. (2.28), Qz is given by (2.25) and (A.l) with p = 0, whereas S@, and 6Qz are easily calculated to be
Hence, Eq. (A.2) is a linear GLM equation for the unknown kernel 6K(x, y). It is solved in the standard manner. From this point on the computations are straightforward, but somewhat lengthy. We only state the result. Denote by cl(x, x) = X(x, x):
X
/5-
-ip+
c* 1Ale+,
-- Ad, 1;
p2 -&
15 -
(* I2 e-2idl
W-i I II - 5* I pcLA2e-id’ ’ 5
where fj2 = 2hx - 4Pt, A, = (A -
5*1 + /J20 - 0,
A, Es l;(h -
5*) + p25*o
- I).
(A.41
The rest of the notation is that of Sec. IIC. The final result is obtained by varying Eq. (2.27) with respect to p and using SK(x, x) = n(x, x) from (A.4). One finds:
ei(d,-d,) . I<-L-*1. SJ3L-0= ?Th(A - (*)”
(1 JP2Y
x (haa* - (1 + p2m*a + A2a*l+ x5*(1 + p2)2- xp2 I i - i* 12>, * 232 4s+
lo4
=
1
1
cc*1 ?A(:- <*y r” (1 + py- Pa* - (1 + CL21 x [-au* + X(1+ p2)a” + p2 / 5 - 5* 121,
. &,-zm,) x , 5 _ [* 12 2 s,s- I@ = 7rh(X le - [*)" 5*2 (1 Jp213 [A(1 + p2) - 2al3 (A.51
As a check of consistency, one may verify that
;s,s2= s,s,s,+ ~[ss$s++ s+s,s-]= 0.
(A-6)
CONTINUOUS
SPIN
127
CHAIN
For the identification of the stability angles and phase shifts, it is more convenient to use a representation for the spin variable that was previously employed for discussions at the operator level [32]. Introduce a complex field 4 through s, = 1 - $J*$fJ,
s, = (2 - +*+)1/z 4
(A.7)
hence,
Notice that our present computation may be written as
is done for S2 = 1. In terms of C#J, the Lagrangian
L = ; f$* &#I - H(S($)).
(A.9
The analogy with our considerations in Section III is, of course, evident. For a symmetric treatment we further use real spinors (3 defined from
6-t 4* 1 s, + s4 - P _ s, - sa = ~ 2 = 2 (1+ S3)l/Z ’ p = 2iq1 + S3)l/2 * (A.10) The solution of the linearized problem is now given by
x’+’ = 6 o! “P0
and
x’-’ zzzs,, OL 0B = LY(+)1**
(A.1 1)
One finds: X (+) =
1 2(1 + S3) S,(S+ + s-1 - (Si. + s-j s,s3 4(1 + &)3/2 I -2i(l + S,) S,(S+ - SJ + i(S+ - SJ SDS31p=. ’ (A*12)
Inserting in (A.12) the value for S given in (2.30) and 6,s from (A.5), the solution of the small oscillations problem is complete. Restoring the vaIue s for the spin, S2 = s2, the stability angles are given by v = &4sh%,
27r T = S[E2+ “2/4S‘q
and the corresponding phase shift by ‘*@ _- <*> ‘)I 6 = 2arg[ ((A
= 4 iarctg ($)
+ arctg [ 2(h EU,,,4$)]\.
(A.14)
As usual, the sum over the stability angles contains divergences originating from the vacuum fluctuations as well as the short-distance singularities of the present continuous theory. The former are trivial to identify, whereas the latter require a more careful study of the renormalization structure of the theory. We have not investigated this question in detail, but it seems that these renormalization counter595/1==/I-9
128
JEVICKI AND PAPANICOLAOU
terms essentially cancel the above contribution and consequently, one obtains no effect from the first quantum correction as was the case in Ref. [23]. Note: We have recently learned about the work of J. Klauder on C-number path integrals for spin. He used coherent states to obtain a path integral representation which is equal to our exp. (3.17). For a careful discussion and earlier references the reader should consult Ref. [33]. ACKNOWLEDGMENT We are grateful to L. D. Faddeev for informative
conversations on this subject.
REFERENCES 1. H. A. BETHE, 2. P&r. 71 (1931), 205. 2. M. L~~SCHER,Nucl. Phys. B 117 (1976), 475. 3. K. NAKAMURA AND T. SAWADA, Phys. Lett. A 48 (1974), 321. 4. M. LAKSHMANAN, TH. W. RUIJGROK, AND C. J. THOMPSON, Physica The Hague A 84 (1976), 577; AND M. LAKSHMANAN, Phys. Lert. A 61 (1977), 53. 5. J. TJON AND J. WRIGHT, Phys. Rev. B 15 (1977), 3470. 6. L. A. TAKHTAJAN, Phys. Lett. A 64 (1977), 235. 7. K. POHLMEYER, Comm. Math. Phys. 46 (1976), 207. 8. A. NEVEU AND N. PAPANICOLAOU, Comm. Math. Phys. 58 (1978), 31. 9. R. DASHEN, B. HASSLACHER,AND A. NEVEU, Phys. Rev. D 11 (1975), 3424. 10. M. L~~SCHERAND K. POHLMEYER, Nucl. Phys., in press. 11. M. L~~SCHER,Nucl. Phys., in press. 12. R. RAJARAMAN, Phys. Rep. C21 (1975), 227. 13. A. JEVICKI, in “The Significance of Non-linearity in the Natural Sciences,” Orbis Scientiae Plenum, New York, 1977. 14. L. D. FADDEEV AND V. E. KOREPIN, Phys. Rep. C 42 No. 1, 1978. 15. R. DASHEN, B. HASSLACHER, AND A. NEVEU, Phys. Rev. D 12 (1975), 2443. 16. F. A. BEREZIN AND M. S. MARINOV, Particle spin dynamics as the Grassman variant of classical mechanics, ITEP Preprint, 1976. 17. J. SCHWINGER, “On Angular Momentum,” AEG Report NYO-3071, 1952. 18. L. D. FADDEEV, Theor. Math. Phys. (USSR) 1 (1969), 3. 19. See, e.g., G. F. VALENTI AND M. LAX, Phys. Rev. B 16 (1977), 4936. 20. A, A. KIRILLOV, Usp. Mat. Nauk. 17 (1962), 57. 21. B. KOSTANT, in Lecture Notes in Mathematics No. 170, Springer-Verlag, New York/Berlin, 1970. 22. J. M. SOURIAU, “Structure des Systemes Dynamiques,” Dunod, Paris, 1970. 23. C. NOHL, Ann. Physics (N.Y.) 96 (1976), 237. 24. S. L. ADLER, Phys. Rev. D 17 (1978), 3212. 25. S. K. WONG, Nuovo Cimento A 55 (1970), 689. 26. I. B. KHRIPLOVICH, Sov. JETP 74 (1978), 1. 27. R. GIL= AND L. MCLERRAN, MIT Preprint, 1978. 28. See however: P. SENJANOVIE, Nucl. Phys. B 116 (1976), 365, and Ref. [16]. 29. L. D. FADDEEV AND V. E. ZAKHAROV, Functional Anal. Appl. 5 (1972), 280. 30. H. FLASCHKA AND A. C. NEWELL, in “Dynamical Systems, Theory and Applications” (Battele Seattle 1974 Rencontres) (J. Moser, Ed.), p. 441, Springer-Verlag, New York, 1975 AND A. G. NEWELL, in “Proceedings, N. S. F. Conference on Solitons, Tucson, Arizona” (H. Flaschka and D. W. McLaughlin, Eds.), 1975. 31. J. P. KEENER AND D. W. MCLAUGHLIN, Arizona preprint, 1976. 32. T. HOLSTEIN AND H. PRIMAKOFF, Phys. Rev. 58 (1940), 1098. 33. J. KLAWER, Path integrals and stationary phase approximations, Bell Lab., preprint, 1978.
