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Quantum reference frames and quantum transformations

M. TOLLER

Dipartimento di Fisica dell’Università di Trento - Trento, Italy INFN, Gruppo collegato di Trento - Trento, Italy

(ricevuto il 5 Settembre 1996; approvato il 10 Dicembre 1996)

Summary. — A quantum frame is defined by a material object following the laws of quantum mechanics. The present paper studies the relations between quantum frames, which are described by some generalization of the Poincaré group. The possibility of using a suitable quantum group is examined, but some arguments are given which show that a different mathematical structure is necessary. Some simple examples in lower-dimensional space-times are treated. They indicate the necessity of taking into account some “internal” degrees of freedom of the quantum frames, that can be disregarded in a classical treatment.

PACS 03.65.Bz – Foundations, theory of measurement, miscellaneous theories (including Aharonov-Bohm effect, Bell inequalities, Berry’s phase).

PACS 11.30.Cp – Lorentz and Poincaré invariance.

1. – Introduction

It has been remarked a long time ago [1] that, from the physical point of view, a frame of reference is defined by a material object of the same nature as the objects that form the system under investigation and the measuring instruments. More recently, some consequences of this idea have been discussed by several authors [2-4]. Then, in principle, one should take into account:

1) The gravitational field generated by the objects that define the frames.

2) Their quantum properties that do not permit an exact determination of their positions and of their velocities.

The gravitational field can be disregarded if the mass of the object that defines the reference frame is sufficiently small and the quantum effects are not important if the mass is sufficiently large. If, in a given physical situation, one can choose a mass that satisfies both these conditions, the material nature of the reference frames becomes irrelevant. Otherwise, we fall into the domain of quantum gravity and the usual geometric concepts are not valid any more [5-8].

In the present work, as in ref. [3], we disregard the gravitational interaction and we consider quantum reference frames with finite mass M. Of course, the physical

phenomena can be described completely by considering the limit M KQ. However, we think that a good understanding of this problem for finite M is preliminary to a treatment taking into account the effects of quantum gravity.

The relations between classical reference frames are described by elements of the Poincaré or of the Galilei group G. Our aim is to find a mathematical structure that describes the relations between quantum frames. A short account of some results is given in [9]. We start from the remark that in the absence of a classical reference frame the observable quantities, which concern both the objects that define a quantum reference frame and the objects described by the theory, have to be invariant under the action of the group G.

Various authors [10-15] have suggested that the quantum aspects of space-time can

be described by a quantum group [16, 17] obtained from a deformation of the

commutative Hopf algebra of the functions defined on the group G. An interpretation of this quantum group in terms of quantum frames seems to be natural, though different points of view can be adopted. This interpretation, however, even in the absence of gravitation, presents two problems:

1) A quantum group cannot describe the “internal” degrees of freedom of the quantum frames and we shall see that they play an essential role.

2) If we have three quantum frames F1, F2and F3, the observables which describe the

relation between F1and F2 cannot be compatible (in the sense of quantum theory)

with the observables which describe the relations between F2 and F3. This fact

cannot be taken into account by a quantum group.

The second problem is an aspect of the “paradox of quantum frames” discussed in ref. [3].

In sect. 2 we introduce a general mathematical structure which describes the relations between non-interacting quantum frames. In sect. 3 we consider the classical (non-quantum) limit and we discuss the connection with the formalism based on Hopf algebras. In sect. 4, 5 and 6 we study some simple examples in zero and one space dimensions, both relativistic and non-relativistic. The role of the “internal” degrees of freedom of the quantum frames is clarified by these examples.

It has been suggested in ref. [8] that in a quantum description of space-time it is necessary to introduce some new degrees of freedom, besides the space-time coordinates. We suggest that they can be interpreted as “internal” degrees of freedom of the quantum frames.

2. – Relations between quantum frames

We consider in a flat space-time a system composed of n objects F0, R , Fn 21. Some

of them define reference frames and the others are the objects described by the theory. In order to simplify the problem, we assume that these objects do not interact mutually, but only with the measuring instruments. This assumption is justified only if we can disregard gravitation, a long-range interaction which involves in a universal way all the kinds of physical objects.

