Fact-checked by Grok 2 weeks ago

Continuous symmetry

Continuous symmetry refers to a property of mathematical objects or physical systems that remain invariant under a continuous family of transformations, such as rotations, translations, or scalings by arbitrary real parameters, rather than discrete steps.^[1] In mathematics, these symmetries are precisely captured by Lie groups, which are smooth manifolds equipped with group operations that model infinitesimal changes and their compositions, providing a framework for analyzing geometric and algebraic structures like rotations in Euclidean space or Lorentz transformations in special relativity.^[2] In physics, continuous symmetries play a foundational role through Noether's theorem, which establishes a one-to-one correspondence between such symmetries of the laws of nature and conserved quantities; for instance, time-translation invariance implies energy conservation, while spatial-translation invariance yields momentum conservation.^[3] This connection, first proven by Emmy Noether in 1918, underpins much of modern theoretical physics, from classical mechanics to quantum field theory, where symmetries dictate the form of Lagrangians and Hamiltonians.^[4] Examples include the rotational symmetry of isotropic media, leading to angular momentum conservation, and the gauge symmetries in electromagnetism, which enforce charge conservation.^[1] The study of continuous symmetries extends beyond invariance to include symmetry breaking, where spontaneous or explicit mechanisms disrupt perfect symmetry, resulting in phenomena like the Higgs mechanism in particle physics or phase transitions in condensed matter.^[5] Lie algebras, derived from the tangent spaces of Lie groups, offer tools to classify these symmetries via infinitesimal generators, enabling computations in diverse fields from robotics to quantum computing.^[6]

Definitions and Concepts

Definition

In mathematics and physics, symmetry refers to a property of a system that remains invariant under certain transformations, meaning the system's essential features are unchanged after applying the transformation.^[5] Continuous symmetry specifically arises when these transformations are parameterized by continuous variables, such as real numbers, allowing for an infinite variety of intermediate states between any two transformations. For instance, rotations around an axis exemplify this, where the angle of rotation can take any value in the real numbers \mathbb{R}, smoothly varying the orientation without discrete jumps.^[1] A classic example is the rotational symmetry of a circle, which remains indistinguishable under rotation by any angle \theta \in \mathbb{R} about its center, illustrating how continuous parameterization enables invariance across a continuum of transformations rather than isolated steps.^[1] This contrasts with discrete symmetries but highlights the seamless nature of continuous ones, often modeled using Lie groups for their smooth structure.^[2] Continuous symmetries are fundamentally tied to the concept of infinitesimal transformations, which are limiting cases of small changes in the transformation parameter, providing the building blocks for the entire group of symmetries. These infinitesimal generators capture the local behavior and facilitate the analysis of how finite transformations emerge from successive small variations.^[7] Mathematically, a symmetry group G acts continuously on a space X if the action map G \times X \to X, defined by (g, x) \mapsto g \cdot x, is a smooth function, ensuring the transformations vary differentiably across the group.^[8]

Discrete vs. Continuous Symmetries

Discrete symmetries refer to transformations that form finite or countable groups, such as reflections or rotations by discrete angles in the cyclic group \mathbb{Z}_n.^[9] These groups consist of a limited number of distinct elements, where the symmetry operations cannot be parameterized continuously but instead occur in isolated steps.^[10] In contrast, continuous symmetries involve transformations that form Lie groups, allowing for smooth, parameterized variations, such as arbitrary rotations or translations.^[9] A fundamental structural difference lies in their group theory: discrete symmetries yield finite-dimensional representations and rely on categorical invariants like character tables for analysis, whereas continuous symmetries permit infinite-dimensional representations and necessitate differential methods, such as Lie algebras, to study infinitesimal generators.^[9] This continuity enables the decomposition of transformations into arbitrarily small changes, facilitating tools like the exponential map from the Lie algebra to the group.^[10] An illustrative example in physics is parity, a discrete symmetry that inverts spatial coordinates (\mathbf{x} \to -\mathbf{x}), forming a finite group like \mathbb{Z}_2, compared to time translation, a continuous symmetry that shifts time by any real amount (t \to t + \epsilon), parameterized by a continuous variable.^[11] While both preserve the form of physical laws in invariant systems, the discrete nature of parity leads to binary outcomes like even or odd parity states, without implying ongoing conservation.^[11] The consequences of this distinction are profound in physics: continuous symmetries generally lead to conserved quantities through Noether's theorem, associating each independent symmetry parameter with a conserved current, whereas discrete symmetries do not typically produce such continuous conservation laws.^[9] For instance, time translation yields energy conservation, but parity invariance does not enforce a similar ongoing quantity.^[11] This disparity underscores why continuous symmetries underpin much of classical and quantum field theory, enabling predictions of long-term dynamical behavior.^[10]