OF PHYSICS
120, 107-128 (1979)
Semi-Classical Spectrum
of the Continuous
Heisenberg
Spin Chain*
A. JEVICKI AND N. PAPANICOLAOU~ The Institute
for
Advanced
Study,
Princeton,
New Jersey
08540
Received August 29, 1978
We develop a path integral formalism that allows for semi-classical quantization of systems with spin degrees of freedom. We apply it to study the continuous Heisenberg spin chain, which has been known to possess interesting classical solutions. The calculated semi-classical spectrum turns out to be essentially exact. We also construct a new infinite series of conservation laws that are nonlocal generalizations of the spin.
I. INTRODUCTION
Semi-classical methods have been vigorously investigated in recent years and applied to a number of field theories. They consist of a stationary phase approximation of the functional integral around non-trivial extrema of the action. These are classical solutions of the underlying non-linear field equations. Barring difficuIties in actually solving the classical equations, the method is relatively straightforward for systems with Bose degrees of freedom. Important limitations arise for systems with Fermi degrees of freedom, which are formulated in terms of anti-commuting fields. Semiclassical methods have been successfully used for fermions only in isolated cases, where an effective Bose field may be introduced. Similar problem arise for spin degrees of freedom, since functional integral representations for spin systems have been so far given in terms of anti-commuting fields. It is the purpose of the present article to construct a functional integral representation for spin systems that involves ordinary (commuting) integration variables and, thereby, provides a scheme in which semi-classical quantization follows the standard technique. Interesting physical systems with spin degrees of freedom are wideIy known and there is little need to give here a detailed enumeration. In this paper we shall concentrate on the continuous Heisenberg spin chain. The theory is defined by the Hamiltonian (1.1)
* Research supported by the Department of Energy under Grant EY-76-S-02-2220. +Present address: Department of Physics, University of California, Berkeley, Calif. 94720.
107 0003-49 16/79/070107-22$05.00/O CopyrIght f?‘i 1979 by Aradrmlr Prrss. Inc. All rights of reproduction in any form rescrwd.
108
JEVICKI AND PAPANICOLAOU
which together with the commutation yields the equation of motion
relations [sa(x), @(a!)] = i@sr(~)
6(x - x’)
s2(x) is fixed to be s(s + 1). The exact spectrum of the discrete isotropic chain was obtained long ago by Bethe [l]. His Ansatz for the spin eigenstates (known as Bethe’s hypothesis) was subsequently applied to obtain the exact solution for a class of interesting generalizations of the original Heisenberg chain. The literature in this direction is too extensive to be cited here. At any rate, these generalizations are not directly relevant to our present work. The remarkable structure of Bethe’s solution can be presently understood as deriving from an interesting property of the Heisenberg chain, namely, its complete integrability. This is most directly implied by the existence of higher conservation laws constructed in [2]. More complete results are available for the classical theory of the continuous chain, Eq. (1.2). Besides the spin-wave solution, localized soluton solutions were recently obtained in [3, 4, 51 and subsequently rederived by systematic inverse scattering methods in [6]. However, the relevance of the above classical solutions to the quantum theory has been as yet unclear. This problem is solved in the present article. Semi-classical quantization of the soiiton solution leads, in fact, to the exact magnon spectrum. Aside from providing an interesting illustration of the method for semi-classical quantization of spin, the continuous ferromagnetic chain appears to be an interesting example in which to test inverse scattering methods. Thus we present here some results concerning the inverse problem of the continuous spin chain. We construct a new infinite series of higher conservation laws that are non-local generalizations of the spin. This establishes a close connection with a large class of two-dimensional field theories that have been solved by inverse scattering methods. These are the nonlinear o-models of [7] and the classical Fermi interactions of [8]. The interesting common feature of these theories is that the origin of the Lax representation is understood to a considerable extent. Augmenting the above class by new systems, such as the ferromagnetic chain, should certainly prove useful in trying to identify the origin of the frequent occurrence of integrable systems. The plan of the article is as follows: Section II provides the necessary background from the underlying classical theory and presents the new series of non-local conservation laws. Our path integral representation for a general spin system is given in Section III. It is used in Section IV in order to carry out a DHN [9] semi-classical calculation of the spectrum of the continuous ferromagnetic chain. The problem of small oscillations around the onesoliton solution is solved in an Appendix. II. CLASSICAL THEORY A. The reduction technique. It was already mentioned in the introduction that the purpose of this section is to provide the necessary calculational background for
CONTINUOUS
SPIN
109
CHAIN
our subsequent semi-classical discussion, and to present some new results concerning the mathematical structure of the theory. In particular, we construct a new series of conservation laws, which are non-local generalizations of the spin. This and previous results on the subject suggest a strong analogy with the situation in the non-linear a-models and the classical Fermi interactions analyzed at length in [7, 81. We should add, however, that our present discussion of the classical theory is not meant to be complete. We begin with a brief outline of the reduction technique originally employed in [4] for the construction of the one-soliton solution of the continuous Heisenberg chain. Let Sa, a = 1, 2, 3 denote the continuous spin variable, a function of space and time, such that S2 = SaSa = 1. We form the following set of orthonormal vectors, to be referred to as the moving trihedral:
ela = &
,
EabcSbS,cc
e2a =
t2Hjl,s
,
e3a = S”,
(2.1) H = @‘zaS,a,
eiaeja= aij ,
where cabcis the usual antisymmetric tensor and the summation convention is implied. We have also used an obvious notation for the differentiation symbol. An arbitrary vector formed out of S” and its derivatives may be expanded in the above complete basis with components that are invariant under O(3) rotations. By a simple induction, it can be shown that all such invariants may be expressed in terms of the fundamental set of invariants s2=
1,
H = $S,“S,“,
QSC “bcsas,bs~,
(2.2)
and their derivatives. In this notation, the following expansion will be useful for our calculations:
ela
f
-
(2$/2
e2a
-
2He3a.