In a quantum treatment, the states of the system are described by the Hilbert space

H(n)4 H07H17 R 7Hn 21.

We assume that the bounded observables are described by Hermitian operators

be-longing to the algebra (1) A(n)4 L ( H(n)) of the bounded operators in H(n). The

observables which concern the object Fk are described by the algebra Ak4 L ( Hk)

and the observables which concern the pair of objects ]Fi, Fk( are described by the

algebra Aik4 L ( Hi7Hk). We define the injective homomorphisms jk(n): AkK A(n)

and j(n)

ik : AikK A(n)by means of the formulas

(2) j(n)

k (Ak) 4B07 R 7 Bn 21, Bk4 Ak Ak, Bj4 1 for j c k ,

(3) jik(n)(Ai7 Ak) 4B07 R 7 Bn 21, Bi4 Ai Ai, Bk4 Ak Ak, Bj4 1

for j c i , j c k .

These homomorphisms formalize the concept of subsystem, which is essential for the following considerations.

We adopt the Heisenberg picture for time evolution. An object in a given Heisenberg state defines a four-dimensional reference frame and not a set of three-dimensional frames depending on time. In particular, the object defines the origin of the time scale with the precision permitted by the indeterminacy relations.

We indicate by G the Poincaré or the Galilei group, or some other space-time

symmetry group. It acts on H(n)_{by means of the unitary representation}

g KU(n)

(g) 4U0(g) 7 U1(g) 7 R 7 Un 21(g) , g  G .

(4)

The description of the system by means of the algebra A(n) _{implicitly assumes the}

existence of an “external” object with a very large mass which defines a classical reference frame. If a classical reference frame is not available, only the G-invariant

elements of A(n)

represent observables. In other words, G has to be considered as a

global gauge group (2

). We indicate by B(n)

the subalgebra of A(n) _{composed of the}

G-invariant elements, namely the elements which commute with all the operators

U(n)(g). It is a von Neumann algebra [20], namely it is symmetric (self-adjoint) and

closed under the weak operator topology.

We also define the algebra Bk of the G-invariant elements of Ak which describes

the “internal” degrees of freedom of the object Fkand the algebra Bik of the invariant

elements of Aik which describes, besides the “internal” degrees of freedom of the

objects Fi and Fk, the “relative” parameters, which define the relative position, time

and velocity of the two objects. These relative parameters are the quantum analogs of the parameters defining the element of G which connects two classical reference

frames. The homomorphisms j(n)

k and jik(n) can be restricted to these invariant

subalgebras. We put

j(n)

k Bk4 Bk(n)%B(n), jik(n) Bik4 Bik(n)4 Bki(n)%B(n).

(5)

(1_{) In a more refined treatment, one should consider a smaller C * algebra A}(n)_%

L ( H(n)), in order to take into account the possible occurrence of superselection rules and the topological properties of the system [18].

(2_{) We do not consider a local Poincaré gauge invariance, which would lead to a theory of}

If i , j , k , are different indices, we have

Bji(n)O Bjk(n)4 Bj(n), n F3 .

(6)

Not all the kinds of objects are suitable for the definition of a reference frame. For instance, a spherically symmetric object cannot specify the directions of the spatial

coordinate axes. If the object Fj defines a reference frame accurately, we expect that

the algebra B(n)

jk describes the object Fk completely and that the following property

holds.

Assumption 1. The algebra B(n)

is generated by the n 21 subalgebras B(n)

jk with j

fixed and k c j.

In the following sections we shall test this assumption in some simple models. Without any loss of generality, we can assume that j 40.

We remind [20] that the von Neumann algebra generated by a symmetric

(self-adjoint) set S of operators is the double commutant S 9. If we indicate by U(n)the

von Neumann algebra generated by all the operators U(n)_{(g), we have}

B(n)4 U(n)8 . (7)

It is often convenient to describe a quantum system in terms of unbounded observables, which correspond to unbounded self-adjoint operators. If S is a set of self-adjoint not necessarily bounded operators, we indicate by S 8 the algebra of the bounded operators which commute with all the spectral projectors of all the operators in S. We say that a self-adjoint operator is associated to a von Neumann algebra if its spectral projectors belong to the algebra.

