fulltext

(1)

Identities and conservation laws in

relativistic Schrödinger theory

M. SORG

II. Institut for Theoretical Physics, University Stuttgart Pfaffenwaldring 57, D70550 Stuttgart, Germany

(ricevuto il 9 Febbraio 1996; approvato l’11 Giugno 1996)

Summary. — The theory of relativistic Schrödinger equations is further extended to include also non-conservation of the rest mass of a material system due to particle creation (annihilation). The corresponding (non-)conservation laws are discussed in detail, together with their close interrelationship to the basic identities of the theory (o Fierz, Bianchi, and bundle identities). As a demonstration of the general theory, the cosmological solutions of the Einstein-Yang-Mills-Higgs equations are constr-ucted over the Robertson-Walker universes.

PACS 04.50 – Gravity in more than four dimensions, Kaluza-Klein theory, unified field theories; alternative theories of gravity.

1. – Introduction

Ironically, the foundation of modern quantum theory, seventy years ago, was not so much celebrated as a triumph but rather was the origin of violent and emotional controversies among the fathers of the new theory [1, 2]. It seems that up to the present day those profound questions and doubts concerning the correct interpretation and meaning of the new theory have not yet found their final solutions. Consequently, there are still several deviating philosophies about the true meaning of quantum theory, including, e.g., the well-known causal interpretation of deBroglie and Bohm [3] or the many-worlds interpretation of Everett et al. [1, 4]. But despite the high intellectual level of some interpretive systems for the new theory, it seems that today most physicists join to a rather simple philosophy known as the Copenhagen interpretation of quantum mechanics which admits only statistical arguments for the explanation of the quantum phenomena. Thus, the latter point of view actually denies the real existence of the causal space-time behaviour of any individual quantum system at any moment of time, so that the definite properties of a microscopic object come about only through the very act of measurement (or observation) itself. Furthermore, the result of such a measurement (or observation) is said to be impredictable in an

(2)

essential way and to obey only statistical laws which, however, are correctly described just by that Copenhagen version of quantum theory.

It belongs to the professional experience of generations of theoretical physicists that such a minimalistic and positivistic approach to quantum mechanics is sufficient in order to solve practical problems (computing energy eigenvalues etc.). On the other hand, such an opportunistic viewpoint was highly unsatisfactory for other people from the very beginning (see the historical debate between the parties of Bohr-Heisenberg-Born-Jordan against Einstein-Planck-deBroglie-Schrödinger [1]). In the course of time, especially Schrödinger appears to have become increasingly unhappy with Born’s purely statistical interpretation of his famous wave equation, whereas he himself liked to consider his own equation as the fluid-dynamic equation of motion for some kind of quantum substance building up the “particle”. Here, the counterargument of his opponents consisted in the experimental evidence that for many-electron systems (e.g., uranium atoms) the particles do not flow together into one single droplet of the questionable quantum fluid but they retain, though being indistinguishable, their individual entity by occupying different orbits according to the Pauli principle. It is only in the special case of Bose-Einstein condensation, where a large number of (bosonic) quanta occupy the same state, that such a many-particle system can be described by one single wave function c(x , t) of fluid-dynamic character (another example is Maxwell’s classical electrodynamics which is accepted as a good approxima-tion to quantum electrodynamics when a large number of photons is involved [5]). It seems that Schrödinger later on surrendered to the persuasive power of these arguments, and he left the field of quantum theory in order to work about Einstein’s General Relativity where he could enjoy the aesthetics of classical-relativistic field theory [6].

However, the long-standing question remains: Is it possible to describe the quantum phenomena also from a fluid-dynamic point of view in the sense of Schrödinger? If yes, the desired approach would have to start again with the 1-particle Schrödinger equation (or with its relativistic generalizations). But, in place of taking the statistical-ensemble route to many-particle physics via the Hilbert tensor product and Fock spaces up to modern quantum field theory [7], one would rather prefer to work with some multi-dimensional fibre bundle over pseudo-Riemannian space-time. As a consequence, the local physics at any event x is described in terms of the corresponding fibre degrees of freedom (cf. here also the somewhat related ideas of ref. [8]). Thus, for an N-particle system, a typical fibre would be (at least) of dimension

N, and the relevant physical quantities would be represented by local operators acting

on that N-dimensional fibre space over any event x. Correspondingly, the simultaneous treatment of two physical systems would be achieved mathematically by resorting to the Whitney sum of the individual tensor and spinor bundles for the separate systems. This is in contrast to the conventional statistical-ensemble approach where one uses the tensor product of the individual Hilbert spaces!

An attempt for the construction of such a Relativistic Schrödinger Theory (RST) has recently been put forward [9-14]. But, as should be evident to anybody, such a new conception will first raise more questions than providing answers. The urgent problems surely will aim here at the mysterious particle-wave duality [11, 12], particle creationOannihilation, origin of the particle masses and charges, interrelationship between quantum behaviour and gravitation, etc. In the present paper, we are especially concerned with the question of particle creationOannihilation within the

(3)

identify that point within the theory where it can be generalized in order to incorporate also matter creationOannihilation processes. If such processes do not occur, the matter system is closed with respect to its total rest mass, and the corresponding closedness condition must be specified in unique mathematical terms. We are treating such closed matter systems in great detail and with frequent exposition of the precise point of extendibility of the theory to non-closed systems. In this way, it may be shown that Schrödinger’s original idea actually is not obsolete but merely requires a more profound mathematics.

More concretely, the point of departure for the productionOdestruction of rest mass is the “conservation equation” within RST. In order to briefly demonstrate the importance of that equation within the present framework, let us briefly display the basic architecture of the theory.

First of all, there is the (N-component) wave function c(x) as a section of the corresponding complex vector bundle. The generalization of the concept of wave function is the intensity matrix I (x). The space-time evolution of these objects is governed by the Hamiltonian 1-form Hm which takes its values in the Lie algebra

SX(N , C) and acts over the typical vector fibre, or matrix fibre respectively, in order to yield the Relativistic Schrödinger Equation for c

iˇc Dmc 4 Hmc ,

(1.1)

or analogously for the (Hermitian) intensity matrix I (4 I)

iˇc Dm I 4 HmQ I 2 I Q Hm

(1.2)

( Hm is the Hermitian adjoint operator of Hm, etc.). The Hamiltonian Hm itself is a

dynamical object whose equations of motion consist of the integrability condition Dm Hn2 Dn Hm1

i

ˇ_c[ Hm, Hn] 4iˇc Fmn (1.3)

and of the conservation equation Dm _H m2 i ˇ_c H m_{Q H} m4 2iˇc( X 1 i G) . (1.4)

As indicated by its very name, it is just the last equation (1.4) which is responsible for the (non-)validity of certain conservation laws, e.g., concerning rest mass M and energy-momentum of the material system. On the other hand, the first dynamical equation (1.3) for Hm establishes the consistency link to the (non-)Abelian gauge field

Fmn(i.e. bundle curvature) and thus guarantees the existence of solutions c(x) for the

RSE (1.1), or (1.2) respectively. Furthermore, the conventional Klein-Gordon form of wave equation is readily deduced from the RSE (1.1) by differentiating that equation once more and applying the conservation equation (1.4), yielding

Dm _D

mc 1 X Qc40 ,

(1.5)

provided we impose the following constraint upon the operator G: GQcf0 ,

(4)

respectively,

GQ I f0 . (1.7)

The ordinary linear Klein-Gordon theory is obtained from the more general wave equation (1.5) by putting X ¨

g

Mc ˇ

h

2 Q 1 , (1.8)

but the more interesting situation (in view of the cosmological applications) is encountered when X is a non-linear function of the scalar density r4 tr I (¨ c c),

i.e.