Mathematical Foundations

Lie Groups and Lie Algebras

A Lie group is defined as a group that is also a smooth manifold, such that the group operations of multiplication and inversion are smooth maps compatible with the manifold structure.^[12] This compatibility ensures that the group structure respects the differential geometry of the manifold, allowing for the study of continuous transformations through calculus.^[13] Lie groups provide the foundational mathematical structure for modeling continuous symmetries, as their elements represent smooth, parameterized families of transformations.^[14] The Lie algebra associated with a Lie group captures the infinitesimal structure of the group and is identified with the tangent space at the identity element.^[15] This vector space encodes the generators of the group, which are the first-order approximations to group elements near the identity, obtained via curves passing through it.^[16] The Lie algebra structure arises from the Lie bracket operation on left-invariant vector fields, which measures the non-commutativity of these infinitesimal transformations. The Lie bracket [X, Y] for two vector fields X and Y on the manifold is defined by

[X, Y] f = X(Y f) - Y(X f)

for any smooth function f, making the space of left-invariant vector fields into a Lie algebra.^[17] A prominent example is the special orthogonal group SO(3), which consists of all 3×3 orthogonal matrices with determinant 1 and represents the continuous symmetries of rotations in three-dimensional Euclidean space. Its Lie algebra \mathfrak{so}(3) is the three-dimensional vector space of skew-symmetric 3×3 matrices, with the Lie bracket corresponding to the cross product; in physics, these basis elements generate the angular momentum operators L_x, L_y, L_z, satisfying [L_i, L_j] = i \hbar \epsilon_{ijk} L_k.^[18] Continuous symmetries in physical or geometric contexts correspond to actions of Lie groups on the underlying spaces, where the group elements induce diffeomorphisms that preserve the system's structure.^[19] The associated Lie algebra then describes the local, infinitesimal behavior of these symmetry transformations through its generators.^[20]

One-Parameter Subgroups

In the theory of Lie groups, a one-parameter subgroup of a Lie group G is defined as a smooth group homomorphism \phi: \mathbb{R} \to G, satisfying \phi(s + t) = \phi(s) \phi(t) for all s, t \in \mathbb{R}, with \phi(0) = e the identity element and the map being differentiable.^[21] This structure captures continuous families of transformations parameterized by a real number, forming a one-dimensional Lie subgroup isomorphic to (\mathbb{R}, +). One-parameter subgroups are intimately connected to the Lie algebra \mathfrak{g} of G through the exponential map \exp: \mathfrak{g} \to G. Specifically, for each X \in \mathfrak{g}, the curve \phi(t) = \exp(tX) defines a one-parameter subgroup, where the exponential map associates to each infinitesimal generator X a flow of group elements. This correspondence is bijective: every one-parameter subgroup arises uniquely from some X \in \mathfrak{g} via this exponentiation, with the tangent vector at the identity \frac{d}{dt} \phi(t) \big|_{t=0} = X.^[21] In matrix Lie groups, this manifests explicitly as \phi(t) = e^{tX}, where the matrix exponential provides the concrete realization.^[22] The evolution of a one-parameter subgroup is governed by the flow equation \frac{d}{dt} \phi(t) = X(\phi(t)), where X denotes the left-invariant vector field on G generated by the Lie algebra element X \in \mathfrak{g}.^[22] This differential equation describes how the infinitesimal action at the identity propagates along the curve, ensuring that \phi(t) remains a homomorphism. Solving this ODE yields the integral curve starting at the identity, uniquely determining the subgroup for each generator X.^[21] In the context of continuous symmetries, one-parameter subgroups illustrate how infinitesimal symmetries—represented by elements of the Lie algebra as vector fields on the manifold—generate finite, global transformations through integration of their flows. This mechanism bridges local tangent space actions to full group elements, enabling the realization of continuous symmetry operations as parameterized paths in G. A fundamental result states that in a compact connected Lie group, every element lies in some one-parameter subgroup. This theorem underscores the generative role of these subgroups, as the exponential map is surjective in this setting, ensuring the entire group is covered by flows from the identity.^[21]

