Admitting and leaving aside - for now - the conceptual and practical advantages of carrying out the statistical counting in an ensemble defined over a separable phase-space, and in view of the dynamical implications of assigning dynamical character to c-number fields, one must consider the question of whether one is justified in thinking that it is possible as all to reformulate - on the basis of variables of a separable phase space rs - a (generic) problem originally posed, by means of a Hamilton functional, say, from a non-separable phase space r.
As a matter of fact for a dynamical problem to be well defined does not require of the existence of a hamiltonian, but it does require of the existence of a lagrangian. For problems admitting a hamiltonian, both lagrangian and hamiltonian formulations of the action principle are equiva lent, in the absence of constraint in the domain of lagrangian variables. The discussion here is confined to such problems.
Now, there is no preferred dynamical language to formulate any given problem (of the kind considered here). As a matter of fact any two
formulations based upon the same lagrangian L, in terms of two related sets of variables - defined over domains of the same dimension and hypervolume - are dynamically equivalent if and only if: (i) The symmetry group of lagrangian invariances is the same, in the sense that for every unitary transformation
of coordinates (and C.C.) in one domain leaving L invariant, there is another transformation of coordinates in the other domain also leaving L invariant, and associated with the same physical properties, e.g. such as displacements of the zero of the time or space scales, or gauge invariance, etc.(ii) if the number (and nature) of well defined and simultaneously measurable con stants of motion is the same and (iii) if the expectation values for these observables are the same.
It is noted that the condition that the hamiltonian (or the number operator) be the same functional for both formulations was not included above, for it is not essential. As a matter of fact if the transformations from lagrangian to hamiltonian variables is not in a one to one correspondence for the variables involved (separable and non-separable) the hamiltonians of
both formulations will be different functionals in general.
These defining properties of dynamical equivalence (in the above sense) are satisfied if both domains of lagrangian variables are related via a unitary transformation. The trouble is that separable and non- separable domains are not (and cannot be) related by canonical, unitary transformation, so these properties must be tested anew.
The main concern here is to investigate whether it is possible to circumvent the trouble raised by the standard formulation of the problem, in the sense that the operators associated with two conserved and well
(*)
defined observables do not commute , namely the hamiltonian and the number operator (and also linear momentum). For simplicity of the argument the attention will be focused here on those invariances associated to conserva tion of energy and total number, a more extensive analysis as to other lagrangian invariances following a similar path. Similarly for the sake of simplicity a non-relativistic langrangian will be considered. This is justified-to some extent - due to the fact that the best known super- responsive phenomena occur in a non-relativistic limit; the treatment,
of coordinates (and C.C.) in one domain leaving L invariant, there is another transformation of coordinates in the other domain also leaving L invariant, and associated with the same physical properties, e.g. such as displacements of the zero of the time or space scales, or gauge invariance, etc.(ii) if the number (and nature) of well defined and simultaneously measurable con stants of motion is the same and (iii) if the expectation values for these observables are the same.
It is noted that the condition that the hamiltonian (or the number operator) be the same functional for both formulations was not included above, for it is not essential. As a matter of fact if the transformations from lagrangian to hamiltonian variables is not in a one to one correspondence for the variables involved (separable and non-separable) the hamiltonians of
both formulations will be different functionals in general.
These defining properties of dynamical equivalence (in the above sense) are satisfied if both domains of lagrangian variables are related via a unitary transformation. The trouble is that separable and non- separable domains are not (and cannot be) related by canonical, unitary transformation, so these properties must be tested anew.
The main concern here is to investigate whether it is possible to circumvent the trouble raised by the standard formulation of the problem, in the sense that the operators associated with two conserved and well
(*)
defined observables do not commute , namely the hamiltonian and the number operator (and also linear momentum). For simplicity of the argument the attention will be focused here on those invariances associated to conserva tion of energy and total number, a more extensive analysis as to other lagrangian invariances following a similar path. Similarly for the sake of simplicity a non-relativistic langrangian will be considered. This is justified-to some extent - due to the fact that the best known super- responsive phenomena occur in a non-relativistic limit; the treatment,
In order to test whether a formulation in terms of non-separable variables is dynamically equivalent to another in terms of separable variables, a relationship must exist between these two sets of variables, say
5 k - 0 . 3 . 1 . I C .