X 4X( r)Q1 (1.9)

(o Klein-Gordon-Higgs equation).

Now we have arrived at the central point of the present paper, namely the conservation equation (1.4) in conjunction with the closedness relation (1.6). In all the preceding papers, the operator G has been put equal to zero which is a very special solution of (1.7) but automatically conserves the rest mass of the material system. Subsequently, it will be demonstrated in great detail that the admittance of a non-vanishing G is just the point where matter creationOannihilation invades the RST. However, for the conservation of rest mass the vanishing of G is only a sufficient but not necessary condition, i.e. we can simultaneously have both rest mass conservation and non-vanishing G. But in order to achieve this goal one must impose a special equation of motion for G besides the closedness relation (1.7), or (1.6) respectively, namely

Dm G4

i

ˇ_c[ GQ Hm2 HmQ G] . (1.10)

Whenever the rest mass of the material system is not conserved, the closing relation (1.7) will not apply, and the dynamical equation for G will couple to the gauge field Fmn,

thus signalling the creation (annihilation) of material particles by annihilation (creation) of gauge bosons. In other words, the convertor G organizes the conversion of massive particles into massless gauge bosons and vice versa. But indeed, in the present paper we want to restrict ourselves to those G-effects which are emerging already for vanishing conversion of matter-energy.

Our procedure is the following.

1.1. Identities. – First, the basic mathematical structure of the theory is investigated in great detail for the case when the convertor G is non-trivial (sect. 2-4). Here, the central point lies in the identities which is even more fundamental than the question of the equations of motion. The reason is that in any case the mathematical objects of the theory must obey some constraints (“bundle identities”) irrespectively of the special kind of equation of motion to be postulated additionally. Evidently, those identities provide us with a general framework which can be filled with the equations of motion in a rather restricted way. Thus, the combined set of identities and dynamical equations yields a highly interrelated system where sometimes the identities imply the equations of motion and vice versa.

(5)

Essentially, there are three kinds of identities: i) the bundle identities (to be obeyed by any section of the corresponding fibre bundle), ii) the Bianchi identity (to be obeyed by the (non-)Abelian gauge field Fmn), iii) the Fierz identity (to be obeyed by

the intensity matrix I (x) in order that the wave function c(x) exists). In sect. 2-4 we explicitly demonstrate the consistency of the identities with the basic equations of motion for RST and exemplify this by considering the field configuration of a Higgs doublet c(x) over Robertson-Walker universes.

1.2. Conservation laws. – Next, a conservation law for the rest mass of a material system is introduced (sect. 5). But because of particle creation (annihilation), this conservation law cannot be of rigorous validity. Rather one is interested in the

additional mathematical conditions which imply the rigorous conservation of the rest

mass and thus exclude particle creation (annihilation). This condition can exactly be formulated as the equation of motion for the convertor G (1.10) in conjunction with the algebraic conditions (1.6) or (1.7). Through inspection of the sources for the energy-momentum density ( M )

Tmn of the material system it becomes evident that the

dynamics for the convertor G plays the crucial part in the theory when matter creation (annihilation) is active (sect. 7). Besides the well-known Lorentz force, being impotent with respect to the production (destruction) of rest mass, there arises a second force term (conversion force) as the source of ( M )Tmn, and it is this conversion force which

accounts for the change of ( M )_T

mn via the changes of the system’s rest mass M.

Even if the rest mass is strictly conserved, there are several non-trivial effects produced by the presence of a non-vanishing convertor G: i) exclusion of the open RW-universe by the Cosmological Principle if applied to the Einstein-Yang-Mills-Higgs equations (sect. 7), and ii) modified expansion dynamics for inflationary cosmology (sect. 8).

2. – The Hamiltonian Hm and field strength Fmn

The importance of the Hamiltonian within the present framework becomes readily manifest when we consider the general problem of motion for matter. Our experimental experience tells us that the state of motion of matter is influenced by the action of some force field Fmn, which by the modern gauge field theories [15] is identified as the

Lie-algebra–valued curvature 2-form in an appropriate fibre bundle over the (pseudo-Riemannian) space-time. But for the time being, we need to know about fibre bundles nothing else than the mere fact that the field strength Fmn must always obey

the well-known Bianchi identity

DlFmn1 Dm Fnl1 Dn Flmf 0 ,

(2.1)

where the covariant derivative in the bundle is defined as usual through Dl Fmnv˜lFmn1 [ Al, Fmn] .

(2.2)

Here Al is the anti-Hermitian bundle connection ( Al4 2 Al), i.e.

Fmn4 ˜m An2 ˜n Am1 [ Am, An] ,

(2.3)

(6)

central question, namely how matter is moving under the action of that field strength Fmn. It is exactly this question which is answered here by reference to the concept of

Hamiltonian 1-form Hm, comparable to the first principles of ordinary quantum

theory.

Suppose matter to be characterized by some object I (intensity matrix [10, 12]) which is considered as a SX (N ,C)-valued Lorentz scalar. Then the above question will be answered by establishing some dynamical link between the intensity matrix I and the (N 3N)-representation of the field strength Fmn. If we agree that this link should

be expressed in form of a differential equation, then this “equation of motion” for I should contain its covariant derivative:

Dm I v¯m I 1 [ Am, I ] .

(2.4)

However, the specific form of the desired equation of motion is not quite arbitrary (apart from its experimental validity) but must obey certain restrictions imposed by the general mathematical logic of space-time. For instance, any section I(x) of the corresponding SX (N ,C)-bundle must satisfy the identity

[ Dm Dn2 DnDm] I f [ Fmn, I ]

(2.5)

and the equation of motion for I must not be in conflict with this requirement. Now it is just the Hamiltonian Hm, a SX (N , C)-valued 1-form, which helps resolving this

difficulty.

Let the Hamiltonian obey the following equation (integrability condition):

.

/

´

Dm Hn2 DnHm1 i ˇ_c[ Hm, Hn] 4iˇc Fmn, ( Dm Hnv˜m Hn1 [ Am, Hn] ) , (2.6)

then the required identity (2.5) is automatically satisfied if we write down the equation of motion for the Hermitian object I (4 I) as

Dm I 4

i

ˇ_c[ I Q Hm2 HmQ I ] . (2.7)

Observe here that the field equation (2.6) simultaneously acts as the necessary and sufficient integrability condition for the equation of motion (2.7), similarly as the Bianchi identity (2.1) is the integrability condition for (2.6), so that we can be sure that the desired solutions I(x) and Hm(x) do actually exist!

