Mathematical Physics Archives - Time Travel, Quantum Entanglement and Quantum Computing https://stationarystates.com/category/mathematical-physics/ Not only is the Universe stranger than we think, it is stranger than we can think...Hiesenberg Sun, 19 Oct 2025 20:55:15 +0000 en-US hourly 1 https://wordpress.org/?v=6.9.1 Temporal Green’s Functions https://stationarystates.com/mathematical-physics/temporal-greens-functions/?utm_source=rss&utm_medium=rss&utm_campaign=temporal-greens-functions Sun, 19 Oct 2025 20:48:46 +0000 https://stationarystates.com/?p=1057 Temporal Green’s Function Temporal Green’s Function The temporal Green’s function is the Green’s function that solves a differential equation involving time — typically the evolution equation of a dynamical system […]

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Temporal Green’s Function

Temporal Green’s Function

The temporal Green’s function is the Green’s function that solves a differential equation involving time — typically the evolution equation of a dynamical system — for a delta-function disturbance in time.
It tells you how the system responds at later (or earlier) times to an instantaneous impulse applied at a specific time.

1. Definition

Consider a linear time-dependent differential operator L_t acting on some function f(t):

    \[ L_t f(t) = s(t), \]

where s(t) is a source term.

The temporal Green’s function G(t, t') is defined as the solution to:

    \[ L_t G(t, t') = \delta(t - t'). \]

Once G(t, t') is known, the solution for any source s(t) is given by:

    \[ f(t) = \int_{-\infty}^{\infty} G(t, t')\,s(t')\,dt'. \]

2. Physical Meaning

G(t, t') represents the response of the system at time t due to a unit impulse applied at time t'.
If the system is **causal**, the Green’s function vanishes for t < t':

    \[ G(t, t') = 0 \quad \text{for} \quad t < t'. \]

This ensures that the response cannot precede the cause.

3. Example 1 – First-Order Decay (Damped System)

Consider the simple equation:

    \[ \frac{df}{dt} + \alpha f = s(t), \]

where \alpha > 0.

The Green’s function satisfies:

    \[ \left(\frac{d}{dt} + \alpha\right) G(t, t') = \delta(t - t'). \]

For a causal solution (G=0 when t < t'):

    \[ G(t, t') = e^{-\alpha (t - t')} \, \Theta(t - t'), \]

where \Theta is the Heaviside step function.

Thus, the general solution is:

    \[ f(t) = \int_{-\infty}^{t} e^{-\alpha (t - t')} s(t')\,dt'. \]

This is essentially the *temporal convolution* of the source with an exponentially decaying kernel — widely used in electronic RC circuits, population decay, or thermal relaxation models.

4. Example 2 – The Free Particle Propagator (Quantum Mechanics)

In quantum mechanics, the Green’s function (often called the propagator) is defined as the solution to:

    \[ \left(i\hbar \frac{\partial}{\partial t} - \hat{H}\right) G(x, t; x', t') = i\hbar\,\delta(x - x')\delta(t - t'). \]

For a free particle with Hamiltonian \hat{H} = \frac{\hat{p}^2}{2m}:

    \[ G(x, t; x', t') = \sqrt{\frac{m}{2\pi i \hbar (t - t')}} \exp\!\left[\frac{i m (x - x')^2}{2\hbar (t - t')}\right] \Theta(t - t'). \]

This temporal Green’s function tells you how a wave packet at position x' and time t' evolves to position x at a later time t.

5. Example 3 – The Wave Equation (Retarded Green’s Function)

For the 1D wave equation:

    \[ \frac{\partial^2 \psi}{\partial t^2} - c^2 \frac{\partial^2 \psi}{\partial x^2} = s(x,t), \]

the temporal Green’s function satisfies:

    \[ \left(\frac{\partial^2}{\partial t^2} - c^2 \frac{\partial^2}{\partial x^2}\right)G(x,t; x',t') = \delta(x-x')\delta(t-t'). \]

The **retarded Green’s function** is:

    \[ G_R(x,t;x',t') = \frac{1}{2c}\,\Theta\!\big(t-t'-|x-x'|/c\big). \]

It represents a pulse traveling at finite speed c: a disturbance at (x',t') affects point x only after sufficient time for the wave to propagate.

6. When the Temporal Green’s Function is Useful

  • Quantum Mechanics: Time-evolution kernels (propagators) for Schrödinger equations.
  • Electrical Circuits: RC and RLC circuit response to impulses.
  • Heat Conduction: Temporal evolution of temperature fields under time-varying sources.
  • Acoustics and Electromagnetism: Retarded potentials and causal field propagation.
  • Control Theory: Impulse response functions in linear dynamical systems.

