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Point transform theory of liquid helium
ANNALS
OF PHYSLCS:
62, 320-342
Point
(1971)
Transform
Theory
of Liquid
Helium*
EUGENE P. GROSS+ Brand&
University,
Waltham,
Massachusetts
02154
Received June 19, 1970
The method of point transformations is used to obtain a microscopically based theory of the excitations of a system of interacting bosons. The theory is governed by a transformed Hamiltonian whose ground state is a noninteracting state and which is quadratic in the new momenta. The form of Hamiltonian is compatible with translational, rotational, and Galilean invariance. If the assumption is made that two-body interactions between the dressed particles predominate, two well-behaved functions remain undetermined. In the absence of a microscopic calculation these functions may be fitted using experimental data for liquid helium. The thermal properties of the system can be calculated as with the less rigorously based Landau-Khalatnikov Hamiltonian. The present theory retains the possibility of describing the Bose-Einstein condensation and quantized circulation. I. INTRODUCTION
In the past thirty-five years, many theories have been constructed to explain the striking properties of superfluid helium [l]. The successful theories divide into several classes.Each classintroduces physical ideas and a mathematical structure which contribute to the understanding of some properties, but slights other aspectsof the behavior of helium. One example is the classof microscopic theories that usethe language of ordinary wavefunctions in configuration space [2]. The Bose-Einstein statistics are kept in the forefront by requiring that the ground state wavefunction be nodeless[3]. The simplest type of wavefunction that describesthe spatial order characteristic of any fluid then yields a reasonablevariational theory of the energy, pressure, and sound speed as a function of density at absolute zero [4]. This configuration space approach can also deal quantitatively with the spectrum of elementary excitations and with the inelastic scattering spectrum [5]. The spatial order of the ground state continues to play a central role in the analysis. It is also possible to discuss the scattering of excitations, the structure of a vortex line [6], the behavior of an * Work supported by U. S. Air Force Office of Scientific Research under grant AFOSR 68 1370B. r Visiting Professor of Physics, Massachusetts Institute of Technology, 1970.
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impurity [7], etc. In each of these problems, one needs to examine special stationary states of the system, rather than ensembles. The configuration method becomes increasingly cumbersome when finite temperature properties are the object of analysis. III practice the theory is taken to the point where contact is made with a pseudomicroscopic phenomenological theory like that of Landau and Khalatnikov [S], or generalizations [9] compatible with the existence of quantized circulation. These excitation gas theories fail to describe the Bose-Einstein phase transition. The configuration space method is of course a valid quantum-mechanical approach, and in principle could be extended to account for the phase transition. In practice there are technical difficulties arising from the need to have a complete set of orthonormal states, which implies kinematical interactions between the excitations. The theory then becomes unwieldy and unappealing. A second class of theories of helium is constructed as a generalization of the theory of weakly interacting boson systems [lo]. It does not concern itself primarily with spatial order and focuses attention on the macroscopic occupancy of a singleparticle state (‘condensate’). One obtains a broad understanding of the dynamics of the superfluid at absolute zero and at finite temperatures in a way that is fully compatible with the Einstein condensation [ll]. This approach can be easily formulated in the language of modern many-body theory [12], thus bringing to bear a wide range of techniques for the finite temperature case. However, the second line of thought cannot yet compete with the configuration space approach in making calculations of the properties of the lowest states of liquid helium. This dualistic state of affairs suggests that transformation theory can be useful [13]. The advantages of the configuration space method can be incorporated in a many-body coordinate transformation that maps the interacting ground state into a noninteracting state. The interacting ground state is the square root of the Jacobian. The Hamiltonian in the new frame is then amenable to treatment by the second type of approach. Arguments can be given for the existence of a point transformation that accomplishes this objective [14]. The form of the new Hamiltonian is severely restricted by symmetry considerations and by the fact that it is the result of a point transformation. One key assumption makes the theory immediately useful. It is that the terms of the new Hamiltonian describing pairwise interactions between the ‘dressed’ particles predominate. This assumption has been verified for the weakly interacting Bose gas [ 151. The transformed Hamiltonian contains two functions of momentum which can be fitted to a limited number of experiments at absolute zero, if we are unable or unwilling to calculate them from first principles. We then have a Hamiltonian describing the excitation gas which can be used in ways similar to that of the Landau-Khalatnikov theory. The new Hamiltonian is, however, soundly grounded microscopically and remains compatible with Einstein condensation and quantized circulation phenomena.
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Section 2 is devoted to general considerations of strategy in using point transformations. The fundamental transformed Hamiltonian is stated in Eq. 6. In Section 3 we discussthe basic features of the Hamiltonian and show that it has the characteristics discussedin this introduction. Section 4 is devoted to the derivation of the Hamiltonian by imposing the requisite symmetry restrictions. In Section 5 we show that some of the well-accepted treatments of the weakly interacting boson problem can be related to the point transform approach. These treatments emergewhen certain drastic approximations are made in the differential equations governing the point transformation.
2. GENERAL
CONSIDERATIONS
In the present section we present the broad strategy of this approach to the theory of many-boson systemsand in particular to liquid helium. It is based on our earlier studies of point transformations. The main steps of the procedure are as follows [14]. In the simplest casethe many-body Hamiltonian is
The system is subjected to a many-body coordinate transformation. For easeof presentation we group the 3N coordinates x,,Ji = I,..., N, (Y = 1,2, 3) as a single vector x. Symbolically, X(E) = f(x),
(2)
where E = 1 represents the fully realized transformation and x(0) = X. In consequence,the momenta transform as P(C) = Bbw
+ &X)Pl.
(3)
The transformed Hamiltonian takes the general form
WP, 4 = H(P(~),4’)) = &PA .Ap + W-4.
(4)
The ‘potential’ W(x) in the new frame is a many-body potential consisting of two parts. One part is the transformed direct potential energy V(X(E)). The second part is of quantum origin and comes from the original kinetic energy. It is a result of the particular momentum ordering chosen for the transformed Hamiltonian. These consideration are of a general nature. The first important step is to decide what the coordinate transformation is to accomplish. For bosons a desirable objective is to reduce the potential W(X) to a constant, independent of x. If we could accomplish this, the new Hamiltonian would have the noninteracting state
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of the many-boson system as an exact eigenstate. For a system enclosed in a periodic cube of volume Q this state is @ == (I/vQ)” in a coordinate diagonal description. It is @ = (&“/t/m) Q0 in a number occupation description. Thus H’@ = E&p, , N) @.
(3
Here we have written the constant to which W is reduced as E,(p,), making the assumption that it will be the ground state of the original Hamiltonian that is mapped into the noninteracting state. Since the transformation is unitary, the eigenvalue is unchanged and is Eo(pO, N) with p. = N/Q. (In the future we will choose units where Q = 1 and fi = 1, unless these factors clarify the argument.) It is clear that this is a desirable objective. The nonconstant part of the potential is pAAp/2M and describes solely the excitation gas. For example, in the calculation of the canonical partition function, the ground state contributes only the factor e-aEc . Thus the pressure of the system is the sum of a contribution due to the ground state and a contribution from the excitation gas. The entropy is due solely to the excited states of W. The extent to which our objective can actually be accomplished will be discussed in the Section 3. For the moment we continue to outline the overall strategy. The new Hamiltonian pAAp/2M has the form pWg,,(x)p, , i.e., the kinetic energy appropriate to particles whose motion is described in a curvilinear coordinate system. Loosely speaking there is a velocity-dependent interaction among the ‘quasiparticles.’ There are a number of general restrictions on the form of the metric tensor. The generator of a suitable point transformation should be symmetric in the particle variables and should commute with the generators of translations, rotations, reflections, etc. Furthermore, the fact that the original Hamiltonian is Galilean-invariant must carry consequences for the form of the metric tensor g&z). Finally, there is one feature resulting from the fact that we are dealing with a many-particle system. The metric tensor can be uniquely decomposed into a constant part, an intrinsically two-body additive interaction, etc. Clearly, the point transformation approach will be of real utility only if the intrinsically many-body terms are of minor importance for many interesting situations. One must make detailed calculations of point transformations for particular systems to understand the limitations of the assumption that at most intrinsic two-body interactions are important. If, however, we proceed hopefully to make this key assumption, the general considerations lead to a Hamiltonian H’ = Eo + ; &a+(p)
a(p) +
M,, = A,,(k)
1 K,@, k.:P.~
P, s> a+(p) a+(q) 4~ - k) 4s + k),
(pm ; qu) (pa -. qB - 2k,).
(6)
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Here we have introduced creation and destruction operators for the quasiparticles, i.e., we have passed from a coordinate diagonal representation in the new frame to an occupation number diagonal representation. The symmetric tensor has the structure
L&k) = As(k)&x9,+ F
AT(k),
N=
1,2,3,
,8=
1,2,3,
(7)
where As(k) and AT(k) are two functions of the absolute value of li. It is clear that HI is Hermitian. The translation invariance is apparent from the restrictions on the wavevectors which are arguments of the creation and destruction operators. Rotational invariance is easily verified. The fact that H1 results from a point transformation on the original Hamiltonian is reflected in the simple dependence on p and q, viz., the new interaction is at most quadratic in these wavevectors. Galilean invariance then yields the more specific form written above. Finally, the restriction to two-body additive interactions finds expression in the fact that the interaction contains only two creation and two destruction operators. In a fully microscopic theory, we have to exhibit a particular coordinate transformation and to calculate AS(k) and AT(k) in terms of the direct two-body potential V(r,J. However, the main point of our approach is that we can sidestep these difficuit calculations and have available a microscopically basedphenomenological Hamiltonian. One can take H1 seriously and calculate a limited number of properties at absolute zero, e.g., the phonon roton spectrum, to fix As(k) and XT(~). We can then undertake calculations to make predictions of other properties at absolute zero and of the entire temperature dependent properties of the system. In the next section this claim is spelledout in detail, and some of the key interesting features of H1 are emphasized. In the following section the assertedform of H1 is derived.