3. – Example: Robertson-Walker universe

In constructing a new theory, it is always helpful to examplify the general ideas by considering a concrete physical situation. (Those readers who do not need a concrete demonstration may readily skip to sect. 4.) As such we may take the Robertson-Walker universe. The reason is because its high symmetry yields a very simple field strength Fmn and, consequently, also a simple Hamiltonian Hm according to the integrability

condition (2.6). This provides us with a nice example for the interplay between the equations of motion and the identities.

(7)

Observe first that for such a homogeneous and isotropic space-time the time-slices (u 4const) of constant cosmological time u carry a 3-dimensional subgeometry of constant curvature which can be described in terms of an SO( 3 ) gauge formalism [16]. The central object of this formalism is the orthonormal triad ]Bim, i 41, 2, 3 (:

B_imBjm4 gij4 diag [21 , 21 , 21 ] , (3.1a) B_imBi n4 Bmn, (3.1b) BmnBnl4 Bml, (3.1c) Bm m4 3 , (3.1d)

which locally spans the tangent space of those 3-surfaces of homogeneity and isotropy. Let their unit normal vector be denoted by bm (4¯mu) so that the pseudo-Riemannian

metric g_mn may be decomposed into the corresponding projectors parallel and orthogonal to the 3-surface u 4const:

gmn4 bmbn1 Bmn, (3.2a) bm_b m4 1 , (3.2b) bm_B mn4 0 . (3.2c)

Owing to homogeneity and isotropy, the covariant derivatives of these objects become very simple, e.g., for the unit normal b_m

˜_mbn4 HBnm,

(3.3)

where H 4H(u) is the Hubble expansion rate at time u. Similarly, the derivative of the triad vectors B_jm is found as [16]

DmBjn4 2(HbnBjm1 CejklBkmBln) ,

(3.4)

where the time function C 4C(u) parametrizes the deviation of the Yang-Mills connection Am (2.2), (2.3) from the surface connection; and the SO( 3 ) gauge covariant

derivative D has been defined as usual:

D_mB_jnv˜_mB_jn_{1 e}jklAkmBln.

(3.5)

Only for technical reasons, let us recast this into Dm Bn4 2

g

Hbn Bm2

i

2C [ Bm, Bn]

h

, (3.6)

where the definition of the ac( 2 )-valued 1-form Bm should be self-evident,

Bm4 Bjmsj (sjR Pauli matrices) .

(3.7)

Correspondingly, the covariant derivative reads here Dm Bnv ˜m Bn1 [ Am, Bn]

(8)

with the ac( 2 ) connection Am being given by Am4 Ajmtj, (3.9a) [tj , tk ] 4ejk ltl

g

i.e. tj4 2 i 2s j

h

_. (3.9b)

3.1. Field strength. – Now let us use these preliminaries in order to construct the field strength Fmn. Having in mind some gauge field configuration which reflects the

Robertson-Walker symmetry of the underlying space-time, we may try the following ansatz: Fjmntjf Fmn4 2 1 4fll[ Bm, Bn] 2 i 2f»[ Bmbn2 Bnbm] , (3.10)

where the homogeneous scalar fields fll, f» must be functions exclusively of the cosmic

time u. Whether this ansatz (3.10) can be taken as the bundle curvature or not is decided by the Bianchi identity (2.1). Therefore, let our ansatz (3.10) be subjected to that requirement (2.1) and then find the following constraint upon the ansatz scalars fll,

f»:

f.ll1 2 Hfll2 2 Cf»f 0

(3.11)

g

f.llv

dfll

du

h

. Surely, the time dependence of the three scalar fields fll, f», C can be chosen in such a

way that the Bianchi condition (3.11) is safely satisfied and, consequently, we may look upon our ansatz Fmn (3.10) as a possible bundle curvature.

Let us remark here that the Bianchi identity (2.1) is of purely kinematical nature and therefore does not imply any dynamical property of the field strength Fmn. For

instance, we could either require the homogeneous ( Jnf 0) or inhomogeneous

( Jnc 0) Yang-Mills equations

Dm Fmn4

4 p

c Jn

(3.12)

which are both compatible with the original Bianchi identity (2.1). For our present example (3.10), the homogeneous case with Jmf 0 would yield [16]

f». 1 2 Hf»1 2 Cfll4 0 ,

(3.13)

and this by no means is in conflict with the Bianchi constraint (3.11). Indeed, the solution of the coupled system (3.11) and (3.13) is readily found as

fll4 f * sin V R2 , (3.14a) f»4 f * cos V R2 , (3.14b) where f

* is some integration constant, R is the scale parameter of the Robertson-Walker metric [17]

ds2_{4 du}2_{2 R}2dl2

u

_{i.e. H 4} R

. R

v

, (3.15)

(9)

and the “angle” V 4V(u) is related to the time function C (3.4) by V. 4 2 C , (3.16) i.e. V(u) 42

u C(u 8) du8. (3.17)

Thus, one may think that a correct solution of the homogeneous Yang-Mills equations (3.12) has been found.

But now comes a subtle point concerning the relationship between kinematical and dynamical constraints: the results (3.14)-(3.17) seem to imply that the Bianchi condition (3.11) together with the homogeneous Yang-Mills equation (3.13) are unable to fix the time function C 4C(u) emerging in our derivative ansatz (3.6) for the ac(2)-valued 1-form Bm (3.7)! However, this supposition is not true, and the reason is that there

exists a further identity which must be obeyed by any 1-form over space-time, such as Bm, Hm, or Jm. This identity for 1-forms is the analogue of that for scalars (2.5) and

reads, e.g. for Bm,

[ DmDl2 DlDm] Bn4 [ Fml, Bn] 2Rsnml Bs,

(3.18) where Rs

nlmis the Riemannian curvature tensor of the tangent bundle over space-time.

For the present case of a Robertson-Walker universe, this object is given by [16] (3.19) Rs nlm4

g

H22 s R2

h

[ gnlg s m2 gnmgsl] 1 1

g

H.₁ s R2

h

[ gnlb s_b m2 gnmbsbl1 gsmblbn2 gslbnbm] ,

where s 40, 61 is the usual topological index (s40: flat, s411: open, s421: closed universe). Now substitute the derivative for Bm (3.6) into the bundle identity

(3.18) and find the following constraint upon the time function C 4C(u):

C21 s R24 fll,

(3.20a)

C._{1HC 4 f}».

(3.20b)

Here it is readily shown that these two kinematical constraints immediately imply the Bianchi condition (3.11) but not the Yang-Mills equation (3.13)! Quite on the contrary, it is just by virtue of the additional constraints (3.20) that the Yang-Mills equation (3.13) becomes now able to fix the time function C(u): introduce an auxiliary variable z through

C 4 z

R, (3.21)

(10)

so that (3.20b) is simplified into

z. 4 Rf»,

(3.22)

and then find by further differentiation

z O 1 H z.12 zz 2 1 s R2 4 0 . (3.23)

This effectively is the homogeneous Yang-Mills equation (3.13) which thus fixes the ansatz scalars fll, f» (3.14) for the field strength Fmn(3.10) via the angle V (3.17) and

the time function C(u) (3.21).