We now examine the variation of the trihedral under space displacements. elementary completeness arguments, one finds:
et, = Gikeka, a =
1,2,3,
0.3)
Using
(2.4)
where the matrix C, is a linear superposition of the standard anti-hermitean generators of O(3), with real coefficients that are functions of the fundamental invariants Hand Q: Cl = w1=z=,
[I”, I*] = @ZC,
{cola} = (0, -(2H)‘i2,
-Q/2HJ.
(2.5)
110
JEVICKI
AND
PAPANICOLAOU
Passing to the spinor representation and using the T-matrices defined by ta = (l/2$@, where ua are the Pauli matrices, C, reads (2H)‘/” -i Q/2H
1’
(2.6)
Our statements so far have been independent of the time evolution of the spin variable. The continuous Heisenberg spin chain is described by the evolution equation qa
=
E abSbs:a
=
[~abCSbSZc]Z
(2.7)
)
which is in the form of a conservation law implying the conservation of spin. Using (2.7) and the reduction technique outlined above, it is easy to find that the time displacement of the trihedral is described by e$ = C2,ikeka. The matrix C, is given, in the spinor representation,
c,
by
(WV’“>m ff, - iQ (2H)lj2 ’ (2H)li2 4H2 Hz + iQ Q2 + WW2h. (2H)l12 ’ - 4H2 (2H)‘/”
__-Q2
1
= 3
/ The integrability
G33)
*
(2.9)
1
condition for the system of Eqs. (2.4), (2.8) results into: (2.10)
or, in an equivalent form: Ht + Q, = 0,
Q,=[H2+
HH,,
- Hz2 - Qz II * H
(2.11)
Since all invariants calz be expressed in terms of H, Q and their space derivatives, Eqs. (2.11) provide a complete description of the dynamics of invariants. They constitute the “reduced” system associated with the continuous spin chain. They are found in the form of local conservation laws, implying the conservation of the energy s dx H and revealing a “higher” conserved charge, namely s dx Q. The latter hasbeen incorrectly identified in [4] as the momentum of the system. A correct definition of a momentum integral was given in [5]. We shall return to it in Section III. We emphasize
CONTINUOUS
SPIN
111
CHAIN
here, however, that it does not belong to the set of conservation laws following from Eqs. (2.11). For example, the next conserved charge is given by Q’ = J-r dx[S&S&
+ $(S:S$)‘].
(2.12)
The charge density appearing in this equation can be reduced in terms of H and Q. One obtains: Q’ = r’:
dx [9H2 + H’2;
Q2 ]
(3.12)
It is interesting to note that in its reduced form the above density is non-polynomial. The characteristic feature of the present series of charge densities is that they are invariant under O(3) rotations. Roughly speaking, they are generalizations of the energy density. An independent series of conservation laws generalizing the spin will be constructed in the following. For the moment, however, we turn our attention to an interesting identification of the system (2.11) made in [4]. It is shown by direct substitution that given a complex field # = aleiP satisfying the non-linear Schrodinger equation: (2.14) ih + 9L!z + H+*#># = 0 the quantities H E a2/2 and Q = a2pZ satisfy the system of Eqs. (2.11). Hence, the “reduced system” of the continuous Heisenberg chain is the non-linear Schrodinger theory, in a manner analogous to the situation in the non-linear u-models and the Gross-Neveu models, where the “reduced systems” are the sine- and sinh-Gordon equations and their appropriate generalizations. Correspondingly, it is natural to expect that the system of Eqs. (2.4, 9) provide the associated “linear” problem (Lax representation). However, the above line of argument is at this point incomplete, as no principle for the introduction of an eigenvalue parameter has been identified. We shall return to this question in later stages of our discussion. B. Non-local conservation laws. An independent method was proposed by Takhtajan in [6]. For this discussion, it is somewhat more convenient to use matrix notation specified by s = SV, s2 = 1.