Summary

The temporal Green’s function G(t, t') is the system’s impulse response in time.
It converts differential equations into integral equations through convolution:

    \[ f(t) = \int G(t, t')\,s(t')\,dt'. \]

Its key property — causality — ensures that G(t,t')=0 for t<t'.
Whether in quantum mechanics, heat diffusion, or classical wave propagation, temporal Green’s functions provide a unifying tool to compute how systems evolve in time under arbitrary driving forces.


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]]> Momentum Space Representation of the 1\r Operator https://stationarystates.com/mathematical-physics/momentum-space-representation-of-the-1r-operator/?utm_source=rss&utm_medium=rss&utm_campaign=momentum-space-representation-of-the-1r-operator Tue, 07 Oct 2025 19:40:25 +0000 https://stationarystates.com/?p=1014   Momentum Space Representation of the \( \frac{1}{r} \) Operator The operator \( \frac{1}{r}, \quad r = |\mathbf{x}| \) plays a central role in quantum mechanics, especially in Coulomb potentials, […]

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Momentum Space Representation of the \( \frac{1}{r} \) Operator

The operator
\( \frac{1}{r}, \quad r = |\mathbf{x}| \)
plays a central role in quantum mechanics, especially in Coulomb potentials, hydrogen-like atoms, and scattering theory.
In momentum space, multiplication by \( \frac{1}{r} \) in position space becomes a convolution with a well-defined kernel.

1. Position-Space Operator

For a wavefunction \( \psi(\mathbf{x}) \), the action of the operator \( \frac{1}{r} \) is

\[
\left( \frac{1}{r} \psi \right) (\mathbf{x}) = \frac{1}{|\mathbf{x}|} \psi(\mathbf{x}).
\]

2. Fourier Transform Conventions

We adopt the standard physics convention for the Fourier transform:

\[
\psi(\mathbf{x}) = \frac{1}{(2\pi)^{3/2}} \int \tilde{\psi}(\mathbf{k})\, e^{i \mathbf{k}\cdot \mathbf{x}} \, d^3k,
\]
\[
\tilde{\psi}(\mathbf{k}) = \frac{1}{(2\pi)^{3/2}} \int \psi(\mathbf{x})\, e^{-i \mathbf{k}\cdot \mathbf{x}} \, d^3x.
\]

The momentum-space operator \( \widetilde{V}(\mathbf{k},\mathbf{k}’) \) associated with
\( V(\mathbf{x}) = \frac{1}{r} \) is defined through

\[
\widetilde{(V \psi)}(\mathbf{k})
= \frac{1}{(2\pi)^{3/2}} \int \frac{1}{r}\, \psi(\mathbf{x})\, e^{-i \mathbf{k}\cdot \mathbf{x}}\, d^3x.
\]

Substituting the inverse Fourier transform of \(\psi(\mathbf{x})\):

\[
\widetilde{(V \psi)}(\mathbf{k})
= \frac{1}{(2\pi)^3} \int d^3x \, \frac{e^{-i \mathbf{k}\cdot \mathbf{x}}}{r}
\int d^3k’ \, \tilde{\psi}(\mathbf{k}’)\, e^{i \mathbf{k}’\cdot \mathbf{x}}.
\]

Reordering integrals gives:

\[
\widetilde{(V \psi)}(\mathbf{k})
= \frac{1}{(2\pi)^3} \int d^3k’ \, \tilde{\psi}(\mathbf{k}’)
\left[ \int d^3x \, \frac{e^{i(\mathbf{k}’ – \mathbf{k})\cdot \mathbf{x}}}{r} \right].
\]

3. Fourier Transform of \( 1/r \)

The inner integral is the Fourier transform of \( 1/r \):

\[
\int_{\mathbb{R}^3} \frac{e^{i \mathbf{q}\cdot \mathbf{x}}}{|\mathbf{x}|} \, d^3x,
\quad \mathbf{q} = \mathbf{k}’ – \mathbf{k}.
\]

This integral evaluates to

\[
\int \frac{e^{i \mathbf{q}\cdot \mathbf{x}}}{|\mathbf{x}|} \, d^3x
= \frac{4\pi}{|\mathbf{q}|^2}.
\]

This result is obtained by switching to spherical coordinates and integrating over the angular and radial parts.