Physical Applications

Noether's Theorem

Noether's theorem establishes a profound connection between continuous symmetries of physical systems and conservation laws, originating from Emmy Noether's 1918 paper "Invariante Variationsprobleme," which addressed questions raised by David Hilbert and Felix Klein regarding the conservation of energy and momentum in the context of general relativity's general covariance.^[23] Noether, working in Göttingen, developed the theorem amid discussions on the invariance properties of variational principles in Einstein's theory, proving that symmetries imply conserved quantities even when the underlying spacetime is curved.^[24] The theorem states that if the action integral S = \int L \, dt of a system, derived from a Lagrangian L(q, \dot{q}, t), is invariant under a continuous symmetry transformation generated by an infinitesimal variation \delta q, then there exists a conserved quantity associated with that symmetry.^[25] In field theory formulations, for a Lagrangian density invariant under a symmetry, a conserved Noether current j^\mu arises such that \partial_\mu j^\mu = 0, where the index \mu runs over spacetime coordinates.^[23] This applies to both spacetime symmetries (like translations or rotations) and internal symmetries (like phase transformations), provided the symmetry leaves the action unchanged up to a total divergence.^[24] To derive the theorem, consider the variation of the Lagrangian under the symmetry transformation. The total variation \delta L decomposes into an explicit change \frac{\partial L}{\partial q} \delta q + \frac{\partial L}{\partial \dot{q}} \delta \dot{q} plus a term from the coordinate transformation itself.^[25] For an on-shell variation (satisfying the equations of motion), the Euler-Lagrange equations \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}} \right) - \frac{\partial L}{\partial q} = 0 imply that the symmetry condition \delta L = \frac{d}{dt} F (for some F, often zero for strict invariance) leads to \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}} \delta q - F \right) = 0.^[23] Thus, for time-independent symmetries, the quantity \frac{\partial L}{\partial \dot{q}} \delta q is conserved along the system's trajectory. In field-theoretic settings, this generalizes to the current j^\mu = \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} \delta \phi - K^\mu, where K^\mu accounts for the divergence term, satisfying the continuity equation.^[25] The conserved charge is given by Q = \int j^0 \, d^3x, which remains constant under time evolution along the symmetry flow, reflecting the invariance.^[23] This charge generates the symmetry transformations via Poisson brackets in Hamiltonian mechanics.^[24] The theorem assumes the symmetry is a quasi-symmetry of the action, meaning the Lagrangian transforms as \delta L = \partial_\mu K^\mu (a total divergence), ensuring the action's variation vanishes upon integration by parts.^[23] It accommodates both finite-dimensional Lie groups (yielding finitely many conserved currents) and infinite-dimensional groups (yielding differential identities), handling spacetime and internal symmetries without requiring the Lagrangian to be explicitly time-independent.^[24]