(2.15)
The equation of motion (2.7) is transctibed into: 1
1
St = i L%&,I = jj Lx &la!* It is easy to show that the following “linear” R, = -iASR,
Rt =
(2.16)
equations: - G [S, S,] $ 2ih2S/ R
(2.17)
112
JEVICKI
AND
PAPANICOLAOU
are compatible in virtue of (2.16) for arbitrary constant X. The system (2.17) is the Lax representation used in [6] for a systematic solution of the inverse problem. Given the reduction of the present theory to the non-linear Schrodinger theory, a procedure that involves O(3) invariant quantities, it is somewhat surprising that a linear problem such as (2.17) exists-its distinguishing feature being that the Lax matrices are realized in terms of O(3) variant spin variables. However, such phenomenon is not unfamiliar. It systematically occurs in the systems analyzed in [7, 81. An equivalen way of stating Eqs. (2.17) is the following: given a solution S of (2.16), the linear system (2.17) may be solved for R, which can be chosen to be a (space-time dependent) unitary matrix. Furthermore, one can show by direct substitution that [R-‘SRI (x + 4ht, t) is also a solution. It may be said that the above construction provides a non-linear implementation of Galilean boosts. Previous experience suggests that a “linear” system of the above nature generates a new series of conservation laws-the non-local conservation laws. They were first constructed in [lo] for the non-linear u-model and can be easily shown to exist in the classical Fermi interactions of [8]. Their significance for the quantum theory was studied in [II]. We briefiy outline the construction of such conservation laws in the present theory and give a few examples. For a concise derivation of the conservation laws, we introduce dual potentials as follows: Denote by J = (1/2i)[S, S,] so that the equation of motion (2.16) reads S, = J, . The current densities S and J also satisfy the “pure gauge” condition
is,+J[s,J]
=o
(2.18)
following merely from S2 = I. Introduce the dual potential Q = .Qv satisfying the compatible equations fin, = S, f& = J. (2.19) With this variable the current conservation reads trivially on,, = fit, , wheras Eq. (2.18) becomes the equation of motion for Sz: (2.20)
iQ,, + $[!2, ) &Qn,]= 0. Correspondingly,
Eqs. (2.17) are transcribed R, = -iihS,R,
which are compatible in virtue Eqs. (2.21) may be used for a At each stage of this recursion, sequence of conservation laws,
into
R, = [-ihs2,
f 2ih28,]R,
(2.21)
of (2.20). Expanding R as R = 1 + AR, + PR, ,.... recursive determination of the coefficients R, , R, . .. . the integrability condition (2.20) is iterated to yield a the first few of which are:
it&., = $A$+) (a,Q),
= (Q&2 - 2&J,
. .. . .
(2.22a)
Expressed in the original variables, they lead to both local and non-local conservation
CONTINUOUS
laws. For instance, the conserved (2.22a) read
SPIN
113
CHAIN
charges corresponding
to the conservation
(2.22b)
plo = i+= d-Y dx’ e(s - s’) S”(x) S”(x’) --a, Q1”
=z c+c
laws
:m rix dY’ e(s - x’) Sb(x) SC(x’) )... . s - rx
19(x - x’) is the step function. As expected the first charge of this series is the integrated spin. We finally note that for ferromagnetic boundary conditions the above integrals diverge in general. They become finite after subtracting their values in the ferromagnetic ground state. This complication will cause no difficulty in our subsequent considerations. The remarkably similar structure of such diverse systems as the non-linear umodels, classical Fermi interactions and the present continuous ferromagnetic chain suggests that there exists a governing fundamental principle, whose full extent remains to be identified. Conserning the present theory, one more step has to be taken, namely, to explain the origin of the eigenvalue parameter in (2.17) and, consequently, introduce an eigenvalue parameter in the linear system of Section A. We shall have additional comments to make in the end of Section 111. C. Calculational details. In the remainder of this section, we turn our attention to more pragmatic considerations, in order to prepare our semi-classical calculation. We summarize here the necessary information concerning the one-soliton solution. Our computations are based on Takhtajan’s inverse problem, Eqs. (2.17). For later purposes, we shall need the explicit form of the kernel of the Gel’fandLevitan-Marchenko (GLM) equation; we therefore, repeat the calculation of [6] in some more detail. The relevant GLM equation reads
0,
(2.23)
for .Y < J’. The matrices Q1 and @, are determined from (2.24) by the relations (2.25)
114
JEVICKI
AND
PAPANICOLAOU
The form (2.24) for F is sufficiently general to cover the one-soliton solution and small fluctuations around it. The time evolution of the scattering data is given by a(h) = const,
5 = const,
b(h, t) = e--4iA2tb(X, O),
m(t) = e-*icetm,
(2.26)
5 is in general complex, Im 5 3 0. In terms of the kernel K, the solution of Eq. (2.16) is given from
S(x, t) = [X(x, x; t) - UJ uJiK(x, x; t) - up,
(2.27)
where o3 is the usual Pauli matrix. The one-soliton solution is obtained by solving the GLM equation with b = 0. The kernel that solves Eq. (2.23) is then
K,(x, Y; t) = Mx; t> ewl&d, (2.28) Here, * P2 = , cm_mi* 12 exp[2i(< - 5*)x - 4i(P - <*3t],
(2.29)
qSl = Re[25x - 412t] + arg m. Inserting these values into (2.27) and taking 5 = v/4 + i(e/2), we obtain the onesoliton solution 1 s, SE cos 6 = 1 2E2 3 + v2/4 cosh2 E(X - vt - x,,) ’
(2.30)
S, = S, + iS2 = sin Bei*, d = ( c2 + f)
t + ; (x - vt) + arctg [c th(x
- vt - x,)] + q.
with appropriate identification of the trivial constants x,, , q,, . Arbitrary value s for the total spin (Sz = s2) is achieved by the substitutions S -+ sS and t -+ st. By an additional resealing sv -+ a, we get S = s{sin 0 cos 4, sin 0 sin #J, cos e}; S2 = s2; cos 8 = 1 r#l = ,(,2
2.52 1 E2+ vy4s2 ' cosh2 ‘(x - vt - x0) ’ + &
) t + % (x - vt) + arctg [+
(2.31) thr(x - vt - %)I + 91 .
CONTINUOUS SPIN CHAIN
115
The physical content of this solution is not difficult to identify. In the rest frame (U = 0), we obtain a quasi-static configuration for which the third component of the spin is time-independent, whereas the total spin undergoes a precession around the third axis with frequency w = SE2. Concerning the space distribution of the spin density, this solution describes a localized excitation from the ferromagnetic ground state. In fact, everywhere but in a small region, the spin points on the (time) average along a fixed direction and possesses the vacuum value S, N 1, which characterizes the ferromagnetic ground state configuration. For a moving soliton (V f 0), this picture remains essentially the same, except for an interesting Doppler, velocity dependent, shift of the frequency: w = s(e2 + v2/4s2). The above intuitive picture will be gradually sharpened, as we will proceed with the semi-classical computation. To conclude this section, we calculate some relevant “observables.” Thus the energy, the total spin along the third direction and the first “higher” charge are given by:
d= I-02+a dx
e = j-r
H(x)
= 4es2,
dx Q(x) = 2Evs2’
The above values refer to the excited spin; the contributions from the ferromagnetic ground state have been subtracted. An important omission in (2.32) is the momentum integral. This should wait for the considerations of the following section.
