Study on Representation of Linear Canonical Transformation in Quantum mechanics Raoelina Andriambololona1, Hanitriarivo Rakotoson2, Ravo Tokiniaina Ranaivoson3

raoelina.andriambololona@gmail.com1; ,infotsara@gmail.com2 ; tokhiniaina@gmail.com3

Theoretical Physics Department

Institut National des Sciences et Techniques Nucléaires (INSTN- Madagascar)

BP 4279 101-Antananarivo –Madagascar, instn@moov.mg

Abstract: Linear canonical transformations are particularly well known in the framework of

signal theory as integral transformations. Particular example of Linear canonical

transformations are the Fourier and Fractional Fourier Transforms. We show that Linear

canonical transformations can be well described in framework of quantum mechanics using

properties of momentum and coordinates operators, linear algebra and group theory.

Keywords: Linear Canonical Transformation, Fractional Fourier Transform, Quantum theory, Quantum operators, Lie algebra

1-Introduction

Let 𝑓 be a function of the time variable 𝑡 and 𝔐 the real matrix

𝔐 = 𝕒 𝕓𝕔 𝕕

Verifying 𝑑𝑒𝑡𝔐 = 𝕒𝕕− 𝕓𝕔 = 1 i.e 𝔐 ∈ 𝑆𝐿(2,ℝ). In framework of signal theory, a Linear Canonical Transformation (LCT) may be defined as the integral transformation [1], [2]

𝐹 𝑢 =12𝜋𝕓 𝑒

!" 𝑓(𝑡)𝑒!!𝕓(!"!

𝕒!!!𝕕!!! )𝑑𝑡

In which 𝜖 is a real constant. For the case

𝕒 = 𝑐𝑜𝑠 𝛼𝕓 = 𝑠𝑖𝑛(𝛼) 𝕔 = −𝑠𝑖𝑛(𝛼) 𝕕 = 𝑐𝑜𝑠(𝛼)

𝑒𝑖𝜖 = 𝑒!!(!!

!!)

we obtain the Fractional Fourier Transformations [1], [3]

𝐹! 𝑢 =1− 𝑖𝑐𝑜𝑡 𝛼

2𝜋 𝑓(𝑡)𝑒![!"# ! (!!!!!! )! !"

!"#(!)]𝑑𝑡

And for the case 𝛼 = !!, we have the Fourier transformation

𝐹!!𝑢 =

12𝜋

𝑓(𝑡)𝑒!!"#𝑑𝑡

It can be shown that in the framework of quantum mechanics, a linear canonical

transformation can be defined as a linear transformation mixing coordinate and momentum operators 𝒙 and 𝒑 and leaving invariant the canonical commutation relation [1].

𝒚 = 𝕒𝒙+ 𝕓𝒑𝒌 = 𝕔𝒙+ 𝕕𝒑

[𝒚,𝒌]! = [ 𝒙,𝒑]! = i

In fact combining the three relation, we can deduce the condition which links the

parameters, 𝕒,𝕓, 𝕔 and 𝕕, of the linear canonical transformation

𝕒𝕕− 𝕓𝕔 = 1

And we may show that the above integral transformation, defining the LCT, corresponds to the law of transformation of wave functions [1]

𝑦 𝜓 =12𝜋𝕓 𝑒

!" 𝑥 𝜓 𝑒!!𝕓(!"!

𝕒!!!𝕕!!! )𝑑𝑥

𝑦 and 𝑥 being respectively the eigenstates of the operators 𝒚 and 𝒙 and 𝜓 a state vecto

r.This analogy justify definition of LCT based on the use of the momentum and coordinates o

perators. This definition gives a natural way to study them and their multidimensional general

ization in framework of quantum theory.