Summarizing: from this example we should learn that whenever we want to deal with 1-forms over space-time (e.g., for the sake of constructing some field strength Fmn

or Hamiltonian Hm), we first have to observe those bundle identities like (2.5) and

(3.18) which itself involve the field strength Fmn again. Similar identities hold for

higher-order forms. The idea now is that the motion of matter in space-time will occur in such a way that the corresponding identities are obeyed (almost) trivially, i.e. the corresponding identities are already implied by the laws of motion. As an example, remember that the Bianchi identity (2.1) for the field strength Fmn is implied by the

equation of motion for the Hamiltonian Hm(2.6) provided Hm obeys the corresponding

bundle identity like (3.18). Similarly, any solution of eq. (2.7) for the intensity matrix I obeys the bundle identity (2.5) provided the Hamiltonian Hm is a solution of (2.6). As a

counterexample, remember that the Yang-Mills equation (3.13) together with the Bianchi condition (3.11) did not automatically imply the bundle identity (3.20)

(

due to its general form (3.18)

)

! This latter example may be taken as a warning that in general it is not sufficient to simply solve the equations of motion under disregard of the basic bundle identities. The equations of motion together with the identities are forming a consistent network which is held together just by the Hamiltonian.

3._{2. Hamiltonian. – After a consistent field strength F}mnhas been found in form of

the ansatz (3.10) subject to the Bianchi condition (3.11) (the Yang-Mills equation (3.13) needs not be satisfied for the following considerations), we can now look for the corresponding Hamiltonian Hm as a solution of the integrability condition (2.6).

However, for this purpose it is actually not necessary to postulate the validity of the Bianchi identity (2.1) separately because this identity itself is a consequence of that integrability condition (2.6). Therefore, it is sufficient to look for some 1-form Hmand a

2-form Fmn which both together fit into eq. (2.6), and then the Bianchi identity (2.1)

must be satisfied automatically for the chosen 2-form Fmn! Thus, for the moment forget

again the Bianchi condition (3.11) and try the following ansatz for the Hamiltonian Hm:

Hm4 ˇc(hbmQ 1 1HQ Bm) .

(3.24)

(11)

i n t o e q . ( 2 . 6 ) a n d f i n d t h e f o l l o w i n g l i n k o f t h e s c a l a r H to th e f i e l d s t r e n g t h p a r a m e t e r s fl l, f»: fll4 24 H( H 1 C) , (3.25a) f»4 22(H . 1HH ) . (3.25b)

But, strange to say, it is again not possible to meet our expectation mentioned above, namely deducing the Bianchi constraint (3.11) from these eqs. (3.25). Rather we arrive at the somewhat weaker condition

f.ll1 2 Hfll2 2 Cf»4 4( f»2 HC 2 C

. ) H . (3.26)

The reason is the same again as for the question of the Yang-Mills equation (3.13): our ansatz for the Hamiltonian 1-form Hm (3.24) must independently obey the bundle

identity (3.18), i.e.

[ DmDl2 DlDm] Hn4 [ Fml, Hn] 2Rsnml Hs.

(3.27)

However, this yields the same conditions (3.20) as for the case of the 1-form Bm. One is

readily convinced of this assertion because Hm (3.24) is composed of multiples of the

1-forms b_m _{and B}_m, and the curvature operator is strictly linear, e.g., [ Dm Dl2 DlDm]( H Bn) 4H[ Dm Dl2 DlDm] Bn,

(3.28) or, similarly

(3.29) [˜_l˜_m_{2 ˜}_m˜_l_](hb_n_{) 4h[˜}_l˜_m_{2 ˜}_m˜_l] b_n_{4 2 hR}s

nlmbsf 2Rsnlm(hbs) .

Thus the constraint (3.20b), now being implied by the identity (3.27) for Hm, indeed

recasts the weak condition (3.26) into the strong Bianchi condition (3.11). Observe, however, that not both conditions (3.20) are necessary in order to transform the weak form (3.26) into the strong Bianchi form (3.11). This result signals that the integrability condition (2.6) and the bundle identity (3.27) are neither completely independent nor

completely equivalent. But in any case, together they safely produce the Bianchi

identity (2.1). Furthermore, for those field configurations of the present example which have non-vanishing imaginary part Hc of H,

H 4Hr1 iHc,

(3.30a)

h 4hr1 ihc,

(3.30b)

i.e. for Hcc 0, the weak condition (3.26) is equivalent to the strong one (3.11) (this

situation is given for an open universe, s 411, see below). But it is only in this special instance that the integrability condition (2.6) per se implies the Bianchi identity (2.1). Clearly, it does not come as a surprise that the Hamiltonian Hmcannot be determined

from the general bundle identity (3.27), which is valid for any section Hm(x) of the

corresponding SX (N ,C) bundle. But it is important to remark that the Hamiltonian cannot be determined from its equation of motion alone under disregard of the corresponding bundle identity.

But after all conditions have been correctly satisfied, we can now determine the scalar H occurring in our Hamiltonian ansatz (3.24). To this end, we merely have to

(12)

TABLE _{I. – Type of universe and associated Hamiltonian H}_m (3.24). s r_* Hr Hc 0 (flat) 0 2 z 2 R 0 11 (open) 6i 2 z 2 R 6 1 2 R 21 (closed) 61 61 2 z 2 R 0

combine the conditions (3.20), due to the general bundle identity (3.27), with the integrability conditions (3.25), and then we find

(C 12H )24 2 s R2. (3.31)

From this result, one readily concludes H 4

r_{* 2 z}

2 R , (3.32)

where the complex constant r

* is related to the three types of RW-universes as given in table I.

Consequently, we can find a Hamiltonian Hm(x) as solution of the integrability

condition (2.6) for all three types of universes when the field strength Fmnon the right

of (2.6) is required to be of the RW-symmetric form (3.10). The second Hamiltonian ansatz coefficient h(x) (3.24) still remains to be determined.

3._{3. Intensity matrix. – After the special forms of the field strength F}_mn and the Hamiltonian Hm are known, we can now go to study the equation of motion for the

intensity matrix _{I (2.7). Observe that this object I safely exists because its} integrability condition (2.5) is guaranteed by part of the dynamical scheme itself, cf. (2.6). For the present case of N42 fibre dimensions, the Hermitian matrix I (4 I) must look as follows:

I 4 1

2(r1 2sjs

j_{) .}

(3.33)

Evidently, the intensity matrix quite generally is a collection of the N2 _intrinsic

densities, i.e. in our special case of two fibre dimensions r 4 tr ( I ) ,

(3.34a)

sj

4 tr ( I Q sj) . (3.34b)

If such a resolution of I into its density constituents is introduced into the equation of motion for I , one naturally gets the dynamical equations for those individual densities. Since these equations are most conveniently expressed in terms of the Hermitian and

(13)

anti-Hermitian parts of the Hamiltonian, we apply the following decomposition of the latter object into the kinetic field Km and localization 1-form Lm:

Hm4ˇc( Km1i Lm) .

(3.35)

For the present case of N 42 fibre dimensions, the (Hermitian) objects Kmand Lmare

further decomposed into

Km4 Km1 1Kjmsj,

(3.36a)

Lm4 Lm1 1Ljmsj,

(3.36b)

and then the dynamical equations for the densities are readily written down as ¯_m_{r 42(rL}_m_{1 s}j_L

jm) ,

(3.37a)

D_msj4 2 ejklKkmsl2 2(rLjm2 sjLm) .