3. PROPERTIES OF THE QUASIPARTICLEHAMILTONIAN The vertex MI4 = +(pa - qJ(ps - q. - 2k,) A,,(k) describes a nonlocal interaction which has an important long-range component and is rather ‘soft’ at small distances.This is in sharp contrast to the original Hamiltonian. It is, therefore, not unreasonable to initiate the discussion of H1 with self-consistent field type approximations. Of course, later a more serious many-body analysis must be undertaken. The diagonal part of HI comesfrom taking p = q -t k and is H,’
= EG + c &n(p)
+ c (- 1) fl,, y k#O
n(q) n(q + k).
(8)
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This form can be used to make a first estimate of the phonon roton spectrum, the energy of a single excitation. The state vector (in the new frame) is
i.e.,
Y(p) = a’(p) a+(O>N-l_ lu, 2/(Nl)! and belongs to N(p) = 1, N(0) == N - 1, and N(p’) = 0 for p’ # p, p’ # 0. The eigenvalue is c(p) = gj
-- (P(p)
+ AT(p)] p”(N - 1).
(9)
Accepting this as an adequate estimate of the eigenfunction of HI, the combination P(p) + h=(p) can be fixed so that c(p) is the experimental phonon roton spectrum for a given density. This means in particular that the combination P(p) + XT(p) has the behavior p-l for p -+ 0. The spectrum then describes phonons at long wavelengths. It is clearly a key concern of the microscopic theory to show that this is really the case. This can be readily proved for a dilute gas of hard spheres or for a system of weakly interacting bosons where the point transformation can be analyzed in detail [I 51. For larger values of p, c(p) goes through a maximum (‘antirotons’) and then through a minimum (rotons). There is no difficulty in fitting h(p) since the excitation spectrum at absolute zero is stable. Energy and momentum conservation forbids distintegration of single excitations. For higher values of p, however, we encounter lifetime effects. In first approximation it is reasonable to view the usual phonon roton spectrum as continuing upward past the roton mimimum to a free particle spectrum. Thus we view the higher branch of the inelastic neutron scattering experiments of Woods and Cowley [16] as a flat ‘optical’ branch which crosses the phonon-roton-particle spectrum. The two branches are split by interactions in higher order. To understand the nature of the upper branch we must study the resonant scattering of elementary excitations. We will now indicate how further predictions can be made if h(p) is fixed as discussed. Let us consider the problem of determining the statistical thermodynamics. The canonical partition function QN = Tr(e-OH) is invariant to unitary transformations. Thus QN(T, Q) = Tr(e-5H’), (10) and one does not need to know the coordinate transformation H1. The diagonal part of H1 may be rewritten in the form H,’
= EG + c &n(p)
-
c + k#O
fl,&)
g+F#o ‘+I + k) 4l). Y#O
that takes H into
n(k) n(O)
(11)
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This form isolates the condensate occupancy. from the more symmetrical form
but is really misleading.
K(P - s) = &4o(P - S)(PU - cl&+3 - 40).
We start
(12)
We note that K(p - q) for fixed finitep is slowly varying when q - 0. Furthermore, K(p - q) + 0, j p - q j -k 0. Thus an appropriate decomposition is HI,’ = E, + C i&f
- 2NK(p);
n(p)
+ c (K(P - 9) - K(P) - K(s)> n(p) n(q).
(13)
If the last term is neglected, the system behaves as a collection of independent excitations with the zero temperature excitation spectrum. Thus the canonical partition function is Q&2
7-1 m e--B.% Tr,v{e-““‘W”).
(14)
The excitation gas is, however, different from that of Landau and Feynman. It is subject to the same number constraint as the ideal Bose gas. This guarantees that there is a Bose-Einstein consideration [17]. The calculation of the thermodynamics requires a bilt of care since the excitation spectrum c(p) depends explicitly on the density. Thus a straightforward application of the formalism of the grand partition function leads to errors. This problem is, however, easily overcome [lS]. According to the rules for using the canonical function the Helmholtz free energy is given as [19] A,,&‘,
T) == --k T log QaV,
(154
and the pressure as p=-
aAN i ?.Q ) T’
To evaluate log Q,,, , we use the saddle point method, Introduce the definition e-BW _ Tr{e-a[~‘(P)“(P)-pNl:,
(15b) W(fi, ,QL, Sz) by (16)
where the trace includes a sum over particle number N. The mean particle number is given by
m = - gJti.
(174
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Then
(17b) and A,@,
T) = E, - Np mr IV’.
(18)
The Bose-Einstein condensation temperature is determined in the usual way by setting p = 0. Above the transition temperature /L is determined by the number constraint, and the internal energy is
c’ = 2 4P).f(P),
Differentiating
the number condition
with respect to T, we find
The behavior at the phase transition is determined by the fact that the zero temperature spectrum at long wavelengths has a phonon character. Then I1 and I2 are finite and nonzero as p approaches zero and a(p,B)/liT is finite. Below the phase transitionp = 0 and c, does not contain the +/3)/2Tterm. Thus c, is discontinuous at the transition. The above is only a preliminary analysis. It shows that the fitting of X(p) by the zero-temperature spectrum provides the information that determines the thermodynamics. A more thorough analysis of the Hamiltonian H1 is needed. The treatment of the part of Hnf that has been neglected leads to results obtained earlier using a similarity transformation [18]. The form of the long wave excitations is altered at the transition temperature and the entire spectrum is weakly temperature dependent. But a reliable theory must also consider the off-diagonal part of H1. There are other properties of the boson system that can be predicted from an analysis of HI without knowing the coordinate transformation. Let us consider spatially inhomogeneous condensates and focus attention on the determination of the energy of a vortex line. This is the simplest example of a class that includes the
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energy of a vortex ring, the energy of interaction of a pair of vortex lines, etc. To prepare for the analysis of these problems we go to a spatial description in terms of quantized fields. Let a)(x) = --&;
u(p) &P’S,
a(p) = &
e-iD.x$(x)
(22) s
dx.
Then EG + &j” - $$+
V$J+ v# dx + j” J&& #+(y) $$-$
- y) /p
#+(y) #(y) 9
C(x)/ dx dy.
(23)
If A,,(k) gives rise to phonons, the spatial transform has long-range behavior, i.e., ( x - y /-2. The interaction represents a long-range interaction between current densities and between a stress density and a particle density. We may search for a vortex line solution by making a Hartree variational choice of the state vector [20] (24) where g(x) has the form
g(x)= f(r) eie, I f3dx
= 1
(25)
in cylindrical coordinates. The angular momentum of the state is Nfi, and is the same in the original frame, since the point transformation generator commutes with the components of the angular momentum operator. It is thus a macroscopically occupied state with quantized circulation. The energy of the state is obtained by solving the integrodifferential equation resulting from functional variations off(r). Of course, we need to know the coordinate transformation to find the actual spatial density and velocity pattern, i.e., the description in terms of the original variables. We do not follow up the detailed study of the vortex line energy here. A related problem is the behavior of a system of bosonsin a cylindrical container rotating with angular velocity w. The appropriate Hamiltonian is H - wJz , which becomes H’ - wJz after point transformation. Thus, the study of the energetics and thermodynamics of the system do not require knowledge of the point transformation.
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Another class of problems that can be studied directly using H1 involves the lifetime and scattering of elementary excitations. Here one must consider the off-diagonal part of H1 to obtain even a first estimate. The off-diagonal part is u =
c
4&)
P&O -- qo - 2k3) U+(P) a+(q) 4~ - k) 4s + W
(26)
P#cl+k
Let us write Then
U = V, + V, , where
V, contains no zero-momentum
Vc= c &pa(ps-- 243) a+(p)
wavevectors.
a+(O) a(p - k) u(k) $ C.C.
(27)
P#k k#O P#O
Note that H1 doesnot contain terms with two zero-momentum creation or destruction operators. This is in stark contrast to the original Hamiltonian, and is a consequence of the strategy of the point transform method. If such off-diagonal terms occurred, the noninteracting ground state would not be an exact eigenstate of HI. In the present approach the Bogolyubov transformation does not play as essential a role, even though scattering of condensate particles does occur. There is no ‘depletion’ problem since the ground state of the quasiparticles is a 100y0 occupied zero-momentum state. The occupancy of the zero-momentum state for the original particles [3] can be calculated if one knows the Jacobian of the point transformation, but is not really an essential quantity. The interaction V, describing a direct scattering of two excitations has an interesting structure in virtue of its inherent long-range interactions. The simplest scattering matrix element is (Pi P - P I vs I P’; P - p’) = 4dP - P’mL - ~A&%
- PO>
(28)
and describestwo quasiparticles p’ and P - p’ going to a pair p and P - p. Since (I&p - p’) is weakly singular as / p - p’ 1+ 0, there is a strong preference for small angle scattering. The interaction V, involves the condensate. It is responsible for the finite lifetime of excitations even at absolute zero, provided that the disintegration is permitted by the conservation laws. It also gives a correction in second-order perturbation theory to the diagonal estimate of the excitation spectrum. The fit of h(p) to the experimental spectrum should be based on a more refined estimate that includes the self-energy corrections arising from this off-diagonal part of the Hamiltonian. It is thus clear that a large number of predictions can be made if one calculates seriously with HI. This microscopically based phenomenological Hamiltonian has a number of distinct advantages as compared to the hydrodynamics theory of
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330
Landau and Khalatnikov [S]. As indicated in this section one does not lose the possibility of describing the lambda transition and one is able to describe the development of quantized circulation. These features are mandatory for any phenomenological Hamiltonian that is correctly based on the original microscopic Hamiltonian.
4. DERIVATION
OF THE HAMILTONIAN
In contrast to our earlier work on point transformations, we here work with the infinitesimal generators of point transformations. The unitary transform is written as eirswhere s is Hermitian. Then
X(E) = U--1(c) XU(E),
(294
d.x _ = i[x(c), s]. de
(29b)
For coordinate transformations s is linear in the momenta. The step of removing the potential energy is most easily illustrated in the exactIy soluble two-body problem, where one can then examine the form of the metric form of the metric tensor. We will limit the discussionto making contact with our previous work. In the center of massframe the one-body problem is defined by H = $
+ V(r).
(30)
The generator of the infinitesimal transform is s = ${p . Vqqr) + V+(r) p).
