Linear Canonical Transformations in Relativistic Quantum Physics Hanitriarivo Rakotoson1, Raoelina Andriambololona2, Ravo Tokiniaina Ranaivoson3, Roland Raboanary 4
[email protected],
[email protected],
[email protected],
[email protected],
[email protected],
[email protected] Theoretical Physics Department1,2,3 Institut National des Sciences et Techniques Nucléaires (INSTN- Madagascar) BP 3901Antananarivo101, Madagascar,
[email protected] Mention Physique et Applications, Faculté des Sciences – University of Antananarivo1,4 Abstract : This work is a review and continuation of our previous works concerning Linear Canonical Transformations (LCT) and Phase Space Representation of Quantum Theory. LCT are known to be transformations generalizing some classical integral transformations like Fourier and fractional Fourier transforms. In quantum theory, they can be defined as linear transformations mixing the coordinate and momentum operators and keeping invariant the canonical commutation relations. The aim of this work is to present a study on these transformations in the framework of relativistic quantum physics using a phase space representation of quantum theory. According to this study, these transformations can be considered as being the elements of a pseudo-symplectic group (2,6).They have unitary and spinorial representations. We have also established that Lorentz transformations may be considered as special cases of these LCTs. From a physical point of view, LCTs can be considered as transformations which permit to define linear mixing between space, time, momentum and energy. Keywords: Linear Canonical Transformation, Representation, Relativistic Quantum Physics, Space-time, Momentum-energy.
1-Introduction According to a study that we have done in [1], most of the main theories in physics have symmetries groups with associated fundamental invariants. For the Newtonian mechanics, the main symmetry group is the Galileo one and the main corresponding invariants are distance, time, mass, and the Newton’s laws. For the theory of special relativity, the main symmetry group is the Lorentz group or the Poincaré group (if one include translations in spacetime) and the invariants are the speed of light, the pseudo-distance in spacetime or more generally any Minkowskian inner product and all the fundamental laws of relativistic dynamics [2], [3]. In relativistic quantum physics, fundamental relations are the canonical commutation relations between spatio-temporal coordinates operators and momentum-energy operators . 1
In natural unit system commonly used in relativistic quantum theory [4-7], in which one takes for the Planck reduced constant ℏ = 1 and the speed of light = 1 , the canonical commutation relations are ,
=
−
=
(1.1)
being the components of the symmetrical bilinear form which defines the Minkowskian inner product. It may be expected that a symmetry group which corresponds to relativistic quantum physics is a group of transformations which leaves invariant these canonical commutation relations (1.1). We show in this work that the group formed by linear canonical transformations (LCTs) satisfy this criterion. In the framework of signal theory, LCTs are known to be integral transformations which generalize some classical integral transformations like Fourier and fractional Fourier transforms. Let be a function of the time variable , representing a signal in representation. A linear canonical transformation which transforms to an another function of a variable (representing the signal in -representation) can be defined by the following relation [8] 1
( )= with , , , and
( )
2
the elements of a matrix
(
)
belonging to the special linear group
=
= The matrix ℳ = group
+
− −
(2, ℝ) (1.3)
= and is a real number. The matrix
(1.2)
−
=1
can be written in the form [9] ℳ
=
being an element of the Lie algebra
(1.4) (2, ℝ)of the Lie
(2, ℝ).
For the case
= 0,
= 0 and
=
− , the relation (1.2) is reduced to a fractional Fourier
transformation [8, 10]. And if we choose then = , we have the Fourier transform. In that case, the variable is the angular frequency = and ( ) = ( ) is the spectral representation of the signal. For a general LCT, the variable is a kind of intermediate variable between the time variable ( = 0) and the angular frequency variable ( = ). These results correspond to the fact that in the framework of signal theory, an LCT can be considered as a transformation in the time-frequency plane. It can be shown that in the framework of quantum mechanics, an LCT can be defined as being a linear transformation which mixes linearly the position operator and the momentum operator and leaving invariant the canonical commutation relations. Indeed, we showed in [8] that we have the following equivalence 2
′= + ′= + ⟺ ⟨ ′| ⟩ = [ ′, ′] = [ , ] =
1 2
⟨ | ⟩
(
)
(1.5)
⟨ ′| ⟩ and ⟨ | ⟩ being the wavefunctions respectively in the ′-representation and in the representation. The second part of the relation (1.5) is just an analog of the relation (1.2). Following this equivalence, one may thus define in a general way an LCT using the position and momentum operators and the canonical commutation relations. In section 2, we will present in details the development of this approach, using a phase space representation of quantum theory introduced and developed in our previous work [8-9, 11-15], by considering the case of LCTs defined on a general pseudo-Euclidian space. The main results are presented and discussed in section 3. We use the natural unit system. As in our previous work referred here, we write the operator in bold letter and corresponding eigenvalues by simple letter. The notation used for tensorial and matricial calculations are those defined and used in [2] With regard to linear canonical transformations, many publications have already been made by various authors about them and their applications. We may for instance quote [16-18].