4. Final Momentum-Space Representation

Substituting the Fourier transform result back gives:

\[
\widetilde{(V \psi)}(\mathbf{k})
= \frac{1}{(2\pi)^3} \int d^3k’ \, \tilde{\psi}(\mathbf{k}’)
\left[ \frac{4\pi}{|\mathbf{k} – \mathbf{k}’|^2} \right].
\]

Thus, the momentum-space kernel of the \( \frac{1}{r} \) operator is

\[
\langle \mathbf{k} | \frac{1}{r} | \mathbf{k}’ \rangle
= \frac{4\pi}{|\mathbf{k} – \mathbf{k}’|^2} \frac{1}{(2\pi)^3}.
\]

Equivalently,

\[
\left[\frac{1}{r}\tilde{\psi}\right](\mathbf{k})
= \int \frac{d^3k’}{(2\pi)^3} \frac{4\pi}{|\mathbf{k} – \mathbf{k}’|^2}\, \tilde{\psi}(\mathbf{k}’).
\]
In position space, \( \frac{1}{r} \) acts multiplicatively; in momentum space, it acts via convolution with a Coulomb-like kernel \( \frac{4\pi}{|\mathbf{k}-\mathbf{k}’|^2} \).

5. Special Case: Coulomb Potential

For a Coulomb potential
\( V(\mathbf{x}) = -\frac{e^2}{4\pi\epsilon_0}\frac{1}{r} \),
the momentum-space kernel is

\[
\widetilde{V}(\mathbf{k}, \mathbf{k}’)
= – \frac{e^2}{4\pi\epsilon_0} \frac{4\pi}{|\mathbf{k} – \mathbf{k}’|^2} \frac{1}{(2\pi)^3}
= – \frac{e^2}{\epsilon_0} \frac{1}{|\mathbf{k} – \mathbf{k}’|^2} \frac{1}{(2\pi)^3}.
\]

This form is central in solving the Schrödinger equation in momentum space for hydrogen and in the Born approximation in scattering theory.

6. Summary Table

Quantity Position Space Momentum Space
Operator \( V(\mathbf{x}) = \frac{1}{r} \) Integral operator with kernel \( \frac{4\pi}{|\mathbf{k} – \mathbf{k}’|^2(2\pi)^3} \)
Action on \( \psi \) Multiply by \( \frac{1}{r} \) Convolution with \( \frac{4\pi}{|\mathbf{k} – \mathbf{k}’|^2} \)
Key Fourier Transform — \( \mathcal{F}\{ 1/r \} = \frac{4\pi}{q^2} \)
The \( \frac{1}{r} \) operator is nonlocal in momentum space, mixing all momenta through a \( 1/|\mathbf{k}-\mathbf{k}’|^2 \) kernel.

 

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]]> Worked examples Kac Algebras, Lie Groups, Lie Algebras https://stationarystates.com/mathematical-physics/worked-examples-kac-algebras-lie-groups-lie-algebras/?utm_source=rss&utm_medium=rss&utm_campaign=worked-examples-kac-algebras-lie-groups-lie-algebras Thu, 12 Jun 2025 16:26:25 +0000 https://stationarystates.com/?p=941 1️⃣ Lie Group → Lie Algebra: Example with Step 1: Define the Lie group = Special Orthogonal Group in 3D Set of all real rotation matrices with: (orthogonality) (special) Step […]

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1⃣ Lie Group → Lie Algebra: Example with \mathrm{SO}(3)

Step 1: Define the Lie group

\mathrm{SO}(3) = Special Orthogonal Group in 3D
Set of all 3 \times 3 real rotation matrices with:

  • R^T R = I (orthogonality)
  • \det R = 1 (special)

Step 2: Parametrize near identity

Take an element R(\theta) \approx I + \theta A + O(\theta^2), with small \theta.
A is in the tangent space — this defines the Lie algebra \mathfrak{so}(3).

Step 3: Compute Lie algebra

Plug R(\theta) into R^T R = I:

    \[ (I + \theta A^T)(I + \theta A) = I \Rightarrow A^T + A = 0 \]

Conclusion: A is antisymmetric 3 \times 3 matrix:

    \[ A = \begin{pmatrix} 0 & -a_3 & a_2 \\ a_3 & 0 & -a_1 \\ -a_2 & a_1 & 0 \end{pmatrix} \]

The algebra \mathfrak{so}(3) is 3-dimensional, basis given by:

    \[ L_1 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{pmatrix}, \quad L_2, \quad L_3 \]

Step 4: Lie bracket

The Lie bracket is matrix commutator:

    \[ [L_i, L_j] = \epsilon_{ijk} L_k \]

where \epsilon_{ijk} is the Levi-Civita symbol.

2⃣ Affine Kac–Moody Algebra as a Generalization

Step 1: Start with finite Lie algebra \mathfrak{g}, e.g. \mathfrak{su}(2)

    \[ [J_i, J_j] = i \epsilon_{ijk} J_k \]

Step 2: Build loop algebra

Take functions of a circle S^1, parameterized by e^{int} (Fourier modes).
Now consider operators J_i(n), \; n \in \mathbb{Z}.