If we indicate by Sik a set of self-adjoint operators associated to Bik(n) and we

have S08 4 U(n), S04

0

and assumption 1 is satisfied for the frame F0.

We say that two subalgebras commute if all the elements of the first subalgebra commute with all the elements of the second one. One can easily see that if the indices

i , k , j , l are all different, the subalgebras Bik(n)and Bjl(n)commute. The following result

shows that in a quantum theory one cannot assume that the subalgebras B(n)

jk and Bjl(n)

commute.

Proposition 1. If the objects F0, F1 and F2 define quantum frames, namely they

satisfy assumption 1, and all the pairs of the subalgebras B01( 3 ), B02( 3 ) and B12( 3 )

commute, then the algebra B( 3 )_{is commutative and describes a classical system.}

In fact, since the subalgebras B01( 3 ) and B02( 3 ) generate B( 3 ), B12( 3 ) commutes with

B( 3 )

. In a similar way we see that B( 3 )

02 commmutes with B( 3 ). A third application of

assumption 1 shows that B( 3 ) _{commutes with itself, namely it is commutative.}

same kind (3_{) which describe n quantum frames and therefore satisfy assumption 1. We}

can simplify the notations by means of the identifications Bk4 B( 1 )and Bik4 B( 2 ). In

order to obtain a compact notation for the various homomorphisms of the algebras B(n)_{, we introduce the sets D}

n4 ] 0 , R , n 2 1 ( and for n F m F 1 the sets Gnm of the

injective functions p : DmK Dn. If p  Gnm, we consider the injective homomorphism

jp: A(m)K A(n)defined by . / ´ jp(A07 R 7 Am 21) 4B07 R 7 Bn 21, Bp(k)4 Ak, Bj4 1 for j  p(Dm) . (10)

These homomorphisms can be restricted to the G-invariant subalgebras, namely we have jp: B(m)K B(n). They have the property

jpjs4 jpis, p  Gnp, s  Gpm, n FpFmF1 . (11) We have j_k(n)_{4 j}p, p  Gn1, p( 0 ) 4k , (12) j(n) ik 4 jp, p  Gn2, p( 0 ) 4i, p( 1 ) 4k . (13)

We introduce the automorphism s of the algebra B( 2 )by means of the formula

s 4jp, p  G2 , 2, p( 0 ) 41 , p( 1 ) 40

(14)

and from eq. (11) we get

j_ki(n)_{s 4jik}(n), s2_{4 1 .}

(15)

The case n 43 is particularly interesting: we have an algebra B( 3 ) _{that contains}

three isomorphic subalgebras B01( 3 ), B02( 3 ) and B12( 3 ), and is generated by any pair of

them. This means, for instance, that an element of B( 3 )

12 , which concerns the relation

between the frames F1 and F2, can be expressed as a weak limit of algebraic

expressions containing elements of B( 3 )

01 and B02( 3 ). In other words, the mathematical

structure of B( 3 ) _{with its three subalgebras describes the composition law of the}

transformations of quantum frames, which can be considered as the quantum version of the multiplication law of the group G. It is not yet clear if the algebras B(n)

with n D3 contain further physically relevant pieces of information. One can remark that the formulation of the associative law requires three transformations and four reference frames.

In the treatment given above the algebras B(n) _{and the homomorphisms j}

p have

been constructed starting from the group G and its unitary representations. One can conceive a more general kind of theory in which the group G does not exist and the relations between quantum reference frames are described directly by the von

Neumann algebras B(n) _{and by the homomorphisms j}

p. Of course, some properties

which follow from the construction based on the group G have to be assumed explicitly. Besides assumption 1, we require:

Assumption 2. The homomorphisms jpare injective and satisfy eq. (11).

(3_{) We assume, however, that they have some different quantum number in order to avoid the}

Assumption 3. If the indices i , k , j , l are all different, the subalgebras B(n)

ik and

Bjl(n) commute.

One can also assume that eq. (6) is valid, possibly after a redefinition of the algebra B( 1 )_{. A special case of eq. (11) is}

jpjik(m)4 j

(n)

p(i), p(k), p  Gnm, n FmF2

(16)

and from assumption 1, we see that the homomorphism jpis uniquely determined by the

homomorphism j(n)

ik and jik(m).