2-Linear Canonical Transformations and their representations According to our papers [8], [9] and [13], the phase space representation of quantum theory introduced in [12] provides an adequate framework for defining and studying TCLs. 2.1- Phase space representation and dispersion operator For the one-dimensional case, the phase space representation of quantum mechanics is based on the introduction of the joint position-momentum states denoted | , , , 〉[12], [13] with a positive integer, the mean value of the coordinate operator in this state, the mean value of the momentum operator and the uncertainty (standard deviation) corresponding to for = 0. If is the uncertainty (standard deviation) corresponding to for = 0, we have the following relation [12], [13] =
1 2
(2.1)
We have explicitly the relations ⟨ ⎧⟨ ⎨⟨ ⎩⟨
, , , ,
, , , ,
, , , ,
| | , | | , |( − |( −
, , ⟩= , , ⟩= ) | , , , ⟩ = (2 + 1)( ) = (2 + 1) ) | , , , ⟩ = (2 + 1)( ) = (2 + 1)ℬ
(2.2)
in which = ( ) and ℬ = ( ) are the dispersions (statistical variances) corresponding respectively to and . The phase space representation of a quantum state | 〉 or an observable consists of using the basis {| , , , 〉} to represent them. Details about this representations are given in our work [12], [13] and [14]. We have also shown that the states | , , , 〉 are eigenstates of operators called dispersions operators: there is the coordinate 3
dispersion operator and the momentum dispersion operator explicit expressions are
also denoted ℶ . Their
1 = [( − ) + ( − ) ] = ℶ 2 ℬ 1 = ℶ = [( − ) + ( − ) ]ℬ 2
(2.3)
The corresponding eigenvalue equations are | , , , 〉 = (2 + 1) | , , , 〉 = (2 + 1)ℬ
(2.4)
Since the operators et = ℶ are proportional, we consider only one of them: From now we call it simply dispersion operator (without the prefix momentum).
=ℶ .
We have also shown in our previous work [12], [13] that if we consider the case of a dimensional pseudo-Euclidean space with signature ( , ), = + , , , and ℬ have multidimensional generalizations , , and ℬ which verify the following relations = ⎧ ⎪ℬ = ⎪ ⎪
1 2 ⎨ 1 = ⎪ℬ 2 ⎪ 1 ⎪ ⎩ ℬ =4 =
(2.5)
1 = 2
are the components of the symmetric bilinear form defining the inner product on the pseudo-Euclidean space. The multidimensional generalization of the dispersion operator ℶ is [13] ℶ
=
1 ( 2
−
)(
−
) + 4ℬ ℬ (
−
)(
−
)
(2.6)
2.2- Linear canonical transformation and pseudo-symplectic group We consider the case of a -dimensional pseudo-Euclidean space with signature ( , ). For the case of relativistic theory, this space is the Minkowski space with the signature (1, 3). For the sake of convenience, one introduces the centered-reduced momentums and coordinates operators and defined from and by the relation [13]
and
= √2
(
−
)
= √2
(
−
)
⟺
verify the same commutation relations as
= √2
+
= √2
+
and
(2.7)
(1.1). In fact, we have 4
,
=2
,(
−
−
) =2
=
(2.8)
According to the relation (1.5), linear canonical transformations can be defined as being linear transformations mixing coordinates and momentum operators and leaving invariant the canonical commutations relations. Given relations (1.1), (1.5) and (2.8), we may then define a linear canonical transformation by the relations =Π =Ξ ,
+Θ +Λ =
(2.9) ,
=
We can deduce that the coefficients Π , Θ , Ξ and Λ must satisfy the relations Π