(3.37b)

Let us point here to a very interesting property of these dynamical equations: forming the density combination DF (Fierz deviation)

DF4 r21 sjsj

g

sjsj4 gjksjskf 2

!

j 41

3

sj_sj

h

_,

(3.38)

one can readily deduce from eqs. (3.37) that the Fierz deviation must vary over space-time in the following way:

¯_m_D_F_{4 4 L}_m_D_F. (3.39)

But this immediately implies that the localization coefficient L_m (3.36b) is a gradient field, say

L_m₄ ¯mL L ,

(3.40)

and then DF is linked to the corresponding amplitude field L(x) according to

DF(x) 4DF , in

g

L(x) Lin

h

4 . (3.41)

Here, xin is some reference point of space-time, and DF , in (Lin) denote the associated

initial values. The importance of the result (3.41) lies in the fact that, for a regular field configuration, the Fierz deviation vanishes everywhere over space-time (DFf 0)

whenever it vanishes at a single event (xin, say). In this case, we readily arrive

therefore at the following identity for the densities:

r2

1 sjsjf 0

(3.42)

(Fierz identity). As we shall see below, such an identity exists for any fibre dimension

N and can be written in this general case as

I24 r I . (3.43)

(14)

noting that it guarantees the existence of a wave function c for the description of matter (see below).

Finally, we want to reassure that the gradient condition (3.40) is consistent with the integrability condition (2.6). But this is readily achieved by simply introducing the Hamiltonian decomposition (3.35) into that integrability condition and separating the Hermitian and anti-Hermitian parts:

Dm Kn2 Dn Km1 i[ Km, Kn] 2i[ Lm, Ln] 4i Fmn,

(3.44a)

Dm Ln2 Dn Lm1 i[ Lm, Kn] 1i[ Km, Ln] 40 .

(3.44b)

Now it is a simple matter to recast this into the coefficient form (3.36) yielding for the kinetic field (provided tr Fmn) 40)

˜_mKn2 ˜nKm4 0 , (3.45a) D_mK_jn_{2 D}_nK_jm_{2 2 e}jkl[KkmKln2 LkmLln] 4 1 2Fjmn, (3.45b)

and similarly for the localization field ˜_mLn2 ˜nLm4 0 ,

(3.46a)

DmLjn2 DnLjm2 2 ejkl[KkmLln2 KknLlm] 40.

(3.46b)

But eq. (3.46a) is nothing else than what we look for, namely the gradient condition (3.40) for the localization coefficient L_m.

So we see that the basic building blocks of any consistent field theory of matter (namely identities, integrability conditions, and equations of motion) fit very well together in Relativistic Schrödinger Theory, and now one can pass to the analysis of the corresponding conservation laws. But first let us clarify the concepts of “wave function” and “intensity matrix”.

4. – Wave function vs. intensity matrix

At first glance, it may appear somewhat preposterious to introduce such an object like the intensity matrix I for the description of quantum matter. Anybody knows that the quantum properties of matter are well described in terms of a wave function c. Why does one not exclusively use the notion of wave function in Relativistic Schrödinger Theory and introduce the additional concept of intensity matrix?

The answer is that the concept of intensity matrix _{I is a most natural} generalization of the wave function c and therefore contains the latter as a special case. Observe, however, that the relationship between I and c within the present framework is not the same as between the “statistical operator” r× and the wave function c in standard quantum mechanics [18]. Whereas in the latter approach r× refers to a statistical ensemble, the intensity matrix I of the Relativistic Schrödinger Theory is intended to describe individual physical systems. This implies the hypothesis that the individual space-time behaviour of quantum matter cannot always be described by a (N-component) wave function c but the more general concept of intensity matrix I must be applied. Whether a given physical system is in a “pure

(15)

state” or in a more general “mixed state” is told us by the densities building up the intensity matrix I (x). If the Fierz identity (3.43) is satisfied by the corresponding densities, we have a pure state, otherwise we have a mixed state of matter. As was shown in the preceding section, both states are kept separate if eq. (2.7) is accepted as the correct equation of motion. Observe also that both types of states refer to the same force field Fmnand to the same Hamiltonian Hm, which carries much more (delocalized)

information about the physical system than the Hamiltonian H× of ordinary quantum statistics. Clearly, it remains to be clarified whether and to what extent the present individualistic description passes over to the usual statistical approach.

Let us now face the question as to which kind of equation of motion the wave function should obey when matter is in a pure state being understood for the moment as a special sub-case of the more general I -states. Surely, this equation can be found by considering the specific form of the intensity matrix I for a pure state, admitted by the Fierz identity (3.43)

I 4c7c , (4.1)

and then applying to this form the general equation of motion for I (2.7). This procedure immediately yields

(iˇc Dmc 2 HmQ c)7c 2c7 (2iˇc Dmc 2c Q Hm) 40 .

(4.2)

But from here we conclude that the wave function c must obey the following equation:

iˇc Dmc 4 HmQ c 1hmc ,

(4.3)

where hm is some ordinary (i.e. R1-valued) 1-form. The latter object hm turns out to be

completely redundant as far as the densities are concerned. For instance, consider the scalar density r for a pure state

r 4 tr I 4 c Qc

(4.4)

and find for its derivative by use of eq. (4.3) ¯_m_{r 4 ( D}_m_{c) Q c 1c Q( D}_m_{c) 4} i

ˇ_c c Q[Hm2 Hm] Q c 42 c Q LmQ c (4.5)

which is the same as if the redundant h_m were not present in eq. (4.3).

In order to reveal the true significance of hm, eq. (4.3) is alternatingly differentiated

once more yielding, by use of the integrability condition (2.6), [ DlDm2 Dm Dl] c 4 FlmQ c 2

i

ˇ_c[˜lhm2 ˜mhl] c . (4.6)

But, on the other hand, the corresponding bundle identity for c reads [ DlDm2 Dm Dl] c f FlmQ c ,

(4.7)

and this implies that h_m must be a gradient field

h_m_{4 ¯}_mh (` ˜_lh_m_{2 ˜}_mh_lf 0 ) . (4.8)

(16)

Thus, eq. (4.3) reads iˇc Dmc 4 HmQ c 1 (¯mh) c (4.9) or iˇc Dmc 84 HmQ c 8 (4.10)

if one resorts to the modified wave function c 8

c 8(x) 4exp

y

ih(x)

ˇ_c

z

Q c(x) . (4.11)

However, the U( 1 ) gauge factor emerging here is completely irrelevant when one forms the physical densities by means of the wave function c 8 and may be omitted completely. We thus arrive at the original RSE (1.1) as the wave function counterpart with respect to the intensity matrix equation (1.2).

Observe here that no new condition had to be applied in order to arrive at this result. This means that a wave function c(x) surely exists whenever an intensity matrix I (x) can be obtained from its equation of motion (1.2). But the necessary and sufficient condition for this is the validity of the integrability condition (2.6), which thus also guarantees the existence of solutions c(x) for the RSE. Though no further integrability conditions for the existence of I (x) or c(x) are needed, one may additionally require the commutativity of the localization field Lm

(

cf. (3.35)

)

[ Lm, Ln] 40

(4.12)

in order to explicitly construct the wave function c(x) [10]. Clearly, such a procedure reveals the commutator condition (4.12) as a purely sufficient one, and solutions will exist also in those cases where the commutativity does not hold (e.g., in the case of the

open universe, s 411, see table I).

