Continuous Symmetries and Integral Transforms

Whenever continuous symmetry transformations — e.g., rotations, translations, scaling, etc. — are implemented on an appropriate Hilbert space, it becomes a unitary operator, expressible as

D[\vec{\xi}] \equiv \exp[-i \xi_A J^A],

for some set of Hermitian generators \{ J^A | A=1,2,3,\dots\} and real parameters \{ \xi^A \}. The eigenstates \{ | \nu \rangle \} of these Hermitian operators can be used to expand arbitrary states in the Hilbert space. Namely, if \{ | x \rangle \} were some original set of complete basis (usually position), completeness relations give schematically:

\langle x | \psi \rangle \sim \sum_\nu \langle x | \nu \rangle \langle \nu | \psi \rangle

\langle \nu | \psi \rangle \sim \sum_x \langle \nu | x \rangle \langle x | \psi \rangle .

As we shall see, these form integral transform pairs.

Fourier Transform and Spatial Translation Symmetry

If |\vec{x}\rangle denotes a position eigenket in Euclidean space endowed with Cartesian coordinates \{ \vec{x} \}, translation is defined by

\mathcal{T}[\vec{a}] | \vec{x} \rangle = | \vec{x} + \vec{a} \rangle.

Local translation symmetry is reflected by the position-independence of the measure in

\langle \vec{x} | \vec{x}' \rangle = \delta^{(D)}[\vec{x}-\vec{x}'] = \langle \vec{x} + \vec{a} | \vec{x}' + \vec{a} \rangle \\ = \langle \vec{x} | \mathcal{T}[\vec{a}]^\dagger \mathcal{T}[\vec{a}] | \vec{x}' \rangle,

for any constant displacement \vec{a}. Global translation symmetry is reflected in the position-independence of the measure in

\mathcal{T}[\vec{a}] \equiv \int_{\mathbb{R}^D} d^D\vec{x}' \cdot | \vec{x}' + \vec{a} \rangle \langle \vec{x}'| ;

which is unitary for the same reason. Now, such a unitary translation operator must then be generated by a Hermitian operator P_i, which is usually dubbed “momentum”:

\mathcal{T}[\vec{a}] =\exp[ -i\vec{a}\cdot\vec{P}] .

By identifying translation with Taylor expansion, the momentum operator is can be seen to be a derivative operator:

\langle \vec{x} | P_i | \psi \rangle = -i \partial_{x^i} \langle \vec{x} | \psi \rangle, \qquad \forall |\psi\rangle .

The momentum operator’s complete set of eigenstates | \vec{k} \rangle are nothing but the plane waves, which read in position space as

\langle \vec{x} | \vec{k} \rangle = \exp[i\vec{k}\cdot\vec{x}],

where the associated eigensystem equation is

-i \partial_{x^i} \langle \vec{x} | \vec{k} \rangle = \langle \vec{x} | P_i | \vec{k} \rangle = k_i \langle \vec{x} | \vec{k} \rangle .

Any state \langle \vec{x} | \psi \rangle, originally written in the position basis, may be expanded in momentum states — i.e., as a sum over the eigenstates of P_i:

(Fourier.I) \psi[\vec{x}] \equiv \langle \vec{x} | \psi \rangle = \int_{\mathbb{R}^D} \frac{d^D \vec{k}}{(2\pi)^D} \langle \vec{x} | \vec{k} \rangle \langle \vec{k} | \psi \rangle \\ = \int_{\mathbb{R}^D} \frac{d^D \vec{k}}{(2\pi)^D} e^{i\vec{k}\cdot\vec{x}} \widetilde{\psi}[\vec{k}] .

The inverse transformation is

(Fourier.II) \widetilde{\psi}[\vec{k}] \equiv \langle \vec{k} | \psi \rangle = \int_{\mathbb{R}^D} d^D \vec{x} \langle \vec{k} | \vec{x} \rangle \langle \vec{x} | \psi \rangle \\ = \int_{\mathbb{R}^D} d^D \vec{x} e^{-i\vec{k}\cdot\vec{x}} \psi[\vec{x}] .

Equations (Fourier.I) and (Fourier.II) are of course the well known Fourier transform pairs.