2-Dispersion operator algebra

Let 𝑛,𝑋,𝑃,∆𝑝 be the quantum state of a particle admitting as wave functions, respectively in momentum and coordinates representations the functions [1],[4]

𝜑! 𝑥,𝑋,𝑃,Δ𝑝 =𝐻!(

𝑥 − 𝑋2Δ𝑥

)

2!𝑛! 2𝜋Δ𝑥𝑒!(

!!!!"! )

!!!"#

𝜑! 𝑝,𝑋,𝑃,Δ𝑝 =12𝜋

𝜑! 𝑥,𝑋,𝑃,Δ𝑝 𝑒!!"#𝑑𝑥 =(−𝑖)!𝐻!(

𝑝 − 𝑃2Δ𝑝

)

2!𝑛! 2𝜋Δ𝑝𝑒!(

!!!!"!)

!!!"(!!!)

i.e 𝑥 𝑛,𝑋,𝑃,Δ𝑝 = 𝜑! 𝑥,𝑋,𝑃,Δ𝑝 , and 𝑝 𝑛,𝑋,𝑃,Δ𝑝 = 𝜑! 𝑝,𝑋,𝑃,Δ𝑝 . 𝐻! being a Hermite polynomial. The momentum operator and coordinates operator mean values and statistical dispersion (variance) corresponding to a state 𝑛,𝑋,𝑃,∆𝑝 are respectively 𝑋,𝑃, 2𝑛 + 1 Δ𝑥 !, (2𝑛 + 1) Δ𝑝 !. The parameters Δ𝑝 and Δ𝑥 verifying the relation

Δ𝑝 Δ𝑥 =12

Let ℬ = Δ𝑝 ! , it can be shown that a state 𝑛,𝑋,𝑃,∆𝑝 is an eigenstate of the operator [1],[4],[5]

ℶ! =12 𝒑− 𝑃 ! + 4 ℬ ! 𝒙− 𝑋 !

with the eigenvalues (2𝑛 + 1) Δ𝑝 !. ℶ! is called momentum dispersion operator. If we introduce the two hermitian operators [5]

ℶ! =12 𝒑− 𝑃 ! − 4 ℬ ! 𝒙− 𝑋 !

ℶ× = ℬ 𝒑− 𝑃 𝒙− 𝑋 + 𝒙− 𝑋 𝒑− 𝑃

It can be shown, using the commutation relation [𝒙,𝒑]! = 𝑖, that the three operators ℶ!, ℶ!

and ℶ× verify the commutation relation

[ℶ!, ℶ!]! = 4𝑖ℬℶ×

[ℶ!, ℶ×]! = −4𝑖ℬℶ!

[ℶ×, ℶ!]! = 4𝑖ℬℶ!

i.e they may be considered as the vector basis of a Lie algebra which can be called dispersion operator algebra [5]. Let be

∆𝑥 = 𝒶, ∆𝑝 = 𝒷, 𝒜 = ∆𝑥 ! = (𝒶)! ℬ = (∆𝑝)! = (𝒷)!

We can introduce the following reduced operator

𝒙 =𝒙− 𝑋2 Δ𝑥

=𝒙− 𝑋2𝒶

=𝒙− 𝑋2𝒜

= 2 Δ𝑝 𝒙− 𝑋 = 2𝒷 𝒙− 𝑋 = 2ℬ 𝒙− 𝑋

𝒑 =𝒑− 𝑃2 Δ𝑝

=𝒑− 𝑃2𝒷

=𝒑− 𝑃2ℬ

= 2 Δ𝑥 𝒑− 𝑃 = 2𝒶 𝒑− 𝑃 = 2𝒜 𝒑− 𝑃

and

ℶ! =ℶ!

4ℬ =14 ((𝒑)

! + (𝒙)𝟐)

ℶ! =ℶ!

4ℬ=14((𝒑)! + (𝒙)𝟐)

ℶ× =ℶ×

4ℬ =14 𝒑𝒙+ 𝒙𝒑

We have the commutation relations

[ℶ!, ℶ!]! = 𝑖ℶ×

[ℶ!, ℶ×]! = −𝑖ℶ!

[ℶ×, ℶ!]! = 𝑖ℶ!