III.
PATH INTEGRAL
FOR SPIN
It is well known that the path integral method provides the most appropriate framework for semi-classical quantization, [12-141. In practice, however, important limitations of the applicability of the method may occur: whereas for Bose fields classical solutions can be used directly to dominate the path integral, in the case of Fermi or spin degrees of freedom additional effort is required. This is mostly due to the fact that the corresponding path integral is usually defined over anti-commuting numbers. Consequently, semi-classical calculations involving fermions have been performed only in a few isolated cases, where it was possible to introduce an effective Bose field (see [ 151). We shall be interested in quantum systems that contain spin degrees of freedom. At the operator level, the procedure is standard. Let 9, CI= 1, 2, 3 be the spin operators satisfying the commutation relations: [9, sib] = i@c!3, 32 = s(s + l), s = 4, l... .
(3.1)
116
JEVICKI
Given a Hamiltonian equation of motion
AND
H = H(s),
PAPANICOLAOU
the time evolution
is dictated by the Heisenberg
$9(r) = f [9. H(S)].
(3.2)
The standard functional integral representation for this system is given in terms of anti-commuting numbers. For example, for spin one-half one has, [16] (3.3) where Q , Q and Q represent generators of a Grassman algebra. Clearly, this representation is not very useful for semi-classical quantization, especially in case of many degrees of freedom. It is difficult to find classical solutions in terms of anti-commuting numbers [28]. In what follows, we describe an alternative path integral for spin that involves ordinary c-numbers and, consequently, is more appropriate for our purpose. We start from the well known harmonic oscillator representation of angular momentum [7],
Here, era are the standard Pauli matrices and we have used the notation
A,+]. [ 21 A+=[Al+, n
d=
Ai and &
2,
are Bose creation-annihilation [Ai
) Al,+]
=
sij
operators: )
1,2.
i,j=
Introducing the total occupation number operator fl = A+& relation 92 =f(-$+
1).
(3.6) one obtains the
(3.7)
Consequently, this description allows for arbitrary value of the total spin. In order to account for a fixed value of the angular momentum, denoted by s, one is forced to work in the subspacecharacterized by the constraint: [IQ - 2S]l) = 0.
(3.8)
A functional integral on this restricted subspaceis obtained by using the modified Hamiltonian H, = H(A+iA)
+ X(AfA - 2S),
(3.9)
CONTINUOUS
SPIN
117
CHAIN
where the Lagrange multiplier X is introduced to enforce the constraint. The presence of a constraint necessitates a gauge condition #(A). In choosing #J(A), the only restriction is that the Poisson bracket
{4(4 A+&.B.
(3.10)
be different from zero. Following the general formalism integral representation of the transition amplitude reads 2 =
s
fi BA:9Ai
of Faddeev [18], the path
n 6[A+(t) A(t) - 24 ’ &#(A)]
i=l
t
x W4(4, WP.~.exp[i/dtL], where the Lagrangian
(3.11)
is given by 2 L = ; A+ z A - H(A+?A).
Next, we introduce the spin integration
(3.12)
variables through the identity
IS%? g S[P(t)
- A++A]
= 1,
(3.13)
and totally eliminate the harmonic oscillator variables A and A*. This is possible, because the &conditions (3.13) together with the gauge condition present in (3.11) provide four equations, which allow for the inversion. We use the following simple gauge condition: $(A) = A, + A; = 0.
(3.14)
It then follows that A, = i(s + S3)1/2,
(3.15) A
2
The Lagrangian (3.12) now becomes L(S, S) =
s,s,
- s,s, s + s,
- H($,
(3.16)
whereas the functional integral reduces to 2 = I 99 v S[Sz(t) - s2] exp [i I dt L(f?, !?)I,
(3.17)
118
JEVICKI
AND
PAPANICOLAOU
which is our final form. Here, S(t) is a c-number integration variable and, consequently, such a representation allows for semi-classical approximations. The classical equations of motion obtained by varying our Lagrangian (3.16), with the constraint sz = s2, are identical in form to the Heisenberg equations satisfied by the spin operators. Conserning the form of this Lagrangian, we mention the following. Classical spin Lagrangians have been proposed by solid state physicists, who have extensively studied the classical theory of rigid magnetic continuum. They all differ from (3.16). It now becomes clear, however, that the form of the Lagrangian depends on the gauge condition. If, instead of (3.14), we choose Re A, = 0 one gets
L’ = _ s2s, - s,s, s - s, The well-known Doering-Gilbert L DG
Lagrangian =
&(&~2 s2 -
- H(s).
(3.18)
[9]
- S2Sl) s,2
-
H(S)
is now given by &(L + L’). All these Lagrangians lead to the same equations of motion, as they differ by a total divergence. Nevertheless, surface terms are important in semiclassical quantization and, for such a purpose, (3.16) will be more convenient. We finally mention that the lack of manifest O(3) invariance in the above expressions is quite analogous to the familiar situation in Quantum Electrodynamics, formulated in a non-covariant gauge. The formalism described above obviously generalizes to systems with many degrees of freedom, like for example, the continuous Heisenberg ferromagnet. Having a Lagrangian, we may now deduce the generator of space translations, namely, the momentum integral. Under the variation $x, t) --+ 3(x + 8X(t), t), the action over a finite time interval [ti , tf] varies according to (3.20)
Hence, the momentum
is given by p = J-+” dx[s + SJ’[s, -m
&s,],
(3.21)
which agrees with the form derived previously in Ref. [5]. It is of particular importance to observe that in contrast to the energy and higher conserved charges, the momentum density is IocalIy gauge dependent. We shall return to this point in the end of this section. We first comment on the validity of our canonical procedure. In fact, in intermediate steps of our derivation, canonical transformations were made, which are usually beset by ordering problems. For a more precise specification of the functional integral, one would need to consider the short-time definition of (3.17). Alternatively, it is more efficient to start from the formalism of geometric quantization [20-221 and
119
CONTINUOUS SPIN CHAIN
derive the precise short-time expression for (3.17). These finer details are not relevant for our subsequent use of (3.17) as we will not be interested in higher order computations. As promised earlier, we return to the classical theory, in view of the canonical procedure developed in this section. At a purely classical level, our preceding discussion may be summarized, or rephrased, asfollows: given a solution A of the classical field equations iAt + [A+T~A]~~ +A + hA = 0, (3.22)
AfA = 2s
the spin density Sa = A++A satisfies the conservation equation (Noether’s equation associated with O(3) invariance)
sta = [@%ss*~]z.