From our assumptions one can derive the following result, that is stronger than proposition 1:

Proposition 2. If for a given choice of the indices i , k , j En, where nF3 and i, k, j are all different, the subalgebras B(n)

jk and Bji(n) commute, then all the algebras B(m)

are commutative.

Since the homomorphisms jp are injective, it is easy to show that the subalgebras

B(m)

jk and Bji(m) commute for any choice of the indices i , k , j Em, where mF3 and

i , k , j are all different. By taking also into account assumption 3, we see that the

subalgebra B(m)

ik commutes with all the subalgebras Bjl(m) with j c i , k and l c j. It

follows from assumption 1, that all the subalgebras B(m)

ik commute with B(m) and

another application of assumption 1 shows that B(m)is commutative. The algebras B( 1 )

and B( 2 )

are isomorphic to subalgebras of B( 3 )_{and are commutative too.}

Proposition 2 means that in a quantum theory the observables that describe a

transformation between the frames Fj and Fi cannot be compatible with the

observables that describe a transformation between the frames Fjand Fk. This fact is

strictly related to the “paradox of quantum frames” discussed in ref. [3].

3. – Classical reference frames

By considering the classical limit of the formalism developed in the preceding section, we clarify the meaning of various concepts and we can understand why the relations between quantum frames cannot be described by a quantum group [16, 17] obtained by a “quantization” of the group G.

In a classical theory the algebras A0, R , An 21and A(n)are commutative and can

be considered as algebras of LQ_{functions (essentially bounded measurable functions)}

on the phase spaces G0, R , Gn 21and

G(n)

4 G03 G13 R 3 Gn 21,

(17)

endowed with their Liouville measures. If we adopt the weak topology defined by the

duality with the spaces L1_{, we are dealing with commutative von Neumann algebras of}

operators acting multiplicatively on the corresponding L2 _{function spaces. The}

elements of A(n)_{are functions of the variables x}

0G0, R , xn 21Gn 21. The subalgebra

Ak(n) contains the functions that depend only on the variable xk and the subalgebra

A(n)

ik contains the functions that depend only on the variables xiand xk.

The Lie group G acts smoothly on the phase spaces preserving the Liouville measures. For this action we use the notation (g , xk) Kgxk, g  G , xkGk. A function

f  B(n) _{has the invariance property}

f (gx0, R , gxn 21) 4f(x0, R , xn 21) .

(18)

We say that G acts freely on G0if, for any x0G0, gx04 x0 implies g 4e (e is the unit

element). We also say that two submanifolds intersect transversally at a point if the vectors tangent to the submanifolds at this point generate the whole tangent space. Then we have:

Proposition 3. If G acts freely on G0 and there is a submanifold G×0%G0 that

intersects transversally every orbit of G0in one point, B(n) is generated by the n 21

subalgebras B(n)

0 k, (k 41, R, n21). Under these conditions assumption 1 is satisfied

by the frame F0.

This result could be obtained under much weaker hypotheses, but our purpose is only to give an example.

In order to sketch a proof, we remark that every x0G0can be written uniquely in

the form x04 gy0, with y0G×0, and this relation defines a diffeomorphism of G0 and

G 3G×0. The Liouville measure on G0is transformed into the product of a left-invariant

measure on G and a suitable measure on G×0. Since we have

f (gy0, x1, R , xn) 4f(y0, g21x1, R , g21xn 21) , f  B(n),

(19)

we can identify the algebra B(n) with the algebra of the LQ _{functions defined on}

G ×

03 G13 R 3 Gn 21. The subalgebra generated by the subalgebras B0k(n)contains all the

products of LQ _{functions of the kind f (y}

0, xk) and their linear combinations. This set

of functions is dense in B(n)_.