[ℶ!,𝒑]! =!!𝑖𝒙

[ℶ!,𝒑]! = − !!𝑖𝒙

[ℶ×,𝒑]! =!!𝑖𝒑

[ℶ!,𝒙]! = − !!𝑖𝒑

[ℶ!,𝒙]! = − !!𝑖𝒑

[ℶ×,𝒙]! = − !!𝑖𝒙

The set ℶ!, ℶ!, ℶ× is also a basis of the dispersion operator algebra. 3-Unitary representation

As we have already seen above, a linear canonical transformation can be defined as a linear transformation mixing the coordinate operator 𝒙 and the momentum operator 𝒑 and leaving invariant the commutator 𝒙,𝒑 = 𝑖 [1], [5]. The operators 𝒙 and 𝒑 are linearly linked with the operators 𝒙 and 𝒑 we may also take a definition of linear canonical transformation as linear transformation mixing 𝒙 and 𝒑

𝒑! = Π𝒑+ Θ𝒙𝒙! = Ξ𝒑+ Λ𝒙

𝒙′,𝒑′ = 𝒙,𝒑 = 𝑖⟺ 𝒑! 𝒙! = 𝒑 𝒙 Π Ξ

Θ Λ , Π ΞΘ Λ = ΠΛ− ΘΞ = 1

The matrix Π Ξ

Θ Λ is an element of the special linear group 𝑆𝐿(2,ℝ). We may write it in the form

Π ΞΘ Λ = 𝑒ℳ = 𝑒

ℳ! ℳ!ℳ! ℳ!

with ℳ an element of the Lie algebra 𝖘𝖑(2,ℝ) of the Lie group 𝑆𝐿(2,ℝ), we have

ΠΛ− ΘΞ = 1⟺ℳ! = −ℳ! ⟺ℳ = ℳ! ℳ!ℳ! −ℳ!

Using the commutation relation between the operators ℶ!, ℶ!, ℶ×,𝒙 and 𝒑, we may also establish an unitary representation of the linear canonical transformation [5]

𝒑! = Π𝒑+ Θ𝒙 = 𝑼𝒑𝑼!

𝒙! = Ξ𝒑+ Λ𝒙 = 𝑼𝒙𝑼!

with

Π ΞΘ Λ = 𝑒ℳ = 𝑒

ℳ! ℳ!ℳ! ℳ! = 𝑒

!!

!!× !!!!!!!!!! !× 𝑼 = 𝑒!(!!ℶ!!!!ℶ!!!×ℶ×)

The unitarity of 𝑼 results from the hermiticity of the operators ℶ!, ℶ!and ℶ×.

4- Conclusion

The properties of momentums and coordinates operators permit to perform a well description of Linear Canonical Transformation in framework of quantum mechanics. The introduction of dispersion operator algebra permits to establish unitary representation.

References

1. RaoelinaAndriambololona, Ravo Tokiniaina Ranaivoson, Rakotoson Hanitriarivo, Wilfrid Chrysante Solofoarisina: Study on Linear Canonical Transformation in a Framework of a Phase Space Representation of Quantum Mechanics, arXiv:1503.02449[quant-ph], International Journal of Applied Mathematics and Theoretical Physics. Vol. 1, No. 1, 2015, pp. 1-8, 2015

2. Tian-Zhou Xu, Bing-Zhao Li: Linear Canonical Transform and Its Applications, Science Press, Beijing, China, 2013.

3. V. Ashok Narayanan, K.M.M. Prabhu, “The fractional Fourier transform: theory, implementation and error analysis”, Microprocessors and Microsystems 27 (2003) 511–521, Elsevier, 2003.

4. Ravo Tokiniaina Ranaivoson: Raoelina Andriambololona, Rakotoson Hanitriarivo, Roland Raboanary: Study on a Phase Space Representation of Quantum Theory, arXiv:1304.1034[quant-ph],International Journal of Latest Research in Science and Technology, ISSN(Online):2278-5299, Volume 2,Issue 2 :Page No.26-35,March-April, 2013

5. Raoelina Andriambololona, Ravo Tokiniaina Ranaivoson, Randriamisy Hasimbola Damo Emile, Rakotoson Hanitriarivo: Dispersion Operators Algebra and Linear Canonical Transformations, arXiv:1608.02268 [quant-ph], 2016