Θ −Θ
Π =0
Ξ
Λ −Λ
Ξ =0
Π
Λ −Ξ
Θ =
(2.10)
If we denote respectively and the 1 × t row matrices representing and ; Π, Θ, Ξ and Λ the × square matrices corresponding to these coefficients and the matrix of the , it follows that relations (2.9) and (2.10) are equivalent to the following matrices relations ( ′ ) Π Ξ ′) = ( (2.11 ) Θ Λ Π Θ
Ξ Λ
0 −
0
0 Π Ξ = − Θ Λ
0
(2.11 )
Π Ξ is an element of the pseudo-symplectic Θ Λ matricial group (2 , 2 ) [13]. The dimension of this group as differentiable manifold is equal to (2 + 1) with = + . Given the relationship between a Lie group and its Π Ξ algebra, the matrix may be written in the form Θ Λ The relation (2.11b) means that a matrix
Π Θ
Ξ = Λ
ℳ
=
ℳ ℳ
ℳ ℳ
with ℳ an element of the Lie algebra (2 , 2 ) of the Lie group (2 , 2 relation (2.11) imply that the matrices ℳ , ℳ , ℳ and ℳ verify the relations ℳ = ℳ ℳ = ℳ ℳ =− ℳ
(2.12) ) . The
(2.13)
5
The dimension of belonging to (2 LCTs is
(2 , 2 ) is equal to (2 + 1). A parameterization of a matrix ℳ , 2 ) which is useful for the construction of unitary representation of
ℳ = ℳ = ℳ =
ℳ ℳ ℳ
⇔
= = =
(2.14)
Given (2.13), this parameterization is equivalent to write ℳ=
ℳ =
−
(2.15)
−
2.3 Dispersion operators algebra and unitary representation of LCTs et ℶ× defined by the relations
We have shown in [13] that the dispersion operators ℶ , ℶ ⎧ℶ ⎪ ⎨ℶ ⎪ × ⎩ℶ
1 ( − )( 2 1 ( − = 2 = ℬ ( − )( =
−
) + 4ℬ ℬ (
−
)(
−
)
−
) − 4ℬ ℬ (
−
)(
−
)
−
)+
(
−
−
(2.16)
)
generate a Lie algebra called dispersion operators algebra. The commutation relations characterizing these operators and this Lie algebra are given in [13]. We introduce the reduced operators ℶ , ℶ and ℶ× defined from ℶ , ℶ and ℶ× by the relations ℶ ℶ ℶ
×
=4
ℶ
=4
ℶ
=4
ℶ
(2.17) ×
Given the relations (2.7), (2.16) and (2.17) , we can deduce the following relations 1 ⎧ℶ = ( 4 ⎪ 1 ℶ = ( 4 ⎨ 1 ⎪ × ⎩ℶ = 4 (
+
)
−
)
+
)
(2.18)
The family {ℶ , ℶ , ℶ× } is a basis of the dispersion operators algebra . We have also establish that there is an isomorphism between the Lie algebra (2 , 2 ) and which allows to define an unitary representation of LCTs. In fact, it can be shown that, Π Ξ one can associate a unitary operator to any matrix = ℳ such as we have the Θ Λ following relations
6
Π Θ
Ξ = Λ
ℳ
,
(
ℶ
ℶ
+Θ +Λ =
(
=
= =Π =Ξ
(
×
=
)
)
(2.19)
×
ℶ× )
×
(2.20)
= =
⟺
,
=
,
(2.21) =
,
=
The relations (2.19) and (2.20) express the isomorphism between (2 , 2 ) and and define a mapping between (2 , 2 ) and the group to which the operator belongs to. This mapping then defines the unitary representation of LCTs given in the relation (2.21). 2.5- Representation of LCTs with special pseudo-orthogonal transformations The representation with special pseudo-orthogonal transformations is necessary in order to construct spinorial representation [9, 14-16]. We introduce the following parameterization for the matrix ℳ of the relation (2.12) ℳ1 = + ℳ2 = − ⎨ℳ3 = + ⎩ ℳ4 = − ⎧
(2.22)
with , , and matrices which satisfy the following relations (which are compatible with the relation 2.13) ⎧ = ⎪ = (2.23) ⎨ = ⎪ =− ( )=0 and ⎩ According to the relations (2.21), (2.11) and (2.12), the LCT of the relation ((2.11) can be written in the form ( ′
′) = (
)
(2.24)
This parameterization is the multidimensional generalization of the relation (1.4). We introduce the operators ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