This relationship is - of course - only one of the many conceivable; how ever, it is a rather simple one, which not only will help to illustrate the method but will eventually be (for different reasons) the one employed here. From (1.3.1.)C one can express the lagrangian
£ (q ,4 ) = 1 I ( q $ k " 4kqk) - H(qk ,qk) ( 1 .3 .2 .)C
as
LtqCq1) , ^ 1)] = LCq1,^)
To prove that L(q ,q ) admits a hamiltonian at all one defines genera lized momenta as usual by p£ = 3L/Sq1 and expresses LCq1,^1) as follows
M q ^ 1) = 1 I . ( p & - C.C.) - H'Cp1^ 1) (1.3.3JC. k,i
a From the action principle one can easily prove - from here - that H' is a constant of motion provided, that neither H or H' are explicit functions of time, but depend upon it only through q(t), p(t) or q1(t^p1(t).
This method is followed in sections 3.1 and 3.2 for the ideal Bose 4
gas and for the interacting problem of He. The results obtained there parallel those of Umezawa^^ and Coniglio and Marinaro*'1^ to some extent; even though the formulations here and there are conceptually different.
Here it is shown that a hamiltonian, H', exists*' ^ the separable phase space picture for both problems. For the interacting problem, however, the
A
hamiltonian H' is not fully diagonal in the L.C.S.R.; this representation (*) Different from the particle hamiltonian, H, in general.
zation of the newly obtained hamiltonian. This parallels the fact that the particle hamiltonian, H, is not fully diagonal in the representations adopted in Refs. (18,66) ; which are particular cases of the more general L.C.S.R. used here.
To achieve diagonalization an iterative scheme must be devized along similar lines as that of Umezawa, but not quite exactly like it. The general idea of this scheme is to use the non-diagonal segment of W' (or of H in Umezawa's method) to generate even more general representations of states. At infinite order of such an iterative procedure Umezawa obtained the 'physical representation', in terms of which H was shown to be diagonal - except for some contributions, Q^, which vanish in the infinite volume limit. Here the proposed strategy is the samer J the structure of the separable phase space is not changed, by the iterative procedure, but only the definition of dynamical variables in terms of the
A
ever more general fields is. In consequence the expression for H1 as a functional of generalized coordinates and momenta is not changed, but the expression in terms of newly obtained fields is. Hopefully one ends up with another physical representation, in terms of which H" should be
diagonal. It is conjectured that the diagonalization of P'-from the present setting-would be more general than Umezawa's; in the sense that terms of the form of Qy (i.e. non-diagonal, but neglegible in the bulk limit) should not arise. This is due to the fact that in Umezawa's approach only low order dangerous (diverging) contributions are cancelled out, namely those proportional to linear and quadratic (non-diagonal) contributions in elementary excitation operators of the zeroth order trial representation. Here, on the other hand, an exact condition of cancellation is obtained and employed to obtain fi'. Such a condition is expressed in terms of the dynamical variables - not in terms of elementary excitation operators of (*)
the zeroth order trial representation - hence it holds at all orders of iteration. Furthermore, it involves not only low order dangerous con tributions (proportional to first and second powers in the dynamical variables) but also high order dangerous contributions (proportional to third and fourth powers). In particular the non-diagonal perturbation is free from these dangerous contributions here, unlike in Umezawa's
A
method. For this reason non-diagonal contributions to H' such as Qy should not arise here, and diagonalization should be proved in the finite volume limit.
Now, as to the number operator in the present scheme, it is noted that this functional, N', is determined from the structure of the separable phase space. This functional is given as the local limit, k' -*-k, of the most general one-object propagator, G(k%k), that can be constructed in rs , namely
G(k',k) = f o U 1
(1.3.40C .
=
K - . i / A
k i,k
It is noted that contributions of the form p^q^ (for i^j) do not arise in
A A
G or N', due to the fact that such contributions make reference to two distinguishable objects.
The number functional in the non-separable phase space, rns>is given by
N = jjpkQk (1 .3 .5 0 C ,
thus,from pk = Ipk »qk = £qk it follows that N' and N are not the same
i K i K functionals, i.e. N = N' - I pjqj i.J (i^j) (1.3.6.)C .
This will prove to amount to a very significant difference. When the linear coherent state representation is introduced and canonical variables p ^ q 1 are defined in terms of the fields of this representation, it will be possible to appreciate that N" is diagonal in L.C.S.R. Furthermore that A
N' is the diagonal part of N in this representation. Hence if the
hamiltonian in the separable phase space picture is diagonal in the L.C.S.R.
A A A
(as is the zeroth order hamiltonian H') the commutativity of and N' is ensured.