(31)
This contains a general function (b(r) and commutes with the angular momentum. Since p = p1 - pZ is the relative momentum, s is unchanged under the Galilean transform p1 ---fp1 + Mu, pz -+ pz + Mv . The coordinates transform as the solutions of
dx,
a+
de --ax,(E)=
- #
+‘(r(E));
x,(O) = ,u, .
(32)
To solve the equations multiply by x, and sum over 01.Then EE--
7% dy do)
(33)
POINT
We are interested k?(l) = Ye>
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in the result for E = 1. The finite transform
can be written
as
Here (35)
In general, the connection betweenf(r) and 4(r) is not simple. The direct potential energy now enters the transformed potential energy as V(r(1 +f)). To illustrate how the potential energy is reduced to a constant, suppose that V(r) is strong and short-ranged, with a range r - 1. If f(r) has a form c(r + 7)-l e-T/*, it falls to zero for r - 1.We suppose7 < c. If 7 < c, the argument r(1 +f(r)) is greater than c when
i.e., over almost all space, except for a small region near the origin. Thus the effective range of the potential has been decreased. The physical effects of the original direct potential are taken up in two places. One is in the new spatial potential arising in the proper ordering of momentum operators. For finite but large I it has been shown that the new spatial energy only affects the scattering of particles of momentum less than Z-l [I 51. The scattering of all other particles comes from the metric tensor. The new kinetic energy is (36)
where L&r) S(r)
= 1-
= --F(r)
(1 + f)-3,
A,,, $- 5”
Z(r),
Z-f!(r) = [I $ (rf)‘]-’
- (1 + f)-“.
(37)
The metric tensor is regular and Fourier analyzable even in the extreme limit 7 - 0. For example, first taking ZL+ co to remove the spatial dependence of the potential energy, we have
When the transformed Hamiltonian is written in the notation of second quantization we find a structure identical to that of our basic many-body Hamil-
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tonian of Eq. (6). The only difference is that P(k) and AT(k) are both determined by the single functionf(r) (and of course are independent of the density). We now discuss the many-body problem. The generator of the point transform has the form
The symmetry restrictions must now be imposed on the generator. It is convenient to confine ourselves to Fourier analyzable potentials, which may, however, be as strong as one likes. Symmetrical functions of coordinates will be described in terms of the density fluctuations
p”(k) = 2 e-ikz,.
(40)
j=l
This is a one-body additive function. Additive constructed from polynomials of the s(k). The local Hermitian current density is
functions of higher order may be
&(x, 1) = %{j,(x, t> +A+(x? t>>,
(41)
where
The Fourier transforms of these non-Hermitian
currents are
j,+(k) = 5 eikxjpj,a,
(43)
j=l
J,(k) = B’&(k) +&+(-WI,
J,+(k) = .I,( -k).
Let us now examine the simplest meaningful additive function. It has the form s2 = c k&k)
J,(k) ,6-k)
+ cc.
J,(k)
+
Hermitian
generator, a two-body
k
=
c
km@) k
K--k)
C
W”0r)
,W
J&-k).
(44)
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It consists of a single current multiplied by a density fluctuation. It is clear that s commutes with the total momentum operator, and if d(k) is a function of 1k 1, with the angular momentum operators. Under the Galilean transform pi
-
pi
+
Mv,
J,(k)
-
J,(k)
+
s is unaltered if d(k) + d*(k) = 0. Next examine a generator consisting of three-body s2 = 1 d,(k, I) J,(k) p”(l - k) p”(-Z)
MG(k),
additive terms. Let + C.C.
(45)
The choice of wavevectors ensures translation invariance. Rotation invariance requires that & contain only k,S$(k2, I”, k . Z) and l&(k2, 12,k . I). Galilean invariance is ensured if d,(k, I) + d,*(-k, 4) = 0. The general coordinate transformation is a sum of s, up to n = N. It contains a large number of freely disposable functions. However, the symmetry requirements can be imposed term by term. To actually perform the transformation we must examine the differential equations that will determine the new coordinates. They are (46) but it is advisable to first study the transformation
= c k . k’d(k’) k’
$(k + k’) ,5(-k’)
of density fluctuations
+ C.C. + i[p”(k, E), s3 + . ..I.
(47)
Because we are working with point transforms the right side is a sum of polynomials in the density fluctuations. In addition $(k, E) commutes with ;(k’, E’) for general k, E, k’ and E’. Thus these are ordinary nonlinear equations. Suppose that these equations can be solved as a sum of polynomials in the p^(Z,0) with an explicit dependence on E. Then we can find the ground state wavefunction for the interacting system when we put the requirement that the wavefunction in the transformed frame @o , is the noninteracting state .PNj2, YJE) ‘To do this it is first convenient current fluctuations,
= U(E) CD0 = U(E) Q-N/2. to set down the commutators
(48) of the density and
kW, J&‘N = k$(k + W, K(k), J&N = k,‘J,(k + k’) - &LoC + k’).
(49)
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Now examine Ye.
We form dYo/&
__
de
and use the fact that S(E) = s(O),
= N(E) s(0) Q-N/2.
(50)
But is(O) * 1 can be evaluated by moving the momenta to the right. Thus is,(O) . 1 = -1
Pd(k){p”(k,
0) ,5(-k,
0) - N} = iY2(0).
(51)
The higher terms s, are again polynomials in the density fluctuations. We thus have the differential equation __ = i(U(E) Y(O) u-l(E)} YG(E), dE
(52)
Y(0) = c LqO). Since Y(0) is a polynomial in {(k, 0) the operation U(E) Y(0) U-~(E) merely converts it into the samepolynomial of $(k, E), viz., Y(E). Thus
Here we find the argument of the exponential by inserting the solution fj(k, E’) which depends explicitly on ij(k, 0) and on E’. We can pursue the program of showing that one can choose the point transform to make the total new potential energy vanish, using either particle coordinates and momenta or the collective densities and currents. This has already been discussedfor the weakly interacting boson caseusing the particle description [14]. The method of demonstration is to arrange the potential energy as a sum of two-body, three-body, etc., additive functions. This is essentially a decomposition in powers of the density fluctuations. Then one shows that there is enough freedom in the generator of a general coordinate transformation to annul all of theseterms. We do not pretend to make this argument rigorous for strongly interacting bosons. Our main interest is to determine the structure of the metric tensor. It is instructive to pursue the matter from the point of view of the collective variables, since the analysis points to deficiencies of numerous approaches to the many-boson problem. Consider the quantity T&k)
=
5 (pJ, j=l
e-i”“j(pj)B
.
(54)
This is a one-body additive tensor function, but contains two powers of the momentum of the sameparticle. It is closely related to the Fourier transform of the
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stress tensor density. In particular the kinetic energy is the trace of a zero wavevector quantity, viz., Cal T,,(O). Thus the total Hamiltonian is (55) To determine how the Hamiltonian T&O, E). We must integrate
transforms
under a point transform
we need
(5’3 The generator s is constructed from a current density fluctuations. Hence we need
multiplied
by a polynomial
in the
c [Ed% 341 = --2k . .J@), ; [T,.,(o), J,(k)]
=-L -2 c k,T,,(k) a
(57) - +
Fp(k)
It is apparent that we must know J,(k, E) and T&k,
- Pj,, + k, y
l ). A
.
(58)
closed set of equations is
$
(k, c) = i[p”(k, E), s],
(594
2
(k, E) = i[J,(k,
(59b)
E), s],
The first equation has already been discussed. It involves density fluctuations of different wavelengths which commute for different values of E. The right side of the current equation is at most linear in the current fluctuations with coefficients that depend on the density fluctuations. The general form of the solution for J,(k, c) is linear in the J,(Z, 0) with coefficients depending on the density fluctuations as well as inhomogeneous terms depending on density coordinates. To appreciate the implications of the last equation we need
[/W, T,,(k’)l = k,j,(k -t k’) + 2kjdk +- W - k’j,(k + JO + 2kA$(k + W, lid4 ~cdk’)l = W& + k’) + 2&T,, - k,‘T,, - k,‘Tti, + k,‘k&(k
+ k’) + ks’(k,’
- 3k,)j,(k
+ k’).
(60) (61)
336
GROSS
It is apparent that the equation for T&k, C) is linear in the stresses with density fluctuation coefficients. In addition there are inhomogeneous terms containing current fluctuations. To obtain the solution we imagine the solutions for the densities and currents known and substitute them. The solution for T&k, e) is then at most linear in the stresses Trs(l, 0) and quadratic in the currentsjJZ, 0). The form of the point-transformed Hamiltonian of course agrees with what we know from the particle point of view. It is quadratic in the particle momenta with a metric tensor that depends on the coordinates. The present analysis in terms of the collective variables shows us that there are two distinct contributions. One is linear in the stresses and the other quadratic in the currents. It is now a simple matter to find the form of the Hamiltonian compatible with our symmetry requirements. We will consider all the conditions except Galilean invariance first. The one-body additive terms must belong to zero wavevector and have the form (62) Next we examine two-body scalar term
additive terms linear in the stresses. There will be a
1 W)
c T,,(k) C-k)
This term is translation and rotation invariant It is clear that there can also be a tensor term C AT(k) + k#O a.5 The two-body
+ C.C.
oi
k#O
(T,,(k)
additive terms involving k;. psW.h@)At+@)> m
,5-k)
and quadratic
+ cc.).
the currents
are
k$o p’(k) a,5
+4k)j,W-
(63) in the momenta.
(64)
(65)
In addition there might be a term p. Ciyj,(0)j,+(O) involving the square of the total momentum. Next we axamine the implications of Galilean invariance. If there is an eigenstate of the original Hamiltonian of energy E, there is a corresponding eigenstate in a frame moving with velocity v of value E+v.P+N$.
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337
Thus under the transformation Pi -
pi + Mv,
MU”
H-tH+v-P+N~.
The same condition can be demanded of the transformed Hamiltonian since the coordinate transformation affects only the description of the relative motion of particles. We will therefore construct combinations of the two-body additive terms that are unchanged under Galilean transformations. To do this note that T&) h(k)
-
G44
+ ~&j,(k)
-
A(k)
+ fih$V4.