(3.23)
In fact, this equation can be derived from (3.22) for ;\ chosen to be an arbitrary real function X(x, t), since the constraint AI-A = 2s may be enforced by appropriately choosing the initial data. Therefore, the equations of motion are form invariant under the Abelian local gauge transformation: A -+ Aeif(z*t), h + h +ft
,
(3.24)
with f(x, t) arbitrary. The spin density and, consequently, the energy and higher conservation laws are locally gauge invariant, whereas the momentum density P = (i/2) A+a”,A transforms as P ---f P - fz . This explains the absence of the momentum density in the reduction of Sec. IIA. A reduction in terms of the field A, however, necessitates three fundamental invariants, namely P, Hand (2. This and the fact that P is locally gauge dependent is, we believe, the origin of the eigenvalue parameter in the inverse problem. In practice the eigenvalue parameter is introduced by the Galilean transformation described in section 1I.B.
IV. SEMI-CLASSICAL
QUANTIZATION
In what follows, we will quantize the classical soliton solution described in Section II and derive the associated energy spectrum. This semi-classical quantization is based on the path integral representation introduced in the preceding section and the general method of Dashen, Hasslacher and Neveu [9]. Consider G(E) = i I” dT eiETTr[e-iA7]. 0
(4.1)
120
JEVICKI
AND
PAPANICOLAOU
The trace involved here has the path integral representation given by
(3.17), with the action
(4.2)
Summation over all periodic orbits is implied. The general features of the following calculation are similar to those of [23], where the semi-classical spectrum of the non-linear Schrodinger theory was studied. In fact, as was explained in Set 11 at the classical level our present theory is related to the Schrodinger theory. However, the spectrum is totally different. The classical solution (2.31) provides infinitely many periodic orbits, if the following requirements are imposed on the parameters: 2rr
T
7t S(E2+ a214~2) - - 1 Lz -.
L’,,L
T’
1 = 1, 2,..., I1 = 0, kl,...
L represents the length of the “space box.” Calculating
.
(4.4)
the action for the configuration
(2.31) we find AC1 = T[--8~s~ + 4sv sin-lfl],
(4.5)
where /3=
(2 + U~,4S31/2 .
In terms of the parameter & the relation (4.3) reads nL 1 1 (1 - P)l/2
This will be enforced by an appropriate form for the trace (4.1) reads:
nL
x6 [ I
- 2(2%-s7)1/2= 0. A-function identity. Hence, the approximate
1 - 2(27r.~r)~~~] AIeiTzE (1 - /32)1/2
x exp I i4snL [ -
B (1 _ 82J1,2 + sin-l /3] 1.
(4.8)
The S-function has been introduced in a manner that will be convenient for subsequent
CONTINUOUS
SPIN
121
CHAIN
calculations. d, summarizes the contribution of symmetry modes and will be evaluated explicitly in later stages. We have, finally, ignored quantum fluctuations in writing (4.8).
In order to perform the Q--and p-integrations, tion of the a-function occurring in (4.8):
we introduce the following representa-
The effective action separates into T- and P-dependent parts. We first perform the T-integration by stationary phase approximation. The stationary point of the corresponding action A, = /[-88~~(27i.m)~/~ + ET]
(4. IO)
occurs at Tl,z _ 4~~(2+~ 0 E
’
and then
Furthermore,
the Gaussian integration
yields the contribution (4.13)
Next, we perform the p-integration occurs at
again by stationary phase. The stationary point (4.14)
PO = P and the corresponding value of the action is A, = AC0 = 4nsL sin-lp..
The corresponding
Gaussian integration p(1 [
(4.15)
provides the contribution -
p3)3/3
2tlsLI.L
112
1.
(4.16)
Putting everything together, (4.8) simplifies to G(E)
= i const * 1 1 dp (1 -Piz)l,z
n*l
. -$ . Id, exp[i(A,O +.
&‘)I.
(4.17)
122
JEVICKI
AND
PAPANICOLAOU
We have omitted the explicit specification of the overall constant in the above expression, as it is irrelevant for the evaluation of the spectrum. We now elaborate on the computation of d, , which comes from the symmetry modes. Since there is a preferred macroscopic direction, namely, the direction of the spin in the ferromagnetic ground state, implied here by the boundary condition (4.18)
/pm &(x, t) = s,
there exist only two collective coordinates: the phase yO and the position parameter x0. In computing Ll, , we have to consider a slightly more general situation described by the boundary conditions d(O) = y’, f)(T) = cpv+ 2?Tz
(4.19)
x(0) = x’, x(T) = xn + nL.
(4.20)
and
The relations (4.3) and (4.4) are then replaced by 277 S(E2 + v2/4s”) =
‘p” - cpj + 2rrE T ’ (4.21)
u =
X” -
$
+
HL
which in turn imply an explicit dependence of the classical action (4.5) on the boundary values {c$‘, y’} and {x”, x’j. d, is then defined by a2AcL a$ Fx”
2ri A, = 271-L det
a2Ac2 ax’
ax”
’
(4.22)
I
After some algebra, one finds: A&.
(4.23)
Inserting this into (4.17), we obtain G(E) = i const L f f s d,u (1 -.fL2)1/2 & n=-cr. I=1 X exp Ii [-2x1(-)
+ 4nsL sin-l p]l.
(4.24)
CONTINUOUS
SPIN
123
CHAIN
The sum over 1 is a geometric series and for the summation summation formula
over n we use the Poisson
(4.25)
n=--m Consequently, (4.24) becomes
2~n x 6 4s sin-l p - 7j.
(
(4.26)
This is our final formula for G(E). The spectrum is now easily obtained by looking at the poles of G(E). They are given by 16s3,u2 E = my
m = 1, 2,...,
(4.27)
with 2rrn 4s sin-l p = L
= pn ,
n = 0, *I,...
.