Scaling Symmetry and the Mellin Transform

Scaling transformation applied to a ‘position eigenket’ | r > 0 \rangle on the positive real line is defined as

D_s[\lambda] | r \rangle \equiv \lambda^{s+1} | \lambda \cdot r \rangle .

Here, the application-specific s and the scaling \lambda are both positive. If the inner product is defined as

\langle \psi_1 | \psi_2 \rangle \equiv \int_0^\infty \langle \psi_1 | r' \rangle \langle r' | \psi_2 \rangle r'^{2s+1} d r'

— leading immediately to the completeness relation

(Mellin.Completeness) \mathbb{I} = \int_0^\infty | r' \rangle \langle r' | r'^{2s+1} d r'

— the dilatation operator D_s[\lambda] itself is in fact unitary. This in turn tells us, since D_s[\lambda] is continuously connected to the identity, it must involve the exponential of a Hermitian generator. In fact, if we parametrize

D_s[\lambda = e^\epsilon] | r \rangle = e^{(s+1)\epsilon} | e^\epsilon \cdot r \rangle ,

we see the exponents add upon group multiplication:

D_s[\lambda = e^\epsilon] D_s[\lambda' = e^{\epsilon'}] = D_s[\lambda'' = e^{\epsilon''}];

i.e., where \epsilon'' \equiv \epsilon+\epsilon'. We therefore write the operator itself as

D_s[\epsilon] \equiv \exp\left[ -i \epsilon \cdot \mathcal{E}_s \right] .

A direct Taylor series expansion in \epsilon of \langle \psi | D_s[\epsilon] | r \rangle = e^{(s+1)\epsilon} \langle \psi | e^{\epsilon} \cdot r \rangle would reveal,

\langle r | \mathcal{E}_s | \psi \rangle = i \left( r \partial_r + s + 1 \right) \langle r | \psi \rangle.

This in turn allows us to solve for its eigenstate | \nu \rangle in the position basis:

\langle r | \nu \rangle = e^{-i \nu - (s+1)},

with the normalization (defined up to constant factors) given by

\langle \nu | \nu' \rangle = 2\pi \delta[\nu - \nu'].

Because \epsilon is a real number, \mathcal{E}_s must indeed be Hermitian; and, hence, the latter’s eigenvalues are guaranteed to be real as well. At this point, we may exploit its complete set of eigenfunctions to expand any state on the half line \mathbb{R}^+:

(Mellin.InverseTransform.v1) \langle r | \psi \rangle = \int_{\mathbb{R}} \frac{d \nu}{2\pi} \langle r | \nu \rangle \langle \nu | \psi \rangle \\ = \int_{\mathbb{R}} \frac{d \nu}{2\pi} r^{-i\nu - (s+1)} \langle \nu | \psi \rangle.

If we define

z \equiv (s+1)+i\nu ,

eq. (Mellin.InverseTransform.v1) may be re-phrased as an integral on the complex z-plane:

(Mellin.InverseTransform.v2) \langle r | \psi \rangle = \int_{-i \infty}^{+i\infty} \frac{d z}{2\pi i \cdot r^{z}} \langle z | \psi \rangle.

The inverse transformation, using the completeness relation in eq. (Mellin.Completeness), is

(Mellin.Transform) \langle \nu | \psi \rangle = \int_{\mathbb{R}^+} d r' \cdot r'^{2s+1} \langle \nu | r' \rangle \langle r' | \psi \rangle \\ = \int_{\mathbb{R}^+} d r' \cdot r'^{z - 1} \langle r' | \psi \rangle .

The equations (Mellin.Transform.v2) and (Mellin.Transform) are of course the Mellin transform pairs. Be aware that these integrals are defined with a specific range of s+1 \equiv \text{Re}[z] in mind.

References

  • J. J. Sakurai, Jim Napolitano, “Modern Quantum Mechanics,” Cambridge University Press; 3rd edition
  • Bertrand, J., Bertrand, P., Ovarlez, J. “The Mellin Transform,” The Transforms and Applications Handbook: Second Edition. Ed. Alexander D. Poularikas; Boca Raton: CRC Press LLC, 2000

Author: Yi-Zen Chu

I am a theoretical physicist, with research interests spanning gravitation and field theory, particle cosmology and Mathematica software development.

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