+ Mv&(k)
+ M”v,z@(k), (66)
It is then clear that we must take PW
= -2P(k),
tLT(k) =: -2P(k),
(67)
so that we obtain the combination ?;” (AS(k) %.a + h’(k) $+) c a.4
lTa,(k) p^(--k) + C.C. - ‘&(k).j,+(k)}.
(68)
This combination also has the property that the one-body terms inherent in the separate pieces drop out. This is frequently a troublesome point in working with collective variables. It is also easy to check that the basic momentum factor ordering demanded of H1 is in fact satisfied. The remaining terms are one-body terms, q,, 1, T,,(O) + poP2. However, we require that an eigenfunction of H with total momentum zero be transformed into an eigenfunction of H1 with the same total momentum. Then p0 = 0 and q. = 1/2M gives the proper behavior under a Galilean transformation. We have thus determined the form of the Hamiltonian, insofar as one considers constant and two-body terms, as
H’ = EC + &jc
a
T,,(o) + 1 (h’(k) a,,, + AT(k) +) k#O e.B
x VA4 /;(-k) + C.C.- 2j,(k)j,+(kN.
(69)
Using the definitions of the collective variables, it is easy to pass to the form where the individual particle coordinates and momenta are displayed. On the other hand, one can also use the formalism of second quantization in which the role of the
338
GROSS
zero-momentum identification
condensate particles appears more clearly. To do this we use the
G(p / k) = 2 6(p, - p) e?‘,
tf u+(p) a(p $- k).
a=1
Then iV4
= 1 Q+(P) 4~ + k), 1,
.L(k)
= z ~ma+(p) 4~ + k). Y
(71)
Finally, T,,(k)
= f
(p,), e-ilc”t(pj)o
= 1 (pi),
(P~)~ e--i2;zi + k&(k)
2=1
(72) -
c (PA P
-t k&l
Q+(P) 4~ $- k).
This leads immediately to the form of H1 written in Section 2. It is clear that one can extend these considerations to delimit the intrinsically three-body terms of HI. The three-body terms involve a stresstensor multiplied by two density fluctuations or two current fluctuations and a density fluctuation.
5. RELATIONSTO OTHERWORK The argument for the basic quasiparticle Hamiltonian H’ has been basedon the general properties of point transformations. We have not described the physical motivation behind the method of point transformations since it has been explained elsewhere [14, 151. We have also avoided making detailed calculations of the actual transformation for model systems,in order to concentrate on questions of logical structure. Our aim is to exhibit one correctly based microscopic theory of the excitations of a Bose system, which can be used to make meaningful calculations for a system of strongly interacting particles. Many of the further considerations based on H1 are closely analogous to the similarity transform approach described elsewhere [18]. This method in its primitive form is not as successfulin treating strongly interacting particles, but does have a similar long-range interaction between excitations, and is able to treat the Bose-Einstein condensation, quantized circulation, etc. We cannot enter here into the detailed calculations of the properties of liquid helium that can be based on HI. It is perhaps appropriate to note the connection with some more familiar approaches to the problem of interacting bosons. Let us
POINT
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LIQUID
339
HELIUM
consider the weakly interacting case and examine on the one hand the elementary Bogolyubov approach [lo] that emphasizes the zero-momentum state, and on the other hand the Bogolyubov-Zubarev approach in terms of density fluctuations [21]. The condensate approach is based on the unitary transformation
Q+ = -Q,
u, = eQ, Q = c o(k) u+(k) d-k) k#O
(73)
u(0) u(0) - C.C.
We have not yet made the further c-number approximation to a(O) that breaks particle number conservation. On the other hand, the part of the density fluctuation containing condensate operators is F(k) ;uy d(O) u(k) + a+( -k)
u(0).
(74)
In the c-number approximation a(O) = d% we have the correct commutation [b(k), d(l)] = 0. The longitudinal current is in the same approximation 2 J&’ !$
(k) ‘a
a+(O) u(k) - a’(-k)
u(O),
rule
(75)
and we have [b(k), 1. J(l)] f
T -6(k
-t l)N, (76)
[k . J(k), 1. J(l)] RZ 0. These approximate We now have
relations replace the correct relations u+(O) u(k) .a f{;(k) a+([-k)
The generator
u(0) e i@(k)
of the Bogolyubov Lx
&k)
algebra.
+ 2k . J(k)/P), - 2k . J(k)/k”}.
transformation
Q - c kdWV,(k)
of the current
B(FW = k-%(k).
(77)
takes the form
+ B(W)
J,(k)), (78)
We see that the B transformation can be viewed as a point transformation with further approximations made in working with the current algebra. From the point of view of the Hamiltonian the condensate theory as usually formulated selects a part of the potential energy as crucial. It does not tamper with the kinetic energy so that when formally pursued it c;an account for Bose-Einstein condensation.
340
GROSS
The elementary density fluctuation theory can likewise be viewed in the light of the point-transformation approach. Here one focuses attention on the commutation rule [{(k),
T &]
= -k
. J(k),/M.
(79)
The density fluctuations and current fluctuations are then taken to obey the approximate commutation rules. One searches for a ‘model’ kinetic energy constructed from the longitudinal current and density fluctuations that agrees with Eq. (79). It is T m k=o &
; {p”(-k)
+ 2k . J+(k)/k”}@(k)
+ 2k . J(k)//?).
(80)
The kinetic and potential energies together now constitute a quadratic form in the current and density fluctuations. We seek eigenfunctions Y@(k)) in the representation where the density fluctuations are diagonal. To diagonalize the quadratic form one must introduce the scale transform ,iT(k) = U-lp(k)U
= ,6(k) A(k), 1 = kJ+(k) A(h-) .
U-lkJ+(k)L
With the assumed commutation rules the generator of the scale transform fact again a simple point transformation. Let us recall that
(81)
is in
j,(k) = Jm@> - 5 p”(k), (82) j,+(k) The approximate
commutation k&o
~ ;kB
= J,+(k) - %,6-k). rules are
[j,(k), j,+(k’)]
e - AG(k + k’), (83)
[iW,
1 LLW”] cl
We can now write the transformed
-
Hamiltonian
-Wk as
+ 0.
POINT
TRANSFORM
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HELIUM
341
where (85) The coefficient of the density fluctuation A(k)
may be annulled by taking
= [k?/(kZ + 4NV(k))]“3
(86)
so that
The ground state wavefunction in the new frame is a constant and Go is the ground state energy. It is the well-known result for the weakly interacting Bose gas. The first excited state is just b(k) and the associated energy is the Bogolyubov excitation spectrum. To find the ground state wavefunction in terms of the original, physical density fluctuations one must apply the unitary operator generating the scale transformation. The calculation is that of Section 4 and leads to the Gaussian distribution in the F(k) for the ground state and to F(k) Y,&(k)) for the excitations. These are the results that substantiate the elementary phonon theory. It is interesting that the final form involves only the current j,(k)j,:(k). With the particle interpretation of j,(k) = EL, pi,,&l+i, this term is in the momentumordered form adopted for the strategy used with a general point transformation. The elementary approaches to the weakly interacting gas thus emerge when crude approximations are made within the framework of the point-transform approach. They involve a simple random phase approximation to the differential equations for the density [59a] and current fluctuations [59b]. In addition, they neglect the contributions of the transverse current fluctuations and ignore subtler aspects connected with the presence of the stress tensor. The approximations lead to a failure to describe one or another essential feature of the behavior of the boson system. Starting with Landau, many attempts have been made to work with the density and current algebra, including the transverse currents. This may be more successful than the primitive theories in treating limited classes of problems such as the lowlying spectrum. However, a model Hamiltonian constructed from currents and densities misses the subtler contribution of the stress tensor, which is particularly important near the lambda transition. The point-transformed Hamiltonian, on the other hand, can be analyzed using a particle or condensate approach, as well as from a current algebra point of view.
342
GROSS REFERENCES
1. F. LONDON, “Superfluids,” Vol. Il. John Wiley and Sons, New York, 1954; J. WII.KS. “The Properties of Liquid and Solid lielium,” Oxford University Press, London, 1967; W. E. KELLER, “Helium 3 and Helium 4,” Plenum Press, New York, 1969; K. R. ATKINS, “Liquid Helium,” Cambridge University Press, London/New York, 1959. 2. E. FEENBERG, “Theory of Quantum Fluids,” Academic Press, New York, 1969; G. V. CHESTER, “Lectures in Theoretical Physics” (K. T. Mahantappa and W. E. Brittin, Eds.), Vol. XIB, p. 253, Gordon and Breach, New York, 1969. 3. 0. PENROSEAND L. ONSAGER, Pl7jss. Rev. 104 (19.56). 576; L. REATTO AND G. V. CHESTER, Phys. Rev. 155 (1967), 88. 4. L. MCMILLAN, Phys. Rev. A 138 (1965), 442; D. SCHIFF AND L. VERLET, Phys. Rev. 160 (1967), 208. 5. R. P. FEYNMAN AND M. COHEN, Phys. Rev. 102 (1956), 1189; M. COHEN AND R. P. FEYNMAN, Phys. Rev. 107 (1957), 13. 6. G. V. CHESTER,R. METZ, AND L. REATTO, Phys. Rev. 175 (1968), 275. 7. T. BURKE, K. MAJOR, AND G. V. CHESTER,Ann. Phys. (New York) 42 (1967), 144. 8. I. M. KHALATNIKOV, “Introduction to the Theory of Superthtidity,” W. A. Benjamin, Inc., New York, 1965. 9. E. P. GROSS, in “Quantum Fluids” (D. F. Brewer, Ed.), p. 275, North Holland Publ. Co., Amsterdam, 1966; D. Aha AND E. P. GROSS, Phys. Rev. 145 (1966). 130. 10. N. N. BOGOLYUBOV, Sov. Phys. J. 11 (1947), 23: E. P. GROSS, “Dynamics of Interacting Bosons in Physics of Many Particle Systems” (E. Meeron, Ed.) p. 231, Gordon and Breach, New York, 1966. 11. K. HUANG, “Statistical Mechanics,” John Wiley and Sons, New York, 1963; V. K. WONG, Ph.D. Thesis, Univ. of California, 1966, UCRL 17159. 12. P. HOHENBERG AND P. MARTIN, Ann. Phys. (New York) 34 (1965), 291; S. T. BELIAEV, Sov. Phys. J. 7 (1958), 289; N. M. HUCENHOLTZ AND D. PINES, Phys. Rev. 116 (1959), 489. 13. E. P. GROSS,“Transformation Theory in Mathematical Methods in Solid State and Superfluid Theory” (R. C. Clark and G. H. Derrick, Eds.), Oliver and Boyd, Edinburgh, 1969. 14. F. M. EGER AND E. P. GROSS,Ann. Phys. (New York) 24 (1963), 63; J. Math. Phys. 6 (1965), 891; 7 (1966), 578. 15. F. M. EGER AND E. P. GROSS,Nuovo Cimento 34 (1964), 1225. 16. A. D. B. WOODS AND R. A. COWLEY, Phys. Rev. Lett. 21 (1968), 787. 17. R B. DINGLE, Advun. Phys. 1 (192), 111. 18. E. P. GROSS, Ann. Phys (New York) 52 (1969), 383. 19. K. HUANG, “Studies in Statistical Mechanics” (J. deBoer and G. Uhlenbeck, Eds.), Vol. 2, North Holland Publ. Co., Amsterdam, 1964. 20. E. P. GROSS, Nuovo Cimento 20 (1961), 454; L. P. PITAEVSKI, Sov. Phys.-JETP 40 (1961), 646; A. L. FETTER, “Lectures in Theoretical Physics” (K. Mahantappa and W. E. Britten, Eds.), Vol. XIB, p. 321, Gordon and Breach, New York, 1969. 21. N. N. BOGOLYUBOV AND D. N. ZUBAREV, Sov. Phys. J. 1 (1955). 83.