(4.28)
Thus we obtain the following dispersion curve (p,, --+ p) MP)
=
1 - cos(p/2s)
8s3
no
3
m = 1,2,...,
(4.29)
which provides the complete excitation spectrum for the continuous Heisenberg ch)in. The classical version of this result is the dispersion relating the basic observables of the theory, the total energy momentum and spin [5]. Calculating the momentum (3.21) over the one-soliton solution (2.31), the table (2.32) is completed to be Q3 =
d = 4a2,
4ES 3 + v2/4P
’
(4.30)
For fixed s, the three observables E, Q3 andp are expressed in terms of two independent parameters, E and a. Therefore, there exists a relation among them, which is easily found to be B =
8~3
1 - cos(p/2s) Q3
Although this procedure does not explain reproduces Eq. (4.29).
’
the quantization
(4.31) of Q3, it essentially
124
JEVICKI
AND
PAPANICOLAOU
As will be shown below, the leasing semi-classical approximation given by (4.29) appears to yield essentially the exact spectrum of the theory. For this reason, any further considerations concerning the quantum corrections will be omitted from our discussion. However, the reader, who might be interested in this issue, will find the solution of the problem of small oscillations in our appendix. It was mentioned already that this semi-classical spectrum agrees with Bethe’s exact solution, which for a discrete spin one-half chain reads 4KJ
= 2J-
1 - cos K, ,11 .
(4.32)
Here, 1”(, = 2rr17/N, rr = 0, f I,... N, N is the total number of sites taken to be large, and m = 1, 2... . To compare with our result for the continuous chain, we first write the classical Lagrangian for the discrete case
(4.33) where the constraint $” = f is assumed and J is a coupling constant. continuum limit we define
where a is the lattice spacing. With the additional J’S. The Lagrangian
To take the
specification
I
(4.35)
(4.33) reads (4.36)
In the continuum limit, (4.36) leads to our previous Lagrangian (4.2). For the special value of the coupling constant (4.35) and writing K, = 27rn/N = (2nn/L) n = pa, Bethe’s formula (4.32) reads 1 I - cos(pa) %,L(P) = 2 m
>
m --: 1. 2,...,
whereas the semi-classical spectrum for spin one-half is obtained from (4.29) by setting s = 1/2a. It is thus found to coincide with the exact spectrum.
CONTINUOUS SPIN CHAIN
125
V. CONCLUSION The successful treatment of the Heisenberg spin model, with the present semiclassical method, encourages one to study other field theories, with internal degrees of freedom, in a similar fashion. Of current interest is the system of static colored quarks coupled to the Yang-Mills field. The color degreesof freedom may be treated using the first functional integral representation (3.3) which then leads to a classical mechanicsover a Grassman algebra. In some senseAdler’s recent proposal [24] may have someconnection with such an approach. However, it appears much simpler to work with the alternative c-number representation (3.17). A semi-classicalapproach would then require the solution of ordinary classical equations, in fact those constructed in [25]. Khriplovich [26] has recently discusseda particular Ansatz for the solution of this complex system of equations. An attempt to understand the quantum meaning of his Ansatz was made in [27]. In our opinion, the interesting question is to search for more general, non-perturbative solutions of the classical equations including static quark sources. The quantization can then be done in the manner discussedin the present paper. Such solutions do not exist at present. However, we have analyzed in this way the old static strong coupling theories and in that case one easily rederives the well-known isobar energy levels. APPENDIX:
SMALL
FLUCTUATIONS
The Gaussian integration requires the solution of the linearized equations, around the one-soliton solution. The construction of the eigenfunctions and, thereby, the identification of the stability angles and phase shifts [9] is performed systematically through the inverse scattering formalism. The method used in the following was implied by the considerations of [29] and later amplified in [30, 311. Specifically, a sohrtion of the non-linear equations is parametrized by p(X) == b(h, 0)/a(h) and the discrete parameters m and 5 (the notation is explained in Section TIC). We denote it by S”(; p(h), m, 5). The one-soliton solution isgiven by S” ; p(h) = 0, m, 5). The linearized equations are solved by the eigenfunctions S,S”, &,Ja and aJ” calculated at p(X) = 0. The variation with respect to the discrete parameters may be performed directly by using the explicit expressions for the one-soliton solution. A less trivial calculation is required for 6,s” /0=0and is described below. Consider the GLM equation (2.23) for the more general kernel (2.24) which is written as
(A.11 Varying (2.23) with respect to p(h) and setting p(X) = 0 (this is implied in the following) we obtain the equation .’
3
(A.3
126
JEVICKI
AND
PAPANICOLAOU
Here, Kr is the one-soliton kernel of Eq. (2.28), Qz is given by (2.25) and (A.l) with p = 0, whereas S@, and 6Qz are easily calculated to be
Hence, Eq. (A.2) is a linear GLM equation for the unknown kernel 6K(x, y). It is solved in the standard manner. From this point on the computations are straightforward, but somewhat lengthy. We only state the result. Denote by cl(x, x) = X(x, x):
X
/5-
-ip+
c* 1Ale+,
-- Ad, 1;
p2 -&
15 -
(* I2 e-2idl
W-i I II - 5* I pcLA2e-id’ ’ 5
where fj2 = 2hx - 4Pt, A, = (A -
5*1 + /J20 - 0,
A, Es l;(h -
5*) + p25*o
- I).
(A.41
The rest of the notation is that of Sec. IIC. The final result is obtained by varying Eq. (2.27) with respect to p and using SK(x, x) = n(x, x) from (A.4). One finds:
ei(d,-d,) . I<-L-*1. SJ3L-0= ?Th(A - (*)”
(1 JP2Y
x (haa* - (1 + p2m*a + A2a*l+ x5*(1 + p2)2- xp2 I i - i* 12>, * 232 4s+
lo4
=
1
1
cc*1 ?A(:- <*y r” (1 + py- Pa* - (1 + CL21 x [-au* + X(1+ p2)a” + p2 / 5 - 5* 121,
. &,-zm,) x , 5 _ [* 12 2 s,s- I@ = 7rh(X le - [*)" 5*2 (1 Jp213 [A(1 + p2) - 2al3 (A.51
As a check of consistency, one may verify that
;s,s2= s,s,s,+ ~[ss$s++ s+s,s-]= 0.