Finally, we want to turn to the question whether the wave function c can be constructed from the intensity matrix I (apart from the redundant U(1) gauge factor), provided the Fierz identities guarantee the existence of c. Let us remark here that this problem has been already solved for Dirac’s spinor field where the fibre dimension is N 44 [19, 20]. Without being able to present a recipe for general N, it may be sufficient here to consider the preceding example for N 42 (sect. 3). In this simple situation, the Fierz identity (3.42) suggests the following angular parametrization of the densities: s1_{4 r sin w cos W ,} (4.13a) s2_{4 r sin w sin W ,} (4.13b) s3 4 r cos w . (4.13c)

(17)

The 2-component wave function c(x) is then readily found in terms of the angular variables w, W as c 4kr

u

cosw 2 exp [iWI] sinw 2 exp [iWII]

v

(W f WII2 WI) . (4.14)

Clearly, the densities (4.13) can be constructed also directly by means of the wave function (4.14) (“bracket densities”)

r 4 c Q c ,

(4.15a)

sj

4 c Q sjQ c (4.15b)

in place of using the intensity matrix I (3.34) (“trace densities”).

5. – Current densities and conservation laws

Experimental evidence tells us that material systems can be classified—or even identified—by certain scalar quantities (e.g., rest mass, charge, spin, etc.) which remain strictly unchanged as long as the system actually does exist. This leads us to the question of the strong conservation laws. Other quantities may be conserved only for closed systems but may be changed through interactions with other physical systems,

e.g., energy, linear momentum, angular momentum etc. (weak conservation laws).

Typically, strongly conserved quantities are (Lorentz) scalars, whereas the weakly conserved ones mostly are not scalars. For instance, if a strongly conserved quantity were a 4-vector, that material system would single out some preferred time or space direction in the universe, and this would undermine the very idea of Relativity Theory. Thus, the strongly conserved quantities should always be Lorentz scalars, and, therefore, they can be defined via a continuity equation of the type

˜m_j m4 0 .

(5.1)

Let us first consider the case of the rest mass M of a physical system and look for the associated mass current (c)jm. The motivation here consists in the observation that

neither the first dynamical equation for the Hamiltonian Hm (2.6) nor that for the

intensity matrix I (2.7) or wave function c (1.1) are sufficient to guarantee the validity of the desired conservation laws. Rather, it is only the mutual logical consistency of the mathematical objects introduced so far which has been safely established up to now. Therefore, we have to complement that the central consistency condition (2.6) by some further dynamical constraint upon Hm, the conservation equation, which presumably

must involve now its source ( Dm _H

m) whereas those consistency relations were mainly

(18)

5._{1. Mass current. – Since it is just the Hamiltonian H}_mwhich carries the dimension of “mass energy”, we naturally try the following ansatz for the mass current:

(c)

jm4

1

2 tr ] I Q Hm1 HmQ I ( fˇctr ( I Q Km) . (5.2)

Provided we are able to establish the continuity equation (5.1) for(c)_j

m, then we could

define an invariant mass M for the physical system through

Mc2 4

C3 (c)_jm_dS m, (5.3)

where the integral over an arbitrary 3-surface C3is independent of the special choice of

C3whenever it yields a finite mass M. Thus, the concept of “mass” enters the theory in

the form of a conservation law.

The next question is now what additional constraint has to be imposed upon the Hamiltonian Hm in order that the continuity equation (5.1) holds for the mass current (c)

jm (5.2). But this question is not difficult to answer: let the differential operator ˜ act

under the trace operation (tr) as the gauge covariant derivative D (i.e. ˜itr 4 triD)

and then find by use of the dynamical equation (2.7) for I ˜m(c)jm4 tr

m

I Q

g

DmHm1 i ˇ_c H m Hm

h

1

g

Dm Hm2 i ˇ_c H m Hm

h

Q I

n

. (5.4)

Therefore, having in mind either a closed system with a truly invariant mass M or a non-closed system with varying M, we require in any case as the second dynamical equation for the Hamiltonian Hm (conservation equation):

Dm _H m2 i ˇ_c H m _H m4 2iˇc( X 1 i G) , (5.5)

where both the mass operator X and the conversion operator G are assumed to be Hermitian: X 4 X, G4 G. One may look upon the non-Hermitian unified object X 1i G as a non-Hermitian mass operator so that G defines the anti-Hermitian part thereof. Such an imaginary part of an otherwise real mass is usually taken as a signal for matter creation or annihilation by the gauge field Fmn, i.e. the gauge field energy is

converted into matter energy and vice versa (conversion). Thus, the object governing that conversion process is just the conversion operator G (convertor).

5.2. Conversion dynamics. – Clearly, for non-trivial conversion of matter energy into gauge field energy and vice versa (o particle creationOannihilation), one will not expect to find the matter system being closed and equipped with a fixed mass M (5.3). Rather, one will suppose that the strict continuity equation (5.1) for the mass current

(c)_j

m has to be weakend into

˜m(c)_j m4 q

(5.6)

with a non-trivial source or sink q(x). Indeed, inserting the conservation equation (5.5) into eq. (5.4) readily yields for that conversion function q(x)

˜m(c)_j

mf q 4ˇctr ] I Q G1 GQ I(.

(19)

As a consequence, we have to demand for closed material systems with vanishing conversion function q(x) that the trace of the anti-commutator of I and G should vanish:

tr ] I Q G1 GQ I( f0 . (5.8)

But for non-closed material systems (q c 0) we are left with the problem of finding the dynamical coupling of the convertor G to the gauge field Fmn.

For the moment, let us be satisfied with the treatment of closed systems (q f 0) and let us look here for an equation of motion for the convertor G such that the closedness condition (5.8) is satisfied for all events x in space-time provided it holds at one event

xin (as an initial condition). The solution to this problem is the following field equation

for G:

Dm G4

i

ˇ_c[ GQ Hm2 HmQ G] . (5.9)

Observe that the bundle identity (2.5), to be obeyed by any (Lorentz) scalar object, is an automatic consequence of the equation of motion (5.9)

[ DmDn2 DnDm] G4 [ Fmn, G] .

(5.10)

Thus, the postulated equation (5.9) safely survives the crucial consistency test, although it is not identical to the dynamical equation for the intensity matrix I (2.7)! But now let us reassure that both dynamical equations for I (2.7) and for G (5.9) do actually admit the required closedness relation (5.8) over the whole space-time. For that purpose, one merely has to combine both equations for I (2.7) and G (5.9) into the analogous equation for the products I Q G and GQ I yielding

Dm( I Q G) 4 i ˇ_c [ I Q G, Hm] , (5.11a) Dm( GQ I ) 4 i ˇ_c [ GQ I , Hm] . (5.11b)

(Observe that both equations are the Hermitian conjugates to one another.) As a consequence, the conversion function q(x) (5.7) must be a constant over all space-time

¯_mq f 0 ,

(5.12)

and this constant may be chosen to be zero. This then recasts eq. (5.6) into the strict continuity equation (5.1) as desired. Observe also that the closing requirement (5.8) is met by the stronger condition

I Q G40 (` GQ I 40) (5.13)

or even by Gf0 which was applied in all the preceding papers.