OF PHYSLCS:
62, 320-342
Point
(1971)
Transform
Theory
of Liquid
Helium*
EUGENE P. GROSS+ Brand&
University,
Waltham,
Massachusetts
02154
Received June 19, 1970
The method of point transformations is used to obtain a microscopically based theory of the excitations of a system of interacting bosons. The theory is governed by a transformed Hamiltonian whose ground state is a noninteracting state and which is quadratic in the new momenta. The form of Hamiltonian is compatible with translational, rotational, and Galilean invariance. If the assumption is made that two-body interactions between the dressed particles predominate, two well-behaved functions remain undetermined. In the absence of a microscopic calculation these functions may be fitted using experimental data for liquid helium. The thermal properties of the system can be calculated as with the less rigorously based Landau-Khalatnikov Hamiltonian. The present theory retains the possibility of describing the Bose-Einstein condensation and quantized circulation. I. INTRODUCTION
In the past thirty-five years, many theories have been constructed to explain the striking properties of superfluid helium [l]. The successful theories divide into several classes.Each classintroduces physical ideas and a mathematical structure which contribute to the understanding of some properties, but slights other aspectsof the behavior of helium. One example is the classof microscopic theories that usethe language of ordinary wavefunctions in configuration space [2]. The Bose-Einstein statistics are kept in the forefront by requiring that the ground state wavefunction be nodeless[3]. The simplest type of wavefunction that describesthe spatial order characteristic of any fluid then yields a reasonablevariational theory of the energy, pressure, and sound speed as a function of density at absolute zero [4]. This configuration space approach can also deal quantitatively with the spectrum of elementary excitations and with the inelastic scattering spectrum [5]. The spatial order of the ground state continues to play a central role in the analysis. It is also possible to discuss the scattering of excitations, the structure of a vortex line [6], the behavior of an * Work supported by U. S. Air Force Office of Scientific Research under grant AFOSR 68 1370B. r Visiting Professor of Physics, Massachusetts Institute of Technology, 1970.
320
POINT
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HELIUM
321
impurity [7], etc. In each of these problems, one needs to examine special stationary states of the system, rather than ensembles. The configuration method becomes increasingly cumbersome when finite temperature properties are the object of analysis. III practice the theory is taken to the point where contact is made with a pseudomicroscopic phenomenological theory like that of Landau and Khalatnikov [S], or generalizations [9] compatible with the existence of quantized circulation. These excitation gas theories fail to describe the Bose-Einstein phase transition. The configuration space method is of course a valid quantum-mechanical approach, and in principle could be extended to account for the phase transition. In practice there are technical difficulties arising from the need to have a complete set of orthonormal states, which implies kinematical interactions between the excitations. The theory then becomes unwieldy and unappealing. A second class of theories of helium is constructed as a generalization of the theory of weakly interacting boson systems [lo]. It does not concern itself primarily with spatial order and focuses attention on the macroscopic occupancy of a singleparticle state (‘condensate’). One obtains a broad understanding of the dynamics of the superfluid at absolute zero and at finite temperatures in a way that is fully compatible with the Einstein condensation [ll]. This approach can be easily formulated in the language of modern many-body theory [12], thus bringing to bear a wide range of techniques for the finite temperature case. However, the second line of thought cannot yet compete with the configuration space approach in making calculations of the properties of the lowest states of liquid helium. This dualistic state of affairs suggests that transformation theory can be useful [13]. The advantages of the configuration space method can be incorporated in a many-body coordinate transformation that maps the interacting ground state into a noninteracting state. The interacting ground state is the square root of the Jacobian. The Hamiltonian in the new frame is then amenable to treatment by the second type of approach. Arguments can be given for the existence of a point transformation that accomplishes this objective [14]. The form of the new Hamiltonian is severely restricted by symmetry considerations and by the fact that it is the result of a point transformation. One key assumption makes the theory immediately useful. It is that the terms of the new Hamiltonian describing pairwise interactions between the ‘dressed’ particles predominate. This assumption has been verified for the weakly interacting Bose gas [ 151. The transformed Hamiltonian contains two functions of momentum which can be fitted to a limited number of experiments at absolute zero, if we are unable or unwilling to calculate them from first principles. We then have a Hamiltonian describing the excitation gas which can be used in ways similar to that of the Landau-Khalatnikov theory. The new Hamiltonian is, however, soundly grounded microscopically and remains compatible with Einstein condensation and quantized circulation phenomena.
322
GROSS
Section 2 is devoted to general considerations of strategy in using point transformations. The fundamental transformed Hamiltonian is stated in Eq. 6. In Section 3 we discussthe basic features of the Hamiltonian and show that it has the characteristics discussedin this introduction. Section 4 is devoted to the derivation of the Hamiltonian by imposing the requisite symmetry restrictions. In Section 5 we show that some of the well-accepted treatments of the weakly interacting boson problem can be related to the point transform approach. These treatments emergewhen certain drastic approximations are made in the differential equations governing the point transformation.
2. GENERAL
CONSIDERATIONS
In the present section we present the broad strategy of this approach to the theory of many-boson systemsand in particular to liquid helium. It is based on our earlier studies of point transformations. The main steps of the procedure are as follows [14]. In the simplest casethe many-body Hamiltonian is
The system is subjected to a many-body coordinate transformation. For easeof presentation we group the 3N coordinates x,,Ji = I,..., N, (Y = 1,2, 3) as a single vector x. Symbolically, X(E) = f(x),
(2)
where E = 1 represents the fully realized transformation and x(0) = X. In consequence,the momenta transform as P(C) = Bbw
+ &X)Pl.
(3)
The transformed Hamiltonian takes the general form
WP, 4 = H(P(~),4’)) = &PA .Ap + W-4.
(4)
The ‘potential’ W(x) in the new frame is a many-body potential consisting of two parts. One part is the transformed direct potential energy V(X(E)). The second part is of quantum origin and comes from the original kinetic energy. It is a result of the particular momentum ordering chosen for the transformed Hamiltonian. These consideration are of a general nature. The first important step is to decide what the coordinate transformation is to accomplish. For bosons a desirable objective is to reduce the potential W(X) to a constant, independent of x. If we could accomplish this, the new Hamiltonian would have the noninteracting state
POINT
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HELIUM
323
of the many-boson system as an exact eigenstate. For a system enclosed in a periodic cube of volume Q this state is @ == (I/vQ)” in a coordinate diagonal description. It is @ = (&“/t/m) Q0 in a number occupation description. Thus H’@ = E&p, , N) @.
(3
Here we have written the constant to which W is reduced as E,(p,), making the assumption that it will be the ground state of the original Hamiltonian that is mapped into the noninteracting state. Since the transformation is unitary, the eigenvalue is unchanged and is Eo(pO, N) with p. = N/Q. (In the future we will choose units where Q = 1 and fi = 1, unless these factors clarify the argument.) It is clear that this is a desirable objective. The nonconstant part of the potential is pAAp/2M and describes solely the excitation gas. For example, in the calculation of the canonical partition function, the ground state contributes only the factor e-aEc . Thus the pressure of the system is the sum of a contribution due to the ground state and a contribution from the excitation gas. The entropy is due solely to the excited states of W. The extent to which our objective can actually be accomplished will be discussed in the Section 3. For the moment we continue to outline the overall strategy. The new Hamiltonian pAAp/2M has the form pWg,,(x)p, , i.e., the kinetic energy appropriate to particles whose motion is described in a curvilinear coordinate system. Loosely speaking there is a velocity-dependent interaction among the ‘quasiparticles.’ There are a number of general restrictions on the form of the metric tensor. The generator of a suitable point transformation should be symmetric in the particle variables and should commute with the generators of translations, rotations, reflections, etc. Furthermore, the fact that the original Hamiltonian is Galilean-invariant must carry consequences for the form of the metric tensor g&z). Finally, there is one feature resulting from the fact that we are dealing with a many-particle system. The metric tensor can be uniquely decomposed into a constant part, an intrinsically two-body additive interaction, etc. Clearly, the point transformation approach will be of real utility only if the intrinsically many-body terms are of minor importance for many interesting situations. One must make detailed calculations of point transformations for particular systems to understand the limitations of the assumption that at most intrinsic two-body interactions are important. If, however, we proceed hopefully to make this key assumption, the general considerations lead to a Hamiltonian H’ = Eo + ; &a+(p)
a(p) +
M,, = A,,(k)
1 K,@, k.:P.~
P, s> a+(p) a+(q) 4~ - k) 4s + k),
(pm ; qu) (pa -. qB - 2k,).