We consider the special case in which all the objects are of the same kind and we put

G04 R 4 Gn 214 G. We also assume that G acts freely and transitively on G. This

means that the objects have no “internal” degrees of freedom and that the manifolds G and G are diffeomorphic. If we choose a point y of G, and we put

f (g0y , g1y , R , gn 21y) 4f×(g021g1, R , g021gn 21) ,

(20)

we can consider B(n)

as a space of functions of n 21 arguments in G. In particular, B( 2 ) is a space of functions on G and we can write

B(n)

4 B( 2 )7 R 7B( 2 )

, (n 21 factors ) . (21)

If we restrict our attention to continuous functions, B( 2 ) _{has a structure of Hopf}

algebra [16, 17] (4). We have

[ jp f ](x0, x1, R , xn 21) 4f(xp( 0 ), xp( 1 ), R , xp(m 21)) , p  Gnm,

(22)

[ jpf×](g1, R , gn 21) 4f×(gp( 0 )21gp( 1 ), R , gp( 0 )21 gp(m 21)) ,

(23)

where in the right-hand side we have to put g04 e. We consider some particular cases:

[sf×](g) 4 f×(g21_{) , f}×

B( 2 ), (24)

namely the automorphism s is the antipode of the Hopf algebra B( 2 )_;

[ j( 3 ) 12 f×](g1, g2) 4 f×(g121g2) , (25) namely j( 3 ) 12 4 (s 7 1 ) D , (26) where D : B( 2 ) K B( 2 )7B( 2 )

4 B( 3 ) is the coproduct of the Hopf algebra. From the formula [ j(n) 0 k f×](g1, R , gn 21) 4 f×(gk) , (27) we obtain j01( 3 )A 4A71 , j02( 3 )A 417A , A  B( 2 ). (28)

The formulas (26) and (28) seem very natural, but their generalization to non-commutative Hopf algebras (quantum groups) is not possible. In fact, we see from eq. (28) that the subalgebras B( 3 )

01 and B02( 3 )commute and it follows from proposition 2

that B( 3 ) and B( 2 ) must be commutative. The formalism proposed in sect. 2 can be

considered as a modification of a quantum group. The coproduct is replaced by the homomorphisms j( 3 )

01 , j02( 3 )and j12( 3 ), which are treated in a symmetric way.

Moreover, a quantum group cannot describe the “internal” degrees of freedom of the quantum frames. In fact, from eqs. (28) and (6) we see that B( 3 )

0 and therefore B( 1 )

contain only the multiples of the unit element.

4. – The simplest example

As a first example of the general formalism, we consider the case in which G contains only the time translations and the objects that define the frames are

essentially clocks. The algebra B(n) is composed of the elements of A(n) which

commute with the Hamiltonian. Time measurements and clocks in quantum mechanics have been discussed by several authors [21-28].

The simplest realization of a clock is a system with one degree of freedom. We

consider a set of n clocks of this kind and we describe the clock Fk by means of the

canonical variables qk, pk. We have (ˇ 41)

[qi, pk] 4idik, (29) H 4

!

This system can be reinterpreted as a single free particle in an n-dimensional space.

The momenta pkand the components of the “angular momentum”

Lik4 qipk2 qkpi

are conserved self-adjoint operators. They satisfy the commutation relations (29) and [Lik, pj] 4idijpk2 idkjpi,

(32)

[Lik, Lrs] 4idirLks2 idisLkr2 idkrLis1 idksLir.

(33)

With the notation introduced in sect. 2, we put Sjk4 ]pj, pk, Ljk(. The set S0

contains all the momenta pk and the quantities L0 k with k 41, R, n21, which

generate the whole rotation group. It follows that the commutant S08 is composed of the

rotation-invariant bounded functions of the momenta pk, namely the bounded functions

of the Hamiltonian H, which form, in the present case, the algebra U(n)_{. We have seen}

that the condition (8) is satisfied and assumption 1 is valid.

The observable qkcan be interpreted as the running time directly read on the clock,

while pkis the rate of the clock. In the classical case, the true time is given by the ratio

Tk4 qkpk21. In the quantum case one can define in a suitable linear subspace of H the

Hermitian operator

Tk4 ( 2 pk)21qk1 qk( 2 pk)21,

(34)

which satisfies the commutation relation

i[Hk, Tk] 41 ,

(35)

or, more exactly,

exp [itHk] Tkexp [2itHk] 4Tk1 t .