= = = =
1
√2 1
√2 1
√2 1
√2
,
,
defined by the following relations
,
⊗
+
⊗
=
⊗
−
⊗
=
⊗
+
⊗
=
⊗
−
⊗
=
1
√2 1
√2 1
√2 1
√2
0 + 0 − 0 + 0 −
− 0 + 0 − 0 + 0
(2.25)
7
, , and being the Pauli matrices. The law of transformation of these operators under the action of an LCT is ( with
the 4 × 4
)=(
′
′
)
(2.26)
square matrix − −
= −
(2.27)
−
It can be verified that is an element of the algebra (2 , 2 ) which is the Lie algebra of the special pseudo-orthogonal group (2 , 2 ). The relation (2.26) defines a representation of the TCLs with special pseudo-orthogonal transformations.. 2.6- Spinorial representation of LCT The representation of LCT with special orthogonal transformations as defined by the relation (2.26) allows the construction of the spinorial representation. In fact, a spin group is just the double cover of a special orthogonal group [9, 19-21]. We consider the operator = with
,
,
and
defined by the relation ⊗
+
⊗
+
⊗
+
⊗
(2.28)
the generators of the Clifford algebra ℭ(2 , 2 ) [9, 19-21]. Let us
consider the LCT defined by the relation (2.24) and let us denote ℊ = ( ℊ is an element of (2 , 2 admits as elements the matrices
)). Let us now define the group of the form =
⊗
=
⊗
Π Ξ = Θ Λ (2 , 2 ) ⊗
which
(2.29)
being the 2 × 2 identity matrix and an element of (2 , 2 ) which is the universal cover of (2 , 2 ) . is an element of the Lie algebra (2 , 2 ) of the group (2 , 2 ) . It is a linear combination of the elements of the family { , , , , , , , } which is a basis of (2 , 2 ) . The spinorial representation of LCTs can be obtained with the introduction of a mapping between (2 , 2 ) and (2 , 2 ) × which associate to any element ℊ of (2 , 2 ) an element of (2 , 2 ) ⊗ according to the relations +
)=(
= (ℊ) ⟺ (
)ℊ = (
)
(2.30)
= It can be shown that this relation is verified if we have [9] =
(
℧
℧
℧⋈
℧⋉ )( ⊗
)
(2.31) 8
with 1 2 1 = 2 1 = 2 1 = ( 4
⎧℧ ⎪ ⎪℧
=
⎨℧ ⎪ ⋈ ⎪ ⎩℧⋉
+
⊗
+
⊗ (2.32)
+
⊗
+
−
−
)⊗
and , , and being the elements of the matrices , , and . The relations (2.30) and (2.31) define the spinorial representation of LCTs. The element of (2 , 2 ) ⊗ ⃑ of the relation (2.31) act on the spinors ,of components , which belongs to the spinorial space of the theory [9] ⃑ =
⃑⇔
=
2.7- LCTs and Lorentz transformations We consider the case of the Minkovski space with the signature (1,3).On one hand, for an LCT, we have the relation (2.11a) ( ′
with
) Π Θ
′) = (
Ξ Λ
(2.33)
Π Ξ verifying the relation (2.11b) Θ Λ Π Θ
Ξ Λ
0 −
0
Π Θ
0 Ξ = − Λ
(2.34)
0
On the other hand, let us consider a Lorentz transformation defined by a special pseudoorthogonal matrix Υ belonging to (1,3). Υ verifies the relation Υ= é (Υ) = 1
(2.35)
The laws of transformation of the row matrices and (representing = Υ 0 ) Υ ′) = ( ⟺( ′ 0 Υ = Υ For the matrices
and
(representing ( ′
′) = (
and
and
) are (2.36)
), we have also ) Υ 0
0 Υ
(2.37)
So we have
9
(
− ′
For the matrices
and
− ) Υ 0
− ′) = ( − (representing
=Υ Υ =Υ Υ
and
0
⟺
=
0
0 Υ
(2.38)
), the laws of transformations are
Υ 0
0 Υ
0
Υ 0
0
0 Υ
(2.39)
From the relation (2.7), We may deduce the following matricial relation ⎧( ⎪
0 √2 0 √2
− ) √2 0 − )=( ) √2 0
)=( −
⎨( − ⎪ ⎩
(2.40)
then the relation (2.38) becomes 0 ) √2 ′ =( 0 √2 ′
(
As we have = deduce from (2.41) that
=
=
′
) = 2(