(6)
324
GROSS
Here we have introduced creation and destruction operators for the quasiparticles, i.e., we have passed from a coordinate diagonal representation in the new frame to an occupation number diagonal representation. The symmetric tensor has the structure
L&k) = As(k)&x9,+ F
AT(k),
N=
1,2,3,
,8=
1,2,3,
(7)
where As(k) and AT(k) are two functions of the absolute value of li. It is clear that HI is Hermitian. The translation invariance is apparent from the restrictions on the wavevectors which are arguments of the creation and destruction operators. Rotational invariance is easily verified. The fact that H1 results from a point transformation on the original Hamiltonian is reflected in the simple dependence on p and q, viz., the new interaction is at most quadratic in these wavevectors. Galilean invariance then yields the more specific form written above. Finally, the restriction to two-body additive interactions finds expression in the fact that the interaction contains only two creation and two destruction operators. In a fully microscopic theory, we have to exhibit a particular coordinate transformation and to calculate AS(k) and AT(k) in terms of the direct two-body potential V(r,J. However, the main point of our approach is that we can sidestep these difficuit calculations and have available a microscopically basedphenomenological Hamiltonian. One can take H1 seriously and calculate a limited number of properties at absolute zero, e.g., the phonon roton spectrum, to fix As(k) and XT(~). We can then undertake calculations to make predictions of other properties at absolute zero and of the entire temperature dependent properties of the system. In the next section this claim is spelledout in detail, and some of the key interesting features of H1 are emphasized. In the following section the assertedform of H1 is derived.
3. PROPERTIES OF THE QUASIPARTICLEHAMILTONIAN The vertex MI4 = +(pa - qJ(ps - q. - 2k,) A,,(k) describes a nonlocal interaction which has an important long-range component and is rather ‘soft’ at small distances.This is in sharp contrast to the original Hamiltonian. It is, therefore, not unreasonable to initiate the discussion of H1 with self-consistent field type approximations. Of course, later a more serious many-body analysis must be undertaken. The diagonal part of HI comesfrom taking p = q -t k and is H,’
= EG + c &n(p)
+ c (- 1) fl,, y k#O
n(q) n(q + k).
(8)
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TRANSFORM
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325
HELIUM
This form can be used to make a first estimate of the phonon roton spectrum, the energy of a single excitation. The state vector (in the new frame) is
i.e.,
Y(p) = a’(p) a+(O>N-l_ lu, 2/(Nl)! and belongs to N(p) = 1, N(0) == N - 1, and N(p’) = 0 for p’ # p, p’ # 0. The eigenvalue is c(p) = gj
-- (P(p)
+ AT(p)] p”(N - 1).
(9)
Accepting this as an adequate estimate of the eigenfunction of HI, the combination P(p) + h=(p) can be fixed so that c(p) is the experimental phonon roton spectrum for a given density. This means in particular that the combination P(p) + XT(p) has the behavior p-l for p -+ 0. The spectrum then describes phonons at long wavelengths. It is clearly a key concern of the microscopic theory to show that this is really the case. This can be readily proved for a dilute gas of hard spheres or for a system of weakly interacting bosons where the point transformation can be analyzed in detail [I 51. For larger values of p, c(p) goes through a maximum (‘antirotons’) and then through a minimum (rotons). There is no difficulty in fitting h(p) since the excitation spectrum at absolute zero is stable. Energy and momentum conservation forbids distintegration of single excitations. For higher values of p, however, we encounter lifetime effects. In first approximation it is reasonable to view the usual phonon roton spectrum as continuing upward past the roton mimimum to a free particle spectrum. Thus we view the higher branch of the inelastic neutron scattering experiments of Woods and Cowley [16] as a flat ‘optical’ branch which crosses the phonon-roton-particle spectrum. The two branches are split by interactions in higher order. To understand the nature of the upper branch we must study the resonant scattering of elementary excitations. We will now indicate how further predictions can be made if h(p) is fixed as discussed. Let us consider the problem of determining the statistical thermodynamics. The canonical partition function QN = Tr(e-OH) is invariant to unitary transformations. Thus QN(T, Q) = Tr(e-5H’), (10) and one does not need to know the coordinate transformation H1. The diagonal part of H1 may be rewritten in the form H,’
= EG + c &n(p)
-
c + k#O
fl,&)
g+F#o ‘+I + k) 4l). Y#O
that takes H into
n(k) n(O)
(11)
326
GROSS
This form isolates the condensate occupancy. from the more symmetrical form
but is really misleading.
K(P - s) = &4o(P - S)(PU - cl&+3 - 40).
We start
(12)
We note that K(p - q) for fixed finitep is slowly varying when q - 0. Furthermore, K(p - q) + 0, j p - q j -k 0. Thus an appropriate decomposition is HI,’ = E, + C i&f
- 2NK(p);
n(p)
+ c (K(P - 9) - K(P) - K(s)> n(p) n(q).
(13)
If the last term is neglected, the system behaves as a collection of independent excitations with the zero temperature excitation spectrum. Thus the canonical partition function is Q&2
7-1 m e--B.% Tr,v{e-““‘W”).
(14)
The excitation gas is, however, different from that of Landau and Feynman. It is subject to the same number constraint as the ideal Bose gas. This guarantees that there is a Bose-Einstein consideration [17]. The calculation of the thermodynamics requires a bilt of care since the excitation spectrum c(p) depends explicitly on the density. Thus a straightforward application of the formalism of the grand partition function leads to errors. This problem is, however, easily overcome [lS]. According to the rules for using the canonical function the Helmholtz free energy is given as [19] A,,&‘,
T) == --k T log QaV,
(154
and the pressure as p=-
aAN i ?.Q ) T’
To evaluate log Q,,, , we use the saddle point method, Introduce the definition e-BW _ Tr{e-a[~‘(P)“(P)-pNl:,
(15b) W(fi, ,QL, Sz) by (16)
where the trace includes a sum over particle number N. The mean particle number is given by
m = - gJti.
(174
POINT
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LIQUID
HELIUM
327
Then
(17b) and A,@,
T) = E, - Np mr IV’.
(18)
The Bose-Einstein condensation temperature is determined in the usual way by setting p = 0. Above the transition temperature /L is determined by the number constraint, and the internal energy is
c’ = 2 4P).f(P),
Differentiating
the number condition
with respect to T, we find
The behavior at the phase transition is determined by the fact that the zero temperature spectrum at long wavelengths has a phonon character. Then I1 and I2 are finite and nonzero as p approaches zero and a(p,B)/liT is finite. Below the phase transitionp = 0 and c, does not contain the +/3)/2Tterm. Thus c, is discontinuous at the transition. The above is only a preliminary analysis. It shows that the fitting of X(p) by the zero-temperature spectrum provides the information that determines the thermodynamics. A more thorough analysis of the Hamiltonian H1 is needed. The treatment of the part of Hnf that has been neglected leads to results obtained earlier using a similarity transformation [18]. The form of the long wave excitations is altered at the transition temperature and the entire spectrum is weakly temperature dependent. But a reliable theory must also consider the off-diagonal part of H1. There are other properties of the boson system that can be predicted from an analysis of HI without knowing the coordinate transformation. Let us consider spatially inhomogeneous condensates and focus attention on the determination of the energy of a vortex line. This is the simplest example of a class that includes the
328
GROSS
energy of a vortex ring, the energy of interaction of a pair of vortex lines, etc. To prepare for the analysis of these problems we go to a spatial description in terms of quantized fields. Let a)(x) = --&;
u(p) &P’S,
a(p) = &
e-iD.x$(x)
(22) s
dx.
Then EG + &j” - $$+
V$J+ v# dx + j” J&& #+(y) $$-$
- y) /p
#+(y) #(y) 9
C(x)/ dx dy.
(23)
If A,,(k) gives rise to phonons, the spatial transform has long-range behavior, i.e., ( x - y /-2. The interaction represents a long-range interaction between current densities and between a stress density and a particle density. We may search for a vortex line solution by making a Hartree variational choice of the state vector [20] (24) where g(x) has the form
g(x)= f(r) eie, I f3dx
= 1
(25)
in cylindrical coordinates. The angular momentum of the state is Nfi, and is the same in the original frame, since the point transformation generator commutes with the components of the angular momentum operator. It is thus a macroscopically occupied state with quantized circulation. The energy of the state is obtained by solving the integrodifferential equation resulting from functional variations off(r). Of course, we need to know the coordinate transformation to find the actual spatial density and velocity pattern, i.e., the description in terms of the original variables. We do not follow up the detailed study of the vortex line energy here. A related problem is the behavior of a system of bosonsin a cylindrical container rotating with angular velocity w. The appropriate Hamiltonian is H - wJz , which becomes H’ - wJz after point transformation. Thus, the study of the energetics and thermodynamics of the system do not require knowledge of the point transformation.
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329
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Another class of problems that can be studied directly using H1 involves the lifetime and scattering of elementary excitations. Here one must consider the off-diagonal part of H1 to obtain even a first estimate. The off-diagonal part is u =
c
4&)
P&O -- qo - 2k3) U+(P) a+(q) 4~ - k) 4s + W
(26)
P#cl+k
Let us write Then
U = V, + V, , where
V, contains no zero-momentum
Vc= c &pa(ps-- 243) a+(p)
wavevectors.
a+(O) a(p - k) u(k) $ C.C.