5.3. Wave equation. – An interesting feature of the stronger condition (5.13) is its consistency with the Fierz identity (3.43). In order to see this, multiply (5.13) from the left (right) by I and apply again the Fierz identity. This means that the existence of a non-trivial convertor G is consistent with the existence of a wave function c and

(20)

the associated wave equation. The latter one is readily deduced from the RSE (1.1) by differentiating once more that equation and applying the conservation equation (5.5) together with the stronger condition (5.13) in the form GQc40:

Dm _D

mc 1 X Qc40 .

(5.14)

Specifying here the mass operator X as X ¨

g

Mc

ˇ

h

2

Q 1 (5.15)

just yields the linear Klein-Gordon equation

DmDmc 1

g

Mc ˇ

h

2 c 40 . (5.16)

Non-linear generalizations hereof may be obtained by choosing the mass operator X as a function of, e.g., the scalar density r 4 c c

X ¨X(r)Q1 , (5.17)

see ref. [14, 16]. Observe also that the consistency of the strong closing relation (5.13) with the Fierz identity does not necessarily imply the Fierz identity for fibre dimension N D2, i.e. for ND2 the material system can well be in a mixed state although (5.13) holds. It is only for fibre dimensions N 41 and N42 that the strong closing relation (5.13) implies the Fierz identity.

Is it possible to relax somewhat the strong closing condition (5.13) towards its weakest form (5.8) by requiring the following intermediate form:

I Q G1 GQ I 40? (5.18)

There is some problem with such a requirement because the derivative of the left-hand side is found as

Dm( I Q G1 GQ I ) 4i[ I Q G1 GQ I , Km] 1 [ GQ I 2 I Q G, Lm] .

(5.19)

Therefore, demanding the validity of (5.18) at one event in space-time does in general not imply its validity over the whole space-time. But in exceptional cases, e.g., for

Lm4 LmQ 1 ,

(5.20)

the intermediate demand (5.18) becomes consistent with (5.19), and the corresponding field configurations surely will conserve the rest mass. If these configurations are pure states, i.e. the Fierz identity (3.43) holds, then the intermediate condition (5.18) is equivalent to the strong one (5.13). Therefore, the intermediate form (5.18) is non-trivial only for mixed states. Observe also that for fibre dimension N 42 the intermediate condition (5.18) is always equivalent to the strong form (5.13). This means that for N 42 there are no mixed states for closed systems with non-vanishing convertor G!

(21)

5.4. Anti-correlations. – Finally, one would like to see what is the difference between the similar (but not identical) equations of motion for I (2.7) and for G (5.9). The main difference becomes readily apparent if we rewrite those equations in terms of the kinetic field Km and localization field Lm:

Dm I 4i[ I , Km] 1 ] I , Lm( ,

(5.21a)

Dm G 4i[ G, Km] 2 ] G, Lm( .

(5.21b)

Observe here that the localization field Lm enters both equations with different sign of

the anti-commutator! Since the field Lm governs the localization properties of the

material system, one may realize here that the G-field will be found to be delocalized whenever I is well localized and vice versa. For instance, consider the scalar density r (4 tr I ) whose derivative was written down in terms of Lm in eq. (4.5). Now compare

this to the analogous derivative of the conversion density G (f tr G) to be immediately deduced from eq. (5.21), i.e.

¯_m_{r 4 tr ( D}_m _{I ) 42 tr ( I Q L}_m) , (5.22a)

¯_m_{G 4 tr ( D}_m _{G) 422 tr ( GQ L}_m) . (5.22b)

For the sake of simplicity, adopt for the moment that specific form (5.20) for Lm and

then find from the present result (5.22), ¯_m_{r 42rL}_m, (5.23a)

¯_m_{G 422GL}_m. (5.23b)

This says that whenever the scalar density r is increasing (decreasing) in some space-time direction, then the conversion density G is decreasing (increasing). Thus, both densities r and G are found to be anti-correlated with respect to their localization properties. Let us inspect this effect a little bit more by resorting to our previous example (sect. 3). (Those readers who do not need a concrete demonstration of all the new concepts may readily skip to sect. 7.)

6. – Example: N 42

Similarly to the decomposition of the intensity matrix I (3.33), the convertor G decomposes for N 42 in the following way:

G4 1

2(G Q 1 2Gjs

j_{) .}

(6.1)

Consequently, the weak condition (5.18) or its strong form (5.13) produce the same constraints

Gr 2Gjsj4 0 ,

(6.2a)

rGj1 Gsj4 0 ,

(22)

which for non-trivial G immediately imply the Fierz identity (3.42). Observe that this identity is then obeyed by both matrices I and G, i.e. we have in addition to (3.42) also the Fierz identity for the convertor G:

G2_{1 G}jGj4 0 ` G24 G G .

(6.3)

Thus, for fibre dimension N 42 the Fierz identity is revealed as an inevitable consequence of the closing relation (5.13) or (5.18). But this result does not mean that there are no mixed states for N 42, it only says that those mixed states must have vanishing convertor ( Gf0). Indeed, such states have been studied already in some previous papers [14, 16]. Observe also that, quite analogously to the discussion of the intensity matrix I (4.1), the Fierz identity (6.3) for G implies the existence of a ghost

state x such that

G4x7x . (6.4)

The closing relation (5.13) then demands the orthogonality of both states c and x, i.e.

c Q x 4 xQ cf0 ,

(6.5)

which however would be invalidated in the case of non-conservation of rest mass. The equation of motion for the ghost state x is then readily deduced again from that of the convertor G (5.9) as

iˇc Dmx 4 HmQ x

(6.6)

which just is consistent with the orthogonality relation (6.5), i.e. ¯_m_{( c Q x) 4 ( D}_m_{c) Q x 1c Q ( D}_m_{x) f 0 .} (6.7)

The ghost nature of the state x (or its generalization G) is expressed by the fact that x does not carry energy-momentum or any other physically observable density. Thus, the ghost objects are pure constraint fields which serve for putting right the dynamics of the properly physical fields (such as I or c), and consequently, the ghost fields drop out when one resorts to the second-order wave equations (5.14) in the case of rest-mass conservation. However, for non-conservation, the orthogonality relations (6.5) or closing relation (5.13), respectively, must be abandoned, and one can no longer have a closed second-order dynamics for the wave function c.

Next, let us make the anti-correlation argument

(

below (5.23)

)

somewhat more precise through expressing both densities r and G by the amplitude field L (3.40). The typical situation for a well-localized state is then naturally given by the energy eigenvalue problem written down in terms of the amplitude fields. In place of the special case for the density derivative ¯_mr (5.23a) one can apply the general

relationship (3.37a) being valid for fibre dimension N 42. Here, the Fierz identity (3.42) suggests the reparametrization of the densities sj by the scalar density r and a unit triplet s×j

(

cf. also (4.13)

)

is easily checked by use of eqs. (3.46b) and (6.9b).