One can also prove that N' is a constant of motion from the canonical equations - properly expressed in terms of generalized poisson brackets -
A A
arising from the action principle, i.e. by showing (H',N'} =0 (1.3.7.)C> where
, A ni _ t 3 A 9 B 9 A 9B
“ 1 J" ^ 1 1
i,k 8q 9p
3P aq
(1.3.80C, for arbitrary functionals defined in the separable phase space.Now, some or all of the new variables p1.q1 may be quantum fields (in general there is a good reason - put forward in Chapter Four - for choosing q1 and p1 to be all quantum fields if p and q are) ; hence to prove that
A A
H', N' are simultaneously observable (besides being both constants of motion) one must show
[H'.N'J = 0 (1.3.9.)C
A A
in addition to (H',N'} = 0, where the commutator refers to the Hillbert space of state amplitudes containing a vacuum for the quantum fields in (p1fq1),i-e. q£|Co> = 0=<Co lp£ for all k and some i. Both (1.3.7,9)C are proven in Chapter Three.
A A
In order to prove that the expectation values of N and N" are the same, it will be shown in §2.4 that the normalizing postulate
where |C^> are linear coherent states, entails
<n|N|n> = <C |N'|C >, 1 n 1 n '
or in terms of the volumes of the phase spaces rns and rs , Vol(Tns) = Vol(rs). Similarly, the same normalization postulate is shown to entail
<n|H|n> = <C |H'|C > ;
11 n 1 n
proving that both number and energy eigenvalues are the same in both form ulations. This together with the fact that the zeroth order hamiltonian,
A A
H', and N' commute and are diagonal in L.C.S.R.^and - in addition - are both constants of motion}suffices to entail dynamical equivalence in the present context.
Let us summarize now some of the implications arising in the present theory, in comparison with those of the works reviewed in §1.2.
The general ideal of current dynamical theories is that of diagonalizing the particle hamiltonian in a non-equivalent representation of the same commutation relations. Here the idea is different; one aims at finding the hamiltonian for new dynamical objects, other than particles. In the former case one finds difficulties in diagonalizing the particle hamiltonian - difficulties of the same nature as arise here in diagonalizing H'; one must
A
go to infinite order within a given iterative scheme to get H (there)
or H' (here) diagonal for interacting problems. Eventually one is content - in
A
current theories - with introducing some approximation into the exact H (neglecting some non diagonal segment). In this work one neglects some contributions from H' in order for this functional to be a diagonal hcmnltonian
(i.e. depending only on generalized coordinates and momenta, but independent of generalized velocities). The main difference is that while the diagonalized
A
particle hamiltonian HQ does not commute with the number of particle operator.
A
The unperturbed hamiltonian (for the new dynamical objects) does commute here with the number operator N"
This entails that from the present view no lagrangian symmetry is broken. The difficulty as to a broken hamiltonian symmetry,and the way in which it
is resolved,points out the cause of trouble: Hamiltonian invariance under non-equivalent transformation; once one replaces the statement of hamiltonian invariance by that of Lagrangian invariance no problem arises.
The standard treatment of the problem - given in the works examined in §1.2 - shows two distinctive features: one is that the dynamical objects are still regarded as partiales (of which elementary excitations are just but convenient variables); the second feature is entailed by the first and is that the hamiltonian is actually regarded as the same particle hamiltonian
A
H, which is then diagonalized in a representation other than its natural one. However, no similar compensation can be justified for N and leads to the pseudo problem of a broken hamiltonian symmetry. If the problem is properly formulated from the outset, in terms of new dynamical objects, no problems arise at all; furthermore, the notion of anomalous averages does not arise, for, the only meaningful averages are those involving variables associated with new objects and anomalous averages for these vanish identically.
The involvement of diagonalization conditions in the standard approach will be seen to have counterparts also in the present approach. The corresponding conditions (enabling here the existence of a hamiltonian) are shown to follow from the fact that action principle is redundant in the formulation of the problem in terms of separable phase space. The removal of the redundancy - which is always admissible on logical grounds - will be shown to have the same effect as the diagonalization conditions, i.e. compensating some contributions which prevent the existence of a hamiltonian. The antecedent of the condition of compensation of dangerous diagrams, enabling here the existence of hamiltonian and ensuring in the standard
approach diagonalization of H, is seen here to arise from the action principle, instead of from the minimal ground state energy, as in existing theories.
The diagonalization conditions in the standard approach involve some conditions concerning the coefficients of non-diagonal contributions in elementary excitation operators. These conditions concern low order powers in elementary excitations, namely linear and quadratic. Here the cancellation is an exact one, involving low as well as high order contri butions (proportional to cubic and quartic contributions).
This amounts to a notable difference; for, now all dangerous diagrams cancel out. The unperturbed hamiltonian, H', as well as the perturbation, are free from dangerous contributions (leading to divergencies of the perturbation expansion). For that reason it is believed that Umezawa's result as to hamiltonian symmetry rearrangement should follow in a more simple and general fashion from the present point of view, and also hold in the finite volume limit, not only in the bulk limit as it presently stands.