(27)
P#k k#O P#O
Note that H1 doesnot contain terms with two zero-momentum creation or destruction operators. This is in stark contrast to the original Hamiltonian, and is a consequence of the strategy of the point transform method. If such off-diagonal terms occurred, the noninteracting ground state would not be an exact eigenstate of HI. In the present approach the Bogolyubov transformation does not play as essential a role, even though scattering of condensate particles does occur. There is no ‘depletion’ problem since the ground state of the quasiparticles is a 100y0 occupied zero-momentum state. The occupancy of the zero-momentum state for the original particles [3] can be calculated if one knows the Jacobian of the point transformation, but is not really an essential quantity. The interaction V, describing a direct scattering of two excitations has an interesting structure in virtue of its inherent long-range interactions. The simplest scattering matrix element is (Pi P - P I vs I P’; P - p’) = 4dP - P’mL - ~A&%
- PO>
(28)
and describestwo quasiparticles p’ and P - p’ going to a pair p and P - p. Since (I&p - p’) is weakly singular as / p - p’ 1+ 0, there is a strong preference for small angle scattering. The interaction V, involves the condensate. It is responsible for the finite lifetime of excitations even at absolute zero, provided that the disintegration is permitted by the conservation laws. It also gives a correction in second-order perturbation theory to the diagonal estimate of the excitation spectrum. The fit of h(p) to the experimental spectrum should be based on a more refined estimate that includes the self-energy corrections arising from this off-diagonal part of the Hamiltonian. It is thus clear that a large number of predictions can be made if one calculates seriously with HI. This microscopically based phenomenological Hamiltonian has a number of distinct advantages as compared to the hydrodynamics theory of
GROSS
330
Landau and Khalatnikov [S]. As indicated in this section one does not lose the possibility of describing the lambda transition and one is able to describe the development of quantized circulation. These features are mandatory for any phenomenological Hamiltonian that is correctly based on the original microscopic Hamiltonian.
4. DERIVATION
OF THE HAMILTONIAN
In contrast to our earlier work on point transformations, we here work with the infinitesimal generators of point transformations. The unitary transform is written as eirswhere s is Hermitian. Then
X(E) = U--1(c) XU(E),
(294
d.x _ = i[x(c), s]. de
(29b)
For coordinate transformations s is linear in the momenta. The step of removing the potential energy is most easily illustrated in the exactIy soluble two-body problem, where one can then examine the form of the metric form of the metric tensor. We will limit the discussionto making contact with our previous work. In the center of massframe the one-body problem is defined by H = $
+ V(r).
(30)
The generator of the infinitesimal transform is s = ${p . Vqqr) + V+(r) p).
(31)
This contains a general function (b(r) and commutes with the angular momentum. Since p = p1 - pZ is the relative momentum, s is unchanged under the Galilean transform p1 ---fp1 + Mu, pz -+ pz + Mv . The coordinates transform as the solutions of
dx,
a+
de --ax,(E)=
- #
+‘(r(E));
x,(O) = ,u, .
(32)
To solve the equations multiply by x, and sum over 01.Then EE--
7% dy do)
(33)
POINT
We are interested k?(l) = Ye>
TRANSFORM
THEORY
OF
LIQUID
331
HELIUM
in the result for E = 1. The finite transform
can be written
as
Here (35)
In general, the connection betweenf(r) and 4(r) is not simple. The direct potential energy now enters the transformed potential energy as V(r(1 +f)). To illustrate how the potential energy is reduced to a constant, suppose that V(r) is strong and short-ranged, with a range r - 1. If f(r) has a form c(r + 7)-l e-T/*, it falls to zero for r - 1.We suppose7 < c. If 7 < c, the argument r(1 +f(r)) is greater than c when
i.e., over almost all space, except for a small region near the origin. Thus the effective range of the potential has been decreased. The physical effects of the original direct potential are taken up in two places. One is in the new spatial potential arising in the proper ordering of momentum operators. For finite but large I it has been shown that the new spatial energy only affects the scattering of particles of momentum less than Z-l [I 51. The scattering of all other particles comes from the metric tensor. The new kinetic energy is (36)
where L&r) S(r)
= 1-
= --F(r)
(1 + f)-3,
A,,, $- 5”
Z(r),
Z-f!(r) = [I $ (rf)‘]-’
- (1 + f)-“.
(37)
The metric tensor is regular and Fourier analyzable even in the extreme limit 7 - 0. For example, first taking ZL+ co to remove the spatial dependence of the potential energy, we have
When the transformed Hamiltonian is written in the notation of second quantization we find a structure identical to that of our basic many-body Hamil-
332
GROSS
tonian of Eq. (6). The only difference is that P(k) and AT(k) are both determined by the single functionf(r) (and of course are independent of the density). We now discuss the many-body problem. The generator of the point transform has the form
The symmetry restrictions must now be imposed on the generator. It is convenient to confine ourselves to Fourier analyzable potentials, which may, however, be as strong as one likes. Symmetrical functions of coordinates will be described in terms of the density fluctuations
p”(k) = 2 e-ikz,.
(40)
j=l
This is a one-body additive function. Additive constructed from polynomials of the s(k). The local Hermitian current density is
functions of higher order may be
&(x, 1) = %{j,(x, t> +A+(x? t>>,
(41)
where
The Fourier transforms of these non-Hermitian
currents are
j,+(k) = 5 eikxjpj,a,
(43)
j=l
J,(k) = B’&(k) +&+(-WI,
J,+(k) = .I,( -k).
Let us now examine the simplest meaningful additive function. It has the form s2 = c k&k)
J,(k) ,6-k)
+ cc.
J,(k)
+
Hermitian
generator, a two-body
k
=
c
km@) k
K--k)
C
W”0r)
,W
J&-k).
(44)
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333
HELIUM
It consists of a single current multiplied by a density fluctuation. It is clear that s commutes with the total momentum operator, and if d(k) is a function of 1k 1, with the angular momentum operators. Under the Galilean transform pi
-
pi
+
Mv,
J,(k)
-
J,(k)
+
s is unaltered if d(k) + d*(k) = 0. Next examine a generator consisting of three-body s2 = 1 d,(k, I) J,(k) p”(l - k) p”(-Z)
MG(k),
additive terms. Let + C.C.
(45)
The choice of wavevectors ensures translation invariance. Rotation invariance requires that & contain only k,S$(k2, I”, k . Z) and l&(k2, 12,k . I). Galilean invariance is ensured if d,(k, I) + d,*(-k, 4) = 0. The general coordinate transformation is a sum of s, up to n = N. It contains a large number of freely disposable functions. However, the symmetry requirements can be imposed term by term. To actually perform the transformation we must examine the differential equations that will determine the new coordinates. They are (46) but it is advisable to first study the transformation
= c k . k’d(k’) k’
$(k + k’) ,5(-k’)
of density fluctuations
+ C.C. + i[p”(k, E), s3 + . ..I.
(47)
Because we are working with point transforms the right side is a sum of polynomials in the density fluctuations. In addition $(k, E) commutes with ;(k’, E’) for general k, E, k’ and E’. Thus these are ordinary nonlinear equations. Suppose that these equations can be solved as a sum of polynomials in the p^(Z,0) with an explicit dependence on E. Then we can find the ground state wavefunction for the interacting system when we put the requirement that the wavefunction in the transformed frame @o , is the noninteracting state .PNj2, YJE) ‘To do this it is first convenient current fluctuations,
= U(E) CD0 = U(E) Q-N/2. to set down the commutators
(48) of the density and
kW, J&‘N = k$(k + W, K(k), J&N = k,‘J,(k + k’) - &LoC + k’).
(49)
334
GROSS
Now examine Ye.
We form dYo/&
__
de
and use the fact that S(E) = s(O),
= N(E) s(0) Q-N/2.
(50)
But is(O) * 1 can be evaluated by moving the momenta to the right. Thus is,(O) . 1 = -1
Pd(k){p”(k,
0) ,5(-k,
0) - N} = iY2(0).
(51)
The higher terms s, are again polynomials in the density fluctuations. We thus have the differential equation __ = i(U(E) Y(O) u-l(E)} YG(E), dE
(52)
Y(0) = c LqO). Since Y(0) is a polynomial in {(k, 0) the operation U(E) Y(0) U-~(E) merely converts it into the samepolynomial of $(k, E), viz., Y(E). Thus
Here we find the argument of the exponential by inserting the solution fj(k, E’) which depends explicitly on ij(k, 0) and on E’. We can pursue the program of showing that one can choose the point transform to make the total new potential energy vanish, using either particle coordinates and momenta or the collective densities and currents. This has already been discussedfor the weakly interacting boson caseusing the particle description [14]. The method of demonstration is to arrange the potential energy as a sum of two-body, three-body, etc., additive functions. This is essentially a decomposition in powers of the density fluctuations. Then one shows that there is enough freedom in the generator of a general coordinate transformation to annul all of theseterms. We do not pretend to make this argument rigorous for strongly interacting bosons. Our main interest is to determine the structure of the metric tensor. It is instructive to pursue the matter from the point of view of the collective variables, since the analysis points to deficiencies of numerous approaches to the many-boson problem. Consider the quantity T&k)
=
5 (pJ, j=l
e-i”“j(pj)B
.
(54)
This is a one-body additive tensor function, but contains two powers of the momentum of the sameparticle. It is closely related to the Fourier transform of the
POINT
TRANSFORM
THEORY
OF LIQUID
335
HELIUM
stress tensor density. In particular the kinetic energy is the trace of a zero wavevector quantity, viz., Cal T,,(O). Thus the total Hamiltonian is (55) To determine how the Hamiltonian T&O, E). We must integrate
transforms
under a point transform
we need
(5’3 The generator s is constructed from a current density fluctuations. Hence we need
multiplied
by a polynomial
in the
c [Ed% 341 = --2k . .J@), ; [T,.,(o), J,(k)]
=-L -2 c k,T,,(k) a
(57) - +
Fp(k)
It is apparent that we must know J,(k, E) and T&k,
- Pj,, + k, y
l ). A
.
(58)
closed set of equations is
$
(k, c) = i[p”(k, E), s],
(594
2
(k, E) = i[J,(k,
(59b)
E), s],
The first equation has already been discussed. It involves density fluctuations of different wavelengths which commute for different values of E. The right side of the current equation is at most linear in the current fluctuations with coefficients that depend on the density fluctuations. The general form of the solution for J,(k, c) is linear in the J,(Z, 0) with coefficients depending on the density fluctuations as well as inhomogeneous terms depending on density coordinates. To appreciate the implications of the last equation we need
[/W, T,,(k’)l = k,j,(k -t k’) + 2kjdk +- W - k’j,(k + JO + 2kA$(k + W, lid4 ~cdk’)l = W& + k’) + 2&T,, - k,‘T,, - k,‘Tti, + k,‘k&(k
+ k’) + ks’(k,’
- 3k,)j,(k
+ k’).