(36)

An argument due to Pauli [21] shows that under these conditions Tk cannot be

self-adjoint. In fact, if it is self-adjoint one can prove that exp [iETk] Hkexp [2iETk] 4Hk2 E

(37)

and this equation contradicts the fact that Hk has a spectrum bounded from below.

Note that if an operator is not self-adjoint, it has no spectral representation and one cannot define bounded functions of it.

The relative time of two clocks is given by

Ti2 Tk4 ( 2 pipk)21Lik1 Lik( 2 pipk)21.

(38)

From the commutation relations (32), we see that if we fix pior pk, the quantity Likand

the relative time are completely undetermined. We see that the “internal” variables pi

play an important role and cannot be eliminated by imposing that they take a fixed value. The same remark holds for the more complicated models described in the following sections.

5. – One-dimensional non-relativistic frames

Now we assume that G is the Galilei group in one space dimension. An object which identifies a reference frame must contain al least two particles. We consider n objects

composed of two free particles and we indicate by Mk the total mass and by mk the

reduced mass of the object Fk. We adopt as canonical variables the centre-of-mass

pkconjugated to qk. The Hamiltonian is

H 4

!

k

Hk, Hk4 ( 2 Mk)21Pk21 221pk2.

(39)

The canonical variables qk and pk have been scaled in order to eliminate the reduced

mass mk. They describe a clock as in the preceding section.

The generators of the time and space translations are H and the total momentum

P 4

!

k

Pk,

(40)

while the Galilei boosts are generated by

K 42

!

!

The G-invariant variables are the internal variables pk, the relative variables Lik

discussed in the preceding section, the relative velocities of the centres of mass

Vik4 Mk21Pk2 Mi21Pi,

(43)

and the relative quantities

Yik4 pk(Xk2 Xi) 2qkVik.

(44)

The quantity

( 2 pk)21Yik1 Yik( 2 pk)214 Xk2 Xi2 TkVik

(45)

can be interpreted as the space coordinate of the origin of the frame Fkmeasured with

respect to the frame Fi.

We treat the Hilbert space H(n) _{as a space of L}2 _{functions defined on the}

2 n-dimensional space P of the variables pkand Pk(we use the same symbol for these

variables and for the corresponding multiplication operators). The operators pkand Vik

are clearly self-adjoint. The operators Lik and Yik are generators of a group of linear

measure-preserving transformations of the space P and they can be considered in a natural way as self-adjoint. One can see by means of a direct calculation that, if one

excludes the lower-dimensional manifold where pk4 0 and Vik4 0, the orbits of this

group have codimension two and are just the manifolds on which the quantities H and

P take constant values.

We put Sik4 ]pi, pk, Lik, Vik, Yik( and we see that S0 contains all the “internal”

variables pkand the relative variables L0 k, V0 kand Y0 k. We replace in the space P the

coordinates Pkby their linear combinations P and V0 k. If we indicate by H 8 the space

of the L2

functions of the variable P and by H 9 the space of the L2 _{functions of the}

other variables qk and V0 k, we have H(n)4 H 8 7 H 9. An operator A  S08 is diagonal

in the variables pkand V0 k, but acts in a general way on the variable P. In other words,

we have a direct integral decomposition [20] of A into operators A(pk, V0 k)  W, where

L0 kand Y0 k, A(pk, V0 k) must be constant on the orbits of the linear group generated by

these operators, namely it depends only on the variable

H × 4 H 2

g

!

h

!

g

!

h

!

In conclusion, the elements of S08 are given as direct integrals of bounded

measurable functions A(H×) with values in W and we can write [20] S08 4 W 7 V,

where V is the algebra of the bounded functions of H× considered as operators on H 9.

Note that the algebra W is generated by the operators P and

K 42 i

!

It follows that S08 is generated by the operators P, K and H×, namely it coincides with

U(n). Also in this case the condition (8) is satisfied and assumption 1 is valid.

6. – One-dimensional relativistic frames

As in the non-relativistic case, we consider n frames, each one defined by means of two free particles, described by two irreducible unitary representations of the Poincaré

group with masses mk8 and mk9. The generators of the Poincaré transformations are

given by H 4

!

!