(A-6)
CONTINUOUS
SPIN
127
CHAIN
For the identification of the stability angles and phase shifts, it is more convenient to use a representation for the spin variable that was previously employed for discussions at the operator level [32]. Introduce a complex field 4 through s, = 1 - $J*$fJ,
s, = (2 - +*+)1/z 4
(A.7)
hence,
Notice that our present computation may be written as
is done for S2 = 1. In terms of C#J, the Lagrangian
L = ; f$* &#I - H(S($)).
(A.9
The analogy with our considerations in Section III is, of course, evident. For a symmetric treatment we further use real spinors (3 defined from
6-t 4* 1 s, + s4 - P _ s, - sa = ~ 2 = 2 (1+ S3)l/Z ’ p = 2iq1 + S3)l/2 * (A.10) The solution of the linearized problem is now given by
x’+’ = 6 o! “P0
and
x’-’ zzzs,, OL 0B = LY(+)1**
(A.1 1)
One finds: X (+) =
1 2(1 + S3) S,(S+ + s-1 - (Si. + s-j s,s3 4(1 + &)3/2 I -2i(l + S,) S,(S+ - SJ + i(S+ - SJ SDS31p=. ’ (A*12)
Inserting in (A.12) the value for S given in (2.30) and 6,s from (A.5), the solution of the small oscillations problem is complete. Restoring the vaIue s for the spin, S2 = s2, the stability angles are given by v = &4sh%,
27r T = S[E2+ “2/4S‘q
and the corresponding phase shift by ‘*@ _- <*> ‘)I 6 = 2arg[ ((A
= 4 iarctg ($)
+ arctg [ 2(h EU,,,4$)]\.
(A.14)
As usual, the sum over the stability angles contains divergences originating from the vacuum fluctuations as well as the short-distance singularities of the present continuous theory. The former are trivial to identify, whereas the latter require a more careful study of the renormalization structure of the theory. We have not investigated this question in detail, but it seems that these renormalization counter595/1==/I-9
128
JEVICKI AND PAPANICOLAOU
terms essentially cancel the above contribution and consequently, one obtains no effect from the first quantum correction as was the case in Ref. [23]. Note: We have recently learned about the work of J. Klauder on C-number path integrals for spin. He used coherent states to obtain a path integral representation which is equal to our exp. (3.17). For a careful discussion and earlier references the reader should consult Ref. [33]. ACKNOWLEDGMENT We are grateful to L. D. Faddeev for informative
conversations on this subject.
REFERENCES 1. H. A. BETHE, 2. P&r. 71 (1931), 205. 2. M. L~~SCHER,Nucl. Phys. B 117 (1976), 475. 3. K. NAKAMURA AND T. SAWADA, Phys. Lett. A 48 (1974), 321. 4. M. LAKSHMANAN, TH. W. RUIJGROK, AND C. J. THOMPSON, Physica The Hague A 84 (1976), 577; AND M. LAKSHMANAN, Phys. Lert. A 61 (1977), 53. 5. J. TJON AND J. WRIGHT, Phys. Rev. B 15 (1977), 3470. 6. L. A. TAKHTAJAN, Phys. Lett. A 64 (1977), 235. 7. K. POHLMEYER, Comm. Math. Phys. 46 (1976), 207. 8. A. NEVEU AND N. PAPANICOLAOU, Comm. Math. Phys. 58 (1978), 31. 9. R. DASHEN, B. HASSLACHER,AND A. NEVEU, Phys. Rev. D 11 (1975), 3424. 10. M. L~~SCHERAND K. POHLMEYER, Nucl. Phys., in press. 11. M. L~~SCHER,Nucl. Phys., in press. 12. R. RAJARAMAN, Phys. Rep. C21 (1975), 227. 13. A. JEVICKI, in “The Significance of Non-linearity in the Natural Sciences,” Orbis Scientiae Plenum, New York, 1977. 14. L. D. FADDEEV AND V. E. KOREPIN, Phys. Rep. C 42 No. 1, 1978. 15. R. DASHEN, B. HASSLACHER, AND A. NEVEU, Phys. Rev. D 12 (1975), 2443. 16. F. A. BEREZIN AND M. S. MARINOV, Particle spin dynamics as the Grassman variant of classical mechanics, ITEP Preprint, 1976. 17. J. SCHWINGER, “On Angular Momentum,” AEG Report NYO-3071, 1952. 18. L. D. FADDEEV, Theor. Math. Phys. (USSR) 1 (1969), 3. 19. See, e.g., G. F. VALENTI AND M. LAX, Phys. Rev. B 16 (1977), 4936. 20. A, A. KIRILLOV, Usp. Mat. Nauk. 17 (1962), 57. 21. B. KOSTANT, in Lecture Notes in Mathematics No. 170, Springer-Verlag, New York/Berlin, 1970. 22. J. M. SOURIAU, “Structure des Systemes Dynamiques,” Dunod, Paris, 1970. 23. C. NOHL, Ann. Physics (N.Y.) 96 (1976), 237. 24. S. L. ADLER, Phys. Rev. D 17 (1978), 3212. 25. S. K. WONG, Nuovo Cimento A 55 (1970), 689. 26. I. B. KHRIPLOVICH, Sov. JETP 74 (1978), 1. 27. R. GIL= AND L. MCLERRAN, MIT Preprint, 1978. 28. See however: P. SENJANOVIE, Nucl. Phys. B 116 (1976), 365, and Ref. [16]. 29. L. D. FADDEEV AND V. E. ZAKHAROV, Functional Anal. Appl. 5 (1972), 280. 30. H. FLASCHKA AND A. C. NEWELL, in “Dynamical Systems, Theory and Applications” (Battele Seattle 1974 Rencontres) (J. Moser, Ed.), p. 441, Springer-Verlag, New York, 1975 AND A. G. NEWELL, in “Proceedings, N. S. F. Conference on Solitons, Tucson, Arizona” (H. Flaschka and D. W. McLaughlin, Eds.), 1975. 31. J. P. KEENER AND D. W. MCLAUGHLIN, Arizona preprint, 1976. 32. T. HOLSTEIN AND H. PRIMAKOFF, Phys. Rev. 58 (1940), 1098. 33. J. KLAWER, Path integrals and stationary phase approximations, Bell Lab., preprint, 1978.