These satisfy:

    \[ [J_i(n), J_j(m)] = i \epsilon_{ijk} J_k(n + m) \]

Step 3: Add central extension

Introduce central charge k:

    \[ [J_i(n), J_j(m)] = i \epsilon_{ijk} J_k(n + m) + n \delta_{ij} \delta_{n + m, 0} k \]

Now the algebra is called an Affine Kac–Moody algebra \widehat{\mathfrak{su}(2)}.

Physical meaning:

  • Modes J_i(n) generate local symmetry transformations along the loop (string or 1D system).
  • Central term k is related to quantum anomaly or level.

3⃣ Example in Quantum Physics: WZW Model

Setup:

Fields g(z, \bar{z}) \in SU(N)
Action has Wess–Zumino term → topological term.

Resulting symmetry:

The model has left and right Affine Kac–Moody algebra symmetries:

    \[ \widehat{\mathfrak{su}(N)}_L \times \widehat{\mathfrak{su}(N)}_R \]

Operators J^a(z) (left currents) satisfy:

    \[ [J^a_n, J^b_m] = i f^{abc} J^c_{n + m} + n \delta^{ab} \delta_{n + m, 0} k \]

Similarly for right-moving sector.

Application:

  • String theory: Worldsheet theory of strings on group manifolds → governed by WZW models → governed by Kac–Moody algebra.
  • Conformal field theory: Classification of CFTs via Kac–Moody symmetries.
  • Statistical physics: 2D critical systems (spin chains, quantum Hall effect).

🧾 Conclusion

We’ve worked through:

  • Lie group \mathrm{SO}(3) → infinitesimal generators → Lie algebra \mathfrak{so}(3).
  • Loop algebra of \mathfrak{su}(2) → central extension → Affine Kac–Moody algebra \widehat{\mathfrak{su}(2)}.
  • WZW model → real-world quantum physics example of Kac–Moody symmetry.

Kac–Moody algebras provide the algebraic backbone for infinite-dimensional symmetry groups in modern quantum physics — essential in string theory, CFT, and 2D quantum systems.

 

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]]> Lie Group Algebras https://stationarystates.com/mathematical-physics/lie-group-algebras/?utm_source=rss&utm_medium=rss&utm_campaign=lie-group-algebras Thu, 12 Jun 2025 16:19:21 +0000 https://stationarystates.com/?p=939   Overview Lie groups and Lie algebras are two deeply connected mathematical structures used to study symmetries in mathematics and physics. They are foundational in areas such as particle physics, […]

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Overview

Lie groups and Lie algebras are two deeply connected mathematical structures used to study symmetries in mathematics and physics. They are foundational in areas such as particle physics, quantum mechanics, and general relativity.

A Lie group is a group that is also a smooth manifold — meaning group operations (multiplication, inversion) are smooth (differentiable) functions.
A Lie algebra is a linearized version of the Lie group — it captures the behavior “near the identity” of the group using algebraic tools.

Analogy:
Lie group = full global symmetry (like rotating an entire object)
Lie algebra = infinitesimal symmetry (like rotating by a tiny angle and seeing how the object responds locally)

Mechanism

Lie Group:

  • Set with smooth structure and group operations.
  • Example: rotations of a sphere → SO(3), the group of 3D rotations.

Lie Algebra:

  • Vector space with an additional operation called the Lie bracket [X, Y] (often a commutator in physics).
  • The Lie algebra describes infinitesimal generators of the Lie group.
  • Example: For SO(3), the Lie algebra so(3) corresponds to infinitesimal rotations — it can be represented by 3×3 antisymmetric matrices.

Relationship:

  • A Lie algebra is derived from the Lie group via tangent space at the identity.
  • The Lie group can often be recovered from the Lie algebra (up to some global issues).

🤩 Examples & Applications

Lie Group Lie Algebra Application
SO(3) — rotations in 3D so(3) — infinitesimal rotations Rigid body dynamics, classical mechanics
SU(2) — spin rotations su(2) Spin in quantum mechanics, qubits
SU(3) — color symmetry su(3) Quantum chromodynamics (QCD), describing quark interactions
Poincaré group Poincaré algebra Special relativity, spacetime symmetries in QFT

Applications:

  • Quantum mechanics: Lie algebras describe angular momentum (su(2)).
  • Quantum field theory: Gauge groups like SU(3), SU(2), U(1) underlie the Standard Model.
  • General relativity: Symmetries of spacetime are described by the Poincaré group.

Where Do Kac–Moody Algebras Fit In?

Kac–Moody algebras are generalizations of Lie algebras:

  • Finite-dimensional Lie algebras → used for compact Lie groups (e.g. SU(2), SU(3)).
  • Kac–Moody algebras → can be infinite-dimensional, defined via generators and relations.
  • Important subclass: Affine Kac–Moody algebras → loop algebras with central extensions.