(
0 √2
) √2 0 ,
=
0 Υ
(2.41)
being the 4 × 4 identity matrix, we can
0
)
Υ 0
0
Υ 0
0 Υ
′ 0
0
Υ 0
0 ′
(2.42)
0 Υ
(2.43)
Taking into account of (2.39), this relation becomes (
) = 2(
)
= 2(
)
Υ 0
0 Υ
Υ 0
0 ( )
Υ 0
0 Υ
0 0 ( ) 0
0 Υ
0
=(
ℬ = ( ) and = ( ) being the matrices representing (relation 2.5). These matrices verify the properties
) 2ℬΥ 0
0 2 Υ and
ℬ =
ℬ = ℬ = 1 ⎨ ⎩ ℬ=4
(2.44) =
⎧
Given these properties, we have 2ℬΥ 0 As ( ) = (and
0 2 Υ ℬ=
= 2Υ
ℬ 0
(2.45)
0 2Υ
=
2Υ
ℬ 0
0 2Υ
ℬ
(2.46)
), we may deduce 10
2ℬΥ 0
0 2 Υ
0 −
0
As Υ is an element of the group becomes 2ℬΥ 0 0 2 Υ
2ℬΥ 0
0 2 Υ
=
(1,3), we have Υ 0 −
0
2ℬΥ 0
0 −Υ Υ
Υ Υ
=
0 2 Υ
Υ 0
(2.47)
so the relation (2.47) =
0 −
0
(2.48)
Comparison of the relations (2.33), (2.34), (2.44) and (2.48) allows us to deduce that a Lorentz transformation is a particular case of LCT.
3-Main results We may list the following main results: i) The phase space representation of quantum theory that we have introduced and developed in our previous works provides an adequate framework for the study of LCTs ii) According to the relations (2.11a) and (2.11b), the LCTs can be considered as the elements of a pseudo-symplectic group (2 , 2 ) which acts on the set of momentum and coordinates operators in an -dimensional pseudo-euclidian space with signature ( , ). For the case of relativistic quantum physics, this space is the 4-dimensional Minkowski space with = 1 and = 3. The LCTs are then the elements of the pseudo-symplectic group (2, 6). iii) According to the relations (2.19), (2.20) and (2.21), a unitary representation of LCTs can be obtained using the dispersion operators algebra. Unitary representation is very important for quantum theory. iv) According to the relations (2.26), (2.27), (2.30) and (2.31), we may have special pseudoorthogonal and spinorial representations of LCTs. v) According to the relations (2.33), (2.34), (2.44) and (2.48), Lorentz transformations are particular case of LCTs
4- Conclusion According to the results that we have obtained in this work, the definition of LCTs as transformations which mix linearly coordinate and momentum operators and leaving invariant the canonical commutation relations (relations 2.9, 2.11a and 2.11b) provides a simple and natural way to study them in the framework of relativistic quantum physics. This study can be performed adequately using the phase space representation of quantum theory. It was shown that LCTs can be considered at the same time as generalization of fractional Fourier transform (which are themselves generalizations of Fourier transforms) and as generalization of Lorentz transformations. On one hand, Fourier transforms can be considered as deeply linked with quantum physics because of the wave - particle duality and they allow to perform the change between the coordinates and momentum representations. And on the other hand it is well known that Lorentz group is the main symmetry group of relativistic theory. It is then natural that the group S (2,6) of LCTs can be considered, as expected, to be a main symmetry group of relativistic quantum physics. The existence of the unitary, the special pseudo-orthogonal and the spinorial representations make this group more adequate for this purpose. 11
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