)

But with both amplitude fields L and

L

¯

at hand, the scalar density r is found from (6.9a) in terms of these fields as

r(x) 4rin

g

L(x) Lin

h

2

u

L¯(x) L ¯ in

v

2 , (6.13)

where xinis some reference event in space-time. A similar result will readily be deduced

also for the conversion density G(x), but let us first infer some remarks about the physical meaning of the amplitude fields L, L¯.

The latter result (6.13) suggests that the scalar density r(x) receives its localization properties just by that product of the amplitude fields L(x) Q L¯(x). Indeed, this supposition can easily be verified by deducing the corresponding second-order equation for the amplitude L Q L¯ of the wave function c such that its analogy to the energy eigenvalue problems of standard quantum mechanics becomes obvious. Remember here that in old quantum theory the energy eigenvalues are obtained by imposing certain boundary conditions upon the real amplitude function of c. Thus, splitting up the conservation equation (5.5) into its (anti-)Hermitian parts yields first

Dm _K m1

]

Lm, Km

(

4 G , (6.14a) Dm _L m1 Lm Lm2 Km Km4 2 X . (6.14b)

(24)

for the kinetic field Km ˜mKm1 2(LmKm2 LjmKjm) 4 1 2G , (6.15a) Dm_K jm1 2(LmKjm1 KmLjm) 42 1 2 Gj (6.15b)

and for the localization field Lm (put X 4XQ1)

˜mLm1 (LmLm2 LjmLjm) 2 (KmKm2 KjmKjm) 42X ,

(6.16a)

Dm_L

jm1 2(LmLjm2 KmKjm) 40 .

(6.16b)

For the sake of gaining some confidence into the present coefficient form, one may write down the mass current (c)_j

m (5.2) as (c)_j

m4 ˇc(rKm1 sjKjm)

(6.17)

which concretizes the continuity equation (5.7) into ˜m(c)_j m4 1 2 ˇ_{c(rG 2G}_j_sj_{) .} (6.18)

Therefore, the weakest closing requirement (5.8) just yields the first half (6.2a) of the stronger version (6.2). However, the point of interest here lies in the localization part (6.16b) from which the following second-order equation for the amplitude L Q L¯ can be deduced after some computational manipulations:

(6.19) _{p (L Q L}¯₎₄ 4 ] 2 s×j_K

jmKm1 KmKm2 KjmKjm2 2 ejkls×kKlmLjm2(»)Ljm (»)Ljm2 X((L Q L ¯

) . Actually, this is the relativistic Schrödinger form of the conventional energy-eigenvalue equations of traditional quantum mechanics. (A simple demonstration for

N 41 has been given in ref. [9].) At spatial infinity one expects the asymptotic form for

stationary solutions of (6.19) to look like 2D(L Q L¯) 4

m

Km_K m2

g

Mc ˇ

h

2

n

(L Q L¯) , (6.20)

where the simplified assumption (5.15) for the mass term X of the linear Klein-Gordon theory has been adopted. Consequently, the amplitude L Q L¯ will drop off along some spatial direction r roughly as

L Q L¯A rnexp

y

₂

o

g

Mc ˇ

h

2

2 KmKmQ r

z

.

(6.21)

Thus, the (Abelian part of the) kinetic field determines the invariant “energy eigenvalue” _E4ˇc

k

Km_K

m, i.e. the “wave vector” Km slips off the mass shell

(25)

then immediately transferred to the scalar density r (6.13). But the conversion density

G will not be spatially concentrated in such a way, as we shall see by a closer inspection

of its link to those amplitude fields L(x) and L¯(x).

Indeed, this link can readily be established by simply observing that the coefficient versions of the equations of motion for I (5.21a) and G (5.21b) look very similar, i.e. for G ¯_m_{G 42 2(GL}_m_{1 G}j_L jm) , (6.22a) DmGj4 2(GLjm2 GjLm) 12ejklKkmGl. (6.22b)

As mentioned above, all terms containing the localization field have their signs reversed here in comparison to the case of the intensities (3.37). But all other conclusions are formally the same, especially the Fierz deviation DF (3.41) becomes

now for the G-matrix (G2 1 GjGj)(x) 4 (G21 GjGj)inQ

g

Lin L(x)

h

4 (6.23)

which in comparison to (3.41) gives a first hint to the local anti-correlations (note the (inverse) fourth power of L(x) in both cases). Similarly, the conversive analogue of the scalar density r(x) (6.13) is found now as

G(x) 4Gin

g

Lin L(x)

h

2

u

L¯(x) L ¯ in

v

2 (6.24)

which contains the inverse second power of the field L(x)

(

but the same of L¯(x)

)

in comparison to the scalar density r(x). Thus, the anti-correlation refers only to L(x) but not to L¯(x)! Observe also that the anti-correlation is accompanied by the fact that the

SO( 3 ) gauge triplets Gj and sj are “anti-parallel”, i.e.

Gj4 2 Gs×j,

(6.25)

cf. (6.2) and (6.8).

Although the present examination of those relativistic eigenvalue problems very clearly suggests the emergence of the spatial anti-correlations, their concrete existence can be verified only by explicit numerical computations. However, this would be equivalent to solving the corresponding eigenvalue problem which is a somewhat difficult task (not to be treated in the present paper). In the present context, we are satisfied with a numerical study of the purely temporal anti-correlations, i.e. we want to consider a Robertson-Walker symmetric field configuration. For such a simplified situation, the Hamiltonian coefficients (3.36) would reflect that symmetry and hence are expected to be proportional to the Hubble flow bm:

K_m_{4 Kb}_m, (6.26a) Kjm4 Kjbm, (6.26b) L_m_{4 Lb}_m, (6.26c) Ljm4 Ljbm. (6.26d)

(26)

Of course, the pre-factors K, Kj, L, Ljare homogeneous scalar fields, e.g., K 4K(u), etc.

Next observe that the intrinsic densities sj_{(x) define a preferred direction in the}

corresponding fibre space, and it will be convenient to decompose all SO( 3 ) gauge objects with respect to that direction, i.e. we put

(

cf. also (6.10)

)

Kj4 2 s×jK ¯ 1(»)Kj (6.27a) Lj4 2 s×jL ¯ 1(»)Lj. (6.27b)

Thus, K¯and L¯are the projections of the kinetic and localization coefficients Kj, Ljonto

the preferred direction

K¯_{4 s}×j

Kj,

(6.28a)

L¯4 s×jLj.

(6.28b)

As a consequence, the time derivative of the scalar density r (3.37a) becomes now

bm¯_m_{r f r}._{4 2 r(L 1 L}¯) (6.29)

which is readily integrated to yield the homogeneous specialization of the general result for r(x) (6.13), namely

r(u) 4rin

g

L(u) Lin

h

2

u

fulltext

Identities and conservation laws in

relativistic Schrödinger theory

g

h

.

/

´

g

h

g

h

g

h

u

v



g

h

g

h

(

)

g

!

h

g

h

y

z

(

)

u

v



m

g

h

g

h

n

g

h

g

h

(

)

(

)

]

(

(

)

(

)

g

h

u

v

]

(

m

g

h

n

y

o

g

h

z

k

g

h

g

h

u

v

(

)

(