Why Kac–Moody Algebras Matter:

  • In quantum physics, especially in conformal field theory (CFT) and string theory, symmetries are not finite-dimensional.
  • Kac–Moody algebras describe the infinite symmetries of 1D systems, conformal symmetries, and strings.
Field Example Role of Kac–Moody Algebra
2D conformal field theory WZW model (Wess–Zumino–Witten model) Symmetry algebra is an affine Kac–Moody algebra
String theory Mode expansions of strings Mode algebra forms Kac–Moody algebra
Statistical physics Critical phenomena Describes symmetries at phase transitions

Interpretations & Implications

  • Lie groups and Lie algebras give us a toolkit to understand both global and local symmetries.
  • Kac–Moody algebras expand this to infinite-dimensional cases — essential when studying systems with infinitely many degrees of freedom (like strings, CFTs).
  • Modern theoretical physics (AdS/CFT, integrable systems, even condensed matter) frequently uses Kac–Moody symmetry to constrain or solve models.

Conclusion

Lie groups describe global continuous symmetries. Their associated Lie algebras describe the infinitesimal generators of these symmetries. Kac–Moody algebras generalize Lie algebras to infinite dimensions, crucial in quantum field theory and string theory where infinite-dimensional symmetries naturally arise.

 

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]]> Abelain Group https://stationarystates.com/mathematical-physics/abelain-group/?utm_source=rss&utm_medium=rss&utm_campaign=abelain-group Fri, 28 Feb 2025 14:33:23 +0000 https://stationarystates.com/?p=783 Z(p∞) = { z ∈ ℂ | zpk = 1 for some integer k ≥ 1 } Proof that Z(p∞) is an Abelian Group We define the set: Z(p∞) = […]

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]]> Z(p∞) = { z ∈ ℂ | zpk = 1 for some integer k ≥ 1 }

Proof that Z(p∞) is an Abelian Group

We define the set:

Z(p∞) = { z ∈ ℂ | zpk = 1 for some integer k ≥ 1 }

We will verify the group axioms under multiplication.

1. Closure

If z1, z2 ∈ Z(p∞), then there exist integers k1, k2 such that:

z1pk1 = 1 and z2pk2 = 1.

Let k = max(k1, k2), then pk is a multiple of both pk1 and pk2. Thus,

(z1 z2)pk = z1pk z2pk = 1,

so z1 z2 ∈ Z(p∞).

2. Associativity

Multiplication in ℂ is associative, so for any z1, z2, z3 ∈ Z(p∞),

(z1 z2) z3 = z1 (z2 z3).

3. Identity Element

The number 1 is a root of unity since 1pk = 1 for all k. Thus, 1 ∈ Z(p∞).

4. Inverses

For any z ∈ Z(p∞), there exists some k such that zpk = 1.

The inverse of z is z-1, which satisfies:

(z-1)pk = (zpk)-1 = 1.

Thus, z-1 ∈ Z(p∞).

5. Commutativity

Since multiplication in ℂ is commutative,

z1 z2 = z2 z1 for all z1, z2 ∈ Z(p∞).

Conclusion

Since Z(p∞) satisfies closure, associativity, identity, inverses, and commutativity, it forms an abelian group under multiplication.

 

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]]> Examples of Taylor SEries versus Fourier Series https://stationarystates.com/mathematical-physics/examples-of-taylor-series-versus-fourier-series/?utm_source=rss&utm_medium=rss&utm_campaign=examples-of-taylor-series-versus-fourier-series Thu, 20 Feb 2025 19:12:05 +0000 https://stationarystates.com/?p=771 Intro Which works better for a given function – a Taylor expansion or a Fourier Expansion?  This post explores the pros and cons of each, using specific examples. Examples of […]

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]]> Intro

Which works better for a given function – a Taylor expansion or a Fourier Expansion?  This post explores the pros and cons of each, using specific examples.

Examples of Taylor and Fourier Series Expansions

1. Polynomial Function: f(x) = x^2

Taylor Series Expansion: x^2 = x^2

Fourier Series Expansion: x^2 = \frac{\pi^2}{3} + \sum_{n=1}^{\infty} \frac{4(-1)^n}{n^2} \cos(nx)

Best Fit: Taylor series

2. Trigonometric Function: f(x) = \sin(x)

Taylor Series Expansion: \sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots

Fourier Series Expansion: \sin(x) = \sum_{n=1, \text{odd}}^{\infty} \frac{4}{n\pi} \sin(nx)

Best Fit: Fourier series

3. Exponential Function: f(x) = e^x

Taylor Series Expansion: e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots

Fourier Series Expansion: Not practical

Best Fit: Taylor series

4. Piecewise Function: f(x) = |x| on [-\pi, \pi]

Taylor Series Expansion: Not possible

Fourier Series Expansion: |x| = \frac{\pi}{2} - \sum_{n=1, \text{odd}}^{\infty} \frac{4}{n^2\pi} \cos(nx)

Best Fit: Fourier series

5. Periodic Step Function: f(x) = \text{sgn}(\sin x)

Taylor Series Expansion: Not possible

Fourier Series Expansion: f(x) = \frac{4}{\pi} \sum_{n=1, \text{odd}}^{\infty} \frac{1}{n} \sin(nx)

Best Fit: Fourier series

Comparison Table

Function Taylor Series Fourier Series Best Fit
x^2 Good (converges well) Works if periodic but inefficient Taylor series
\sin(x) Good for small x Best for periodic representation Fourier series
e^x Excellent (globally convergent) Poor (unless forced periodicity) Taylor series
|x| Not possible Works well (some Gibbs effect) Fourier series
\text{sgn}(\sin x) Not possible Best option (Gibbs phenomenon) Fourier series

 

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]]> Taylor Series versus Fourier Series for a function https://stationarystates.com/mathematical-physics/taylor-series-versus-fourier-series-for-a-function/?utm_source=rss&utm_medium=rss&utm_campaign=taylor-series-versus-fourier-series-for-a-function Thu, 20 Feb 2025 16:40:27 +0000 https://stationarystates.com/?p=768 . Domain of Representation Taylor Series: Works best for local approximations around a single point (Maclaurin series if centered at zero). Fourier Series: Represents a function over an entire interval […]

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]]> . Domain of Representation
  • Taylor Series: Works best for local approximations around a single point (Maclaurin series if centered at zero).
  • Fourier Series: Represents a function over an entire interval (typically [−L,L][-L, L] or [0,2π][0, 2\pi]).

Key Differences Between Taylor Series and Fourier Series

Aspect Taylor Series Fourier Series
Nature of Expansion Uses polynomials from derivatives at a single point. Uses sines and cosines (or complex exponentials) over an interval.
Convergence Conditions Requires infinite differentiability and a valid radius of convergence. Requires periodicity and Dirichlet conditions for convergence.
Domain of Representation Local approximation around a single point. Represents a function over an entire interval.
Basis Functions Powers of (x – a). Sinusoids (sines and cosines) or complex exponentials.
Handling of Discontinuities Poor handling; requires smoothness. Can approximate discontinuous functions (with Gibbs phenomenon).
Applications Local function approximation, differential equations, numerical analysis. Signal processing, wave analysis, heat conduction, quantum mechanics.

 

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]]> Functions ONLY definable by their integrals – with applications https://stationarystates.com/mathematical-physics/functions-only-definable-by-their-integrals-with-applications/?utm_source=rss&utm_medium=rss&utm_campaign=functions-only-definable-by-their-integrals-with-applications Sat, 01 Feb 2025 11:54:25 +0000 https://stationarystates.com/?p=748 Functions ONLY Defined by Their Integrals 1. The Gamma Function , for . Applications: Generalization of factorials: . Used in probability distributions and statistical mechanics. Found in Feynman integrals in […]

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]]> Functions ONLY Defined by Their Integrals

1. The Gamma Function \Gamma(x)

\Gamma(x) = \int_0^\infty t^{x-1} e^{-t} \, dt, for x > 0.

Applications:

  • Generalization of factorials: \Gamma(n) = (n-1)!.
  • Used in probability distributions and statistical mechanics.
  • Found in Feynman integrals in quantum physics.

2. The Beta Function B(x, y)

B(x, y) = \int_0^1 t^{x-1} (1 - t)^{y-1} dt.

Applications:

  • Related to the Gamma function via B(x, y) = \frac{\Gamma(x) \Gamma(y)}{\Gamma(x + y)}.
  • Used in Bayesian statistics and machine learning.

3. The Error Function \operatorname{erf}(x)

\operatorname{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} dt.

Applications:

  • Used in Gaussian probability distributions.
  • Appears in heat and diffusion equations.

4. The Fresnel Integrals S(x) and C(x)

S(x) = \int_0^x \sin(t^2) dt, C(x) = \int_0^x \cos(t^2) dt.

Applications:

  • Wave optics and diffraction patterns.
  • Radar signal processing.

5. The Dirichlet Integral

\int_0^\infty \frac{\sin t}{t} dt = \frac{\pi}{2}.

Applications:

  • Fourier analysis and signal processing.

6. The Bessel Functions J_n(x)

J_n(x) = \frac{1}{\pi} \int_0^\pi \cos(n t - x \sin t) dt.

Applications:

  • Solutions to differential equations in cylindrical coordinates.
  • Used in electromagnetics and fluid dynamics.