!

(

)

(

)

The commutation relations are

[H , P] 40, [H, K] 4iP, [P, K] 4iH (50)

and similar relations hold for the generators acting on the various subsystems.

It is convenient to describe the object Fk by means of the variables Hk, Pk, Kk and

pk4 Pk8 Hk9 2 Pk9 Hk8 , qk4 Hk8 Kk9 2 Hk9 Kk8 , rk4 Kk8 Pk9 2 Kk9 Pk8 ,

(51)

which are connected by the identities

Hk22 Pk24 (mk8 )21 (mk9 )21 2( (mk8 mk9 )21 pk2)1 /24 Mk2,

(52)

pkKk1 qkPk1 rkHk4 0 .

We have the commutation relations

.

/

´

.

/

´

(

)

(

)

(

)

These relations and the non-relativistic limit suggest the interpretation of the quantities

T

×

k4 t 2 Tk, X×k4 ( 2 pk)21rk1 rk( 2 pk)21,

(56)

where Tkis given by eq. (34), as the time and space coordinates of the origin of the frame

Fkmeasured in a classical frame. The relative coordinates of the frames Fkand Fiare

given by . / ´ T × k2 T×i4 ( 2 pipk)21Qik1 Qik( 2 pipk)21, X × k2 X×i4 ( 2 pipk)21Rik1 Rik( 2 pipk)21, (57) where Qik4 qipk2 qkpi, Rik4 pirk2 pkri. (58)

These quantities commute with H and P and satisfy the commutation relations [K , Qik] 42 iRik, [K , Rik] 42 iQik.

(59)

It follows that the quantities

Lik4 (Hi1 Hk) Qik2 (Pi1 Pk) Rik,

(60)

Yik4 (Hi1 Hk) Rik2 (Pi1 Pk) Qik

(61)

are Poincaré invariant. They are proportional to the relative time and space coordinates

of the frames Fiand Fkmeasured in the center-of-mass frame of these two objects. Note

that the symbols which appear in the present section have not exactly the same meaning as in the preceding sections.

Another Poincaré-invariant quantity is the relative rapidity of two frames which is given by Vik4 zk2 zi, zi4 tanh21 Pi Hi . (62)

As in the preceding section, we put Sik4 ]pi, pk, Vik, Lik, Yik( and S0contains the

operators pk, V0 k, L0 kand Y0 k. We describe the vectors of the Hilbert space H(n) by

Lorentz-invariant measure dm 4

»

»

₍

₎

Then pkand Vikare self-adjoint multiplication operators. Since we have

K 8k4 2 iH 8k ¯ ¯_{P 8}k , _{K 9}k4 2 iH 9k ¯ ¯_{P 9}k , (64)

we see that Lik and Yik are first-order differential operators with smooth coefficients.

They generate one-parameter groups of (non-linear) measure-preserving morphism of the compact set in P defined by the inequality H GC. These diffeo-morphisms can be extended to the whole manifold P and they define continuous groups of unitary operators in H(n)_{. It follows that the generators L}

ikand Yikare self-adjoint.

A detailed analysis of the group of diffeomorphism generated by L0 kand Y0 kshows

that, if we exclude the lower-dimensional manifold where all the variables pkvanish, the

orbits have codimension two and are characterized by constant values of H and P. The argument proceeds in close analogy with the non-relativistic case. We introduce in the space P the coordinates pk, V0 kand

z 4tanh21 P

H

(65)

and we write H(n)

4 H 8 7 H 9, where H 8 contains the L2 functions of z and H 9

contains the L2

functions of the other coordinates. An operator A  S08 is diagonal in the

variables pkand V0 kand can be represented as a direct integral of a function with values

in the algebra W of the bounded operators in H 8. This function is constant on the orbits and therefore depends only on the quantity

M2_{4 H}2_{2 P}2₄

!

ik

MiMkcosh Vik.

(66)

We have S08 4 W 7 V, where V contains the bounded functions of M2 and W is

generated by the operators z and

K 42 i ¯

¯z .

(67)

Then S08 is generated by the operators z, K and M2, and it coincides with U(n). The

condition (8) is satisfied and assumption 1 is valid.

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