(60) (61)
336
GROSS
It is apparent that the equation for T&k, C) is linear in the stresses with density fluctuation coefficients. In addition there are inhomogeneous terms containing current fluctuations. To obtain the solution we imagine the solutions for the densities and currents known and substitute them. The solution for T&k, e) is then at most linear in the stresses Trs(l, 0) and quadratic in the currentsjJZ, 0). The form of the point-transformed Hamiltonian of course agrees with what we know from the particle point of view. It is quadratic in the particle momenta with a metric tensor that depends on the coordinates. The present analysis in terms of the collective variables shows us that there are two distinct contributions. One is linear in the stresses and the other quadratic in the currents. It is now a simple matter to find the form of the Hamiltonian compatible with our symmetry requirements. We will consider all the conditions except Galilean invariance first. The one-body additive terms must belong to zero wavevector and have the form (62) Next we examine two-body scalar term
additive terms linear in the stresses. There will be a
1 W)
c T,,(k) C-k)
This term is translation and rotation invariant It is clear that there can also be a tensor term C AT(k) + k#O a.5 The two-body
+ C.C.
oi
k#O
(T,,(k)
additive terms involving k;. psW.h@)At+@)> m
,5-k)
and quadratic
+ cc.).
the currents
are
k$o p’(k) a,5
+4k)j,W-
(63) in the momenta.
(64)
(65)
In addition there might be a term p. Ciyj,(0)j,+(O) involving the square of the total momentum. Next we axamine the implications of Galilean invariance. If there is an eigenstate of the original Hamiltonian of energy E, there is a corresponding eigenstate in a frame moving with velocity v of value E+v.P+N$.
POINT
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HELIUM
337
Thus under the transformation Pi -
pi + Mv,
MU”
H-tH+v-P+N~.
The same condition can be demanded of the transformed Hamiltonian since the coordinate transformation affects only the description of the relative motion of particles. We will therefore construct combinations of the two-body additive terms that are unchanged under Galilean transformations. To do this note that T&) h(k)
-
G44
+ ~&j,(k)
-
A(k)
+ fih$V4.
+ Mv&(k)
+ M”v,z@(k), (66)
It is then clear that we must take PW
= -2P(k),
tLT(k) =: -2P(k),
(67)
so that we obtain the combination ?;” (AS(k) %.a + h’(k) $+) c a.4
lTa,(k) p^(--k) + C.C. - ‘&(k).j,+(k)}.
(68)
This combination also has the property that the one-body terms inherent in the separate pieces drop out. This is frequently a troublesome point in working with collective variables. It is also easy to check that the basic momentum factor ordering demanded of H1 is in fact satisfied. The remaining terms are one-body terms, q,, 1, T,,(O) + poP2. However, we require that an eigenfunction of H with total momentum zero be transformed into an eigenfunction of H1 with the same total momentum. Then p0 = 0 and q. = 1/2M gives the proper behavior under a Galilean transformation. We have thus determined the form of the Hamiltonian, insofar as one considers constant and two-body terms, as
H’ = EC + &jc
a
T,,(o) + 1 (h’(k) a,,, + AT(k) +) k#O e.B
x VA4 /;(-k) + C.C.- 2j,(k)j,+(kN.
(69)
Using the definitions of the collective variables, it is easy to pass to the form where the individual particle coordinates and momenta are displayed. On the other hand, one can also use the formalism of second quantization in which the role of the
338
GROSS
zero-momentum identification
condensate particles appears more clearly. To do this we use the
G(p / k) = 2 6(p, - p) e?‘,
tf u+(p) a(p $- k).
a=1
Then iV4
= 1 Q+(P) 4~ + k), 1,
.L(k)
= z ~ma+(p) 4~ + k). Y
(71)
Finally, T,,(k)
= f
(p,), e-ilc”t(pj)o
= 1 (pi),
(P~)~ e--i2;zi + k&(k)
2=1
(72) -
c (PA P
-t k&l
Q+(P) 4~ $- k).
This leads immediately to the form of H1 written in Section 2. It is clear that one can extend these considerations to delimit the intrinsically three-body terms of HI. The three-body terms involve a stresstensor multiplied by two density fluctuations or two current fluctuations and a density fluctuation.
5. RELATIONSTO OTHERWORK The argument for the basic quasiparticle Hamiltonian H’ has been basedon the general properties of point transformations. We have not described the physical motivation behind the method of point transformations since it has been explained elsewhere [14, 151. We have also avoided making detailed calculations of the actual transformation for model systems,in order to concentrate on questions of logical structure. Our aim is to exhibit one correctly based microscopic theory of the excitations of a Bose system, which can be used to make meaningful calculations for a system of strongly interacting particles. Many of the further considerations based on H1 are closely analogous to the similarity transform approach described elsewhere [18]. This method in its primitive form is not as successfulin treating strongly interacting particles, but does have a similar long-range interaction between excitations, and is able to treat the Bose-Einstein condensation, quantized circulation, etc. We cannot enter here into the detailed calculations of the properties of liquid helium that can be based on HI. It is perhaps appropriate to note the connection with some more familiar approaches to the problem of interacting bosons. Let us
POINT
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339
HELIUM
consider the weakly interacting case and examine on the one hand the elementary Bogolyubov approach [lo] that emphasizes the zero-momentum state, and on the other hand the Bogolyubov-Zubarev approach in terms of density fluctuations [21]. The condensate approach is based on the unitary transformation
Q+ = -Q,
u, = eQ, Q = c o(k) u+(k) d-k) k#O
(73)
u(0) u(0) - C.C.
We have not yet made the further c-number approximation to a(O) that breaks particle number conservation. On the other hand, the part of the density fluctuation containing condensate operators is F(k) ;uy d(O) u(k) + a+( -k)
u(0).
(74)
In the c-number approximation a(O) = d% we have the correct commutation [b(k), d(l)] = 0. The longitudinal current is in the same approximation 2 J&’ !$
(k) ‘a
a+(O) u(k) - a’(-k)
u(O),
rule
(75)
and we have [b(k), 1. J(l)] f
T -6(k
-t l)N, (76)
[k . J(k), 1. J(l)] RZ 0. These approximate We now have
relations replace the correct relations u+(O) u(k) .a f{;(k) a+([-k)
The generator
u(0) e i@(k)
of the Bogolyubov Lx
&k)
algebra.
+ 2k . J(k)/P), - 2k . J(k)/k”}.
transformation
Q - c kdWV,(k)
of the current
B(FW = k-%(k).
(77)
takes the form
+ B(W)
J,(k)), (78)
We see that the B transformation can be viewed as a point transformation with further approximations made in working with the current algebra. From the point of view of the Hamiltonian the condensate theory as usually formulated selects a part of the potential energy as crucial. It does not tamper with the kinetic energy so that when formally pursued it c;an account for Bose-Einstein condensation.
340
GROSS
The elementary density fluctuation theory can likewise be viewed in the light of the point-transformation approach. Here one focuses attention on the commutation rule [{(k),
T &]
= -k
. J(k),/M.
(79)
The density fluctuations and current fluctuations are then taken to obey the approximate commutation rules. One searches for a ‘model’ kinetic energy constructed from the longitudinal current and density fluctuations that agrees with Eq. (79). It is T m k=o &
; {p”(-k)
+ 2k . J+(k)/k”}@(k)
+ 2k . J(k)//?).
(80)
The kinetic and potential energies together now constitute a quadratic form in the current and density fluctuations. We seek eigenfunctions Y@(k)) in the representation where the density fluctuations are diagonal. To diagonalize the quadratic form one must introduce the scale transform ,iT(k) = U-lp(k)U
= ,6(k) A(k), 1 = kJ+(k) A(h-) .
U-lkJ+(k)L
With the assumed commutation rules the generator of the scale transform fact again a simple point transformation. Let us recall that
(81)
is in
j,(k) = Jm@> - 5 p”(k), (82) j,+(k) The approximate
commutation k&o
~ ;kB
= J,+(k) - %,6-k). rules are
[j,(k), j,+(k’)]
e - AG(k + k’), (83)
[iW,
1 LLW”] cl
We can now write the transformed
-
Hamiltonian
-Wk as
+ 0.
POINT
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HELIUM
341
where (85) The coefficient of the density fluctuation A(k)
may be annulled by taking
= [k?/(kZ + 4NV(k))]“3
(86)
so that
The ground state wavefunction in the new frame is a constant and Go is the ground state energy. It is the well-known result for the weakly interacting Bose gas. The first excited state is just b(k) and the associated energy is the Bogolyubov excitation spectrum. To find the ground state wavefunction in terms of the original, physical density fluctuations one must apply the unitary operator generating the scale transformation. The calculation is that of Section 4 and leads to the Gaussian distribution in the F(k) for the ground state and to F(k) Y,&(k)) for the excitations. These are the results that substantiate the elementary phonon theory. It is interesting that the final form involves only the current j,(k)j,:(k). With the particle interpretation of j,(k) = EL, pi,,&l+i, this term is in the momentumordered form adopted for the strategy used with a general point transformation. The elementary approaches to the weakly interacting gas thus emerge when crude approximations are made within the framework of the point-transform approach. They involve a simple random phase approximation to the differential equations for the density [59a] and current fluctuations [59b]. In addition, they neglect the contributions of the transverse current fluctuations and ignore subtler aspects connected with the presence of the stress tensor. The approximations lead to a failure to describe one or another essential feature of the behavior of the boson system. Starting with Landau, many attempts have been made to work with the density and current algebra, including the transverse currents. This may be more successful than the primitive theories in treating limited classes of problems such as the lowlying spectrum. However, a model Hamiltonian constructed from currents and densities misses the subtler contribution of the stress tensor, which is particularly important near the lambda transition. The point-transformed Hamiltonian, on the other hand, can be analyzed using a particle or condensate approach, as well as from a current algebra point of view.
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GROSS REFERENCES
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