7. The Airy Function \operatorname{Ai}(x)

\operatorname{Ai}(x) = \frac{1}{\pi} \int_0^\infty \cos \left( \frac{t^3}{3} + xt \right) dt.

Applications:

  • Quantum mechanics and tunneling problems.
  • Optics and wavefront analysis.

8. The Riemann Zeta Function \zeta(s)

\zeta(s) = \frac{1}{\Gamma(s)} \int_0^\infty \frac{x^{s-1}}{e^x - 1} dx, for s > 1.

Applications:

  • Number theory and prime distribution.
  • Quantum field theory.

9. The Lambert W Function W(x)

Defined by W(x)e^{W(x)} = x, with integral representation:

W(x) = \int_0^\infty \frac{dt}{(t+1)e^{t+x}}.

Applications:

  • Used in combinatorics and graph theory.
  • Appears in quantum mechanics and delay differential equations.

10. The Polylogarithm Function \operatorname{Li}_s(x)

\operatorname{Li}_s(x) = \int_0^\infty \frac{x^t}{t^s} dt.

Applications:

  • Found in quantum field theory and statistical mechanics.
  • Used in cryptography and information theory.

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]]> Functions Defined by Their Integrals – with applications https://stationarystates.com/mathematical-physics/functions-defined-by-their-integrals-with-applications/?utm_source=rss&utm_medium=rss&utm_campaign=functions-defined-by-their-integrals-with-applications Sat, 01 Feb 2025 03:59:28 +0000 https://stationarystates.com/?p=746 Functions Defined by Their Integrals Functions that are defined by their integrals often arise in fields like physics, probability theory, and engineering, where the direct formulation of a function may […]

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]]> Functions Defined by Their Integrals

Functions that are defined by their integrals often arise in fields like physics, probability theory, and engineering, where the direct formulation of a function may be complex or not easily expressible in closed form. Below are some examples of such functions, along with real-world applications:

1. The CDF (Cumulative Distribution Function) of a Probability Distribution

Definition: The CDF of a random variable X is defined as the integral of the probability density function (PDF) fX(x) from -∞ to x:

        FX(x) = ∫-∞x fX(t) dt

Application: In statistics and probability theory, CDFs are used to model the probability that a random variable takes a value less than or equal to a given value. For example, the CDF is used in risk analysis and decision-making under uncertainty (e.g., calculating the likelihood of an event occurring within a certain range).

2. The Green’s Function in Differential Equations

Definition: Green’s function G(x, s) is a solution to a boundary value problem that is defined as the integral of the forcing term f(x) over the domain. For a linear differential operator L and boundary conditions, the solution to the equation L u(x) = f(x) can be written as:

        u(x) = ∫ G(x, s) f(s) ds

Application: In electromagnetism and heat conduction, Green’s functions are used to solve problems related to how fields (electric, magnetic, or temperature) propagate in various media. For example, in electromagnetic field theory, Green’s functions describe how a current distribution generates a magnetic field.

3. The Fourier Transform

Definition: The Fourier transform f̂(k) of a function f(x) is defined as:

        f̂(k) = ∫-∞∞ f(x) e-ikx dx

Application: Fourier transforms are extensively used in signal processing to analyze frequencies in time-domain signals. For instance, in audio processing, the Fourier transform is used to decompose sound signals into their constituent frequencies, enabling tasks like filtering and compression.

4. The Potential Function in Physics

Definition: The potential function V(x) in physics can be defined as the integral of the force F(x), where the force is the negative gradient of the potential:

        V(x) = - ∫ F(x) dx

Application: In classical mechanics, the potential function is used to describe the potential energy in systems like gravitational fields or electric fields. For example, in planetary motion, the gravitational potential function defines the energy that governs the movement of planets in space.

5. The Convolution Integral in Signal Processing

Definition: The convolution of two functions f(x) and g(x) is defined as:

        (f * g)(x) = ∫-∞∞ f(t) g(x - t) dt

Application: In image processing, convolution is used to apply filters, such as edge detection or blurring, to images. In audio processing, convolution is used to simulate the response of a system to an input signal, such as reverberation effects in music.

6. The Laplace Transform

Definition: The Laplace transform of a function f(t) is given by:

        ℒ{f(t)} = F(s) = ∫0∞ e-st f(t) dt

Application: In control theory and systems engineering, the Laplace transform is used to analyze the behavior of dynamic systems, such as electrical circuits or mechanical systems. It helps in solving differential equations that describe these systems and analyzing their stability and response.

7. The Radon Transform

Definition: The Radon transform is an integral transform that takes a function defined on a plane and integrates it along straight lines. It is defined as:

        Rf(θ, t) = ∫ℓt,θ f(x) ds

Application: The Radon transform is the mathematical foundation for computed tomography (CT) scans. In medical imaging, it is used to reconstruct images of the interior of a body from X-ray projections taken at different angles.

8. The Heaviside Step Function (Unit Step Function)

Definition: The Heaviside function H(x) is typically defined as:

        H(x) = ∫-∞x δ(t) dt

Application: The Heaviside function is used in control systems and signal processing to model switches or events that occur at specific times. For example, it can represent the turning on or off of a switch in electrical circuits or the onset of a signal.

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]]> Finite Abelian Groups and Applications to Quantum Physics https://stationarystates.com/mathematical-physics/finite-abelian-groups-and-applications-to-quantum-physics/?utm_source=rss&utm_medium=rss&utm_campaign=finite-abelian-groups-and-applications-to-quantum-physics Sat, 11 Jan 2025 03:17:43 +0000 https://stationarystates.com/?p=734 Finite Abelian Groups and Applications to Quantum Physics What Are Finite Abelian Groups? A finite abelian group is a group with the following properties: Closure: For any , . Associativity: […]

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]]> Finite Abelian Groups and Applications to Quantum Physics

What Are Finite Abelian Groups?

A finite abelian group is a group G with the following properties:

  • Closure: For any a, b \in G, a \cdot b \in G.
  • Associativity: For all a, b, c \in G, (a \cdot b) \cdot c = a \cdot (b \cdot c).
  • Identity: There exists an identity element e \in G such that a \cdot e = e \cdot a = a for all a \in G.
  • Inverses: For every a \in G, there exists an a^{-1} \in G such that a \cdot a^{-1} = e.
  • Commutativity: For all a, b \in G, a \cdot b = b \cdot a.

If the group has a finite number of elements, it is called finite.

Structure of Finite Abelian Groups

The Fundamental Theorem of Finite Abelian Groups states that every finite abelian group G can be expressed as a direct product of cyclic groups of prime power order:

    \[ G \cong \mathbb{Z}_{n_1} \times \mathbb{Z}_{n_2} \times \cdots \times \mathbb{Z}_{n_k}, \]

where n_1, n_2, \dots, n_k are integers greater than 1.

Examples of Finite Abelian Groups

  • Cyclic Groups:
    • \mathbb{Z}_n, the integers modulo n under addition.
    • Example: \mathbb{Z}_6 = \{ 0, 1, 2, 3, 4, 5 \} with addition modulo 6.
  • Direct Product of Cyclic Groups:
    • \mathbb{Z}_2 \times \mathbb{Z}_2, the Klein four-group:

          \[ \{(0, 0), (1, 0), (0, 1), (1, 1)\}. \]

    • \mathbb{Z}_2 \times \mathbb{Z}_4:

          \[ \{(0, 0), (1, 0), (0, 1), (1, 1), (0, 2), (1, 2), (0, 3), (1, 3)\}. \]

  • Additive Group of Finite Fields: The set of elements of a finite field \mathbb{F}_q under addition forms a finite abelian group.
  • Root of Unity Groups: The n-th roots of unity \{e^{2\pi i k / n} \mid k = 0, 1, \dots, n-1\} under multiplication.

Applications to Quantum Physics

1. Quantum Mechanics and Symmetry

  • Discrete Symmetries: Finite abelian groups often describe symmetries of quantum systems, such as the Klein four-group \mathbb{Z}_2 \times \mathbb{Z}_2, which can describe symmetries in molecular structures or lattice vibrations.
  • Conservation Laws: The symmetries of a system are associated with conserved quantities, often modeled using finite abelian groups.

2. Quantum Computing

  • Quantum Gates: The structure of finite abelian groups is crucial in algorithms like Shor’s algorithm, where periodicity plays a significant role.
  • Quantum Error Correction: Stabilizer codes, used in error correction, leverage abelian group structures to define subspaces.

3. Topological Phases of Matter

  • Abelian Anyons: Quasiparticles in topological systems exhibit abelian statistics, modeled by finite abelian groups.
  • Fractional Quantum Hall Effect: Finite abelian groups describe the ground state degeneracies and quasiparticle statistics of these systems.

4. Crystallography and Solid-State Physics

  • Lattice Symmetries: Finite abelian groups classify vibrational modes (phonons) and electronic band structures.
  • Bloch’s Theorem: Translational symmetry, often modeled as \mathbb{Z}_n, leads to quantized energy levels in the form of Bloch waves.

Conclusion

Finite abelian groups provide the mathematical foundation for understanding symmetry, periodicity, and conserved quantities in quantum systems. They play a crucial role in quantum computing, error correction, and the study of topological phases of matter, highlighting the deep connections between algebra and the physical world.

 

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