Conservation Laws from Symmetries

In classical mechanics, Noether's theorem establishes a direct correspondence between continuous symmetries of the Lagrangian and conserved quantities, providing a systematic derivation of fundamental conservation laws.^[26] One of the most prominent examples is time-translation symmetry, which arises when the laws of physics are invariant under shifts in time, implying that the Lagrangian L has no explicit time dependence. In this case, the theorem yields the conservation of energy, where the Hamiltonian H = \sum_i p_i \dot{q}_i - L remains constant along the system's dynamical trajectory.^[4] This conserved quantity represents the total energy of the system and holds for any time-independent potential, underscoring the uniformity of temporal evolution in isolated systems.^[26] Spatial translation symmetry, reflecting the homogeneity of space, similarly leads to momentum conservation via Noether's theorem. When the Lagrangian is independent of the coordinates \mathbf{q}_i, the total linear momentum \mathbf{p} = \sum_i \frac{\partial L}{\partial \dot{\mathbf{q}}_i} is conserved.^[4] This result applies to systems in uniform gravitational or electromagnetic fields where forces derive from potentials that do not explicitly vary with position, ensuring that the center-of-mass motion proceeds at constant velocity. Rotational symmetry, characteristic of isotropic environments, further implies the conservation of angular momentum. For a Lagrangian invariant under rotations, the total angular momentum \mathbf{L} = \sum_i \mathbf{r}_i \times \mathbf{p}_i is preserved, as demonstrated by the theorem's application to infinitesimal angular transformations.^[26] These laws collectively govern the dynamics of particle systems, from planetary orbits to molecular vibrations, without relying on ad hoc assumptions.^[4] The framework extends naturally to relativistic field theories, where continuous symmetries under the Poincaré group—encompassing translations and Lorentz transformations—generate conservation laws through the stress-energy tensor T^{\mu\nu}. Noether's theorem dictates that this tensor is conserved, satisfying \partial_\mu T^{\mu\nu} = 0 on-shell, which encodes the local conservation of energy (\nu=0) and momentum (\nu=j) in spacetime.^[27] The canonical form of T^{\mu\nu} arises from the field's variation under infinitesimal coordinate shifts, with improvements like the Belinfante symmetrization ensuring gauge invariance in theories such as electromagnetism or scalar fields. This tensor plays a central role in general relativity, sourcing the gravitational field via Einstein's equations.^[27] In quantum mechanics, the implications of these symmetries manifest through unitary representations on the Hilbert space, where continuous transformations are implemented by operators U satisfying U^\dagger U = I.^[28] The infinitesimal generators of these symmetries are Hermitian observables that commute with the Hamiltonian, [G, H] = 0, ensuring conservation under time evolution via the Schrödinger equation i\hbar \frac{d}{dt} |\psi\rangle = H |\psi\rangle. For time-translation symmetry, the Hamiltonian itself serves as the generator, dictating the phase evolution e^{-iHt/\hbar} of energy eigenstates.^[28] Analogously, the momentum operator generates spatial translations and angular momentum generates rotations, with these observables' expectation values remaining constant, thereby bridging classical conservation laws to quantum probabilities.^[28]

Examples and Illustrations

Rotational and Translational Symmetries

Translational symmetry refers to the invariance of a physical system under arbitrary displacements in space, mathematically expressed as the transformation \mathbf{x} \to \mathbf{x} + \mathbf{a} for any constant vector \mathbf{a}.^[4] In free space, this symmetry is continuous, implying that the laws of physics are homogeneous and independent of position, which underpins the uniformity of physical properties across empty space.^[29] For example, the motion of a free particle follows the same equations regardless of its starting location, reflecting this spatial homogeneity.^[4] In contrast, crystalline solids exhibit only discrete translational symmetry, where invariance holds for specific lattice translations rather than arbitrary ones, representing a spontaneous breaking of the continuous symmetry present in the underlying atomic interactions.^[30] Rotational symmetry involves invariance under arbitrary rotations in three-dimensional space, governed by the special orthogonal group SO(3), which parameterizes all proper rotations.^[31] A prominent example is spherical symmetry in central force problems, such as gravitational attraction, where the force depends solely on the distance between bodies and points radially, leaving the system's dynamics unchanged under any rotation about the center.^[32] In the Kepler problem, describing the motion of planets orbiting the Sun under inverse-square gravity, this rotational invariance leads to the conservation of orbital angular momentum, ensuring that the plane of the orbit remains fixed while the area swept by the radius vector is constant over time.^[33] These conserved quantities arise from the underlying symmetries, as formalized in conservation laws.^[34] To visualize these symmetries, consider the infinitesimal generators represented as vector fields: for translations, the field is constant and uniform, resulting in parallel flow lines with zero curl, indicating no local rotation.^[35] In contrast, infinitesimal rotations produce a vector field proportional to \boldsymbol{\omega} \times \mathbf{r}, where \boldsymbol{\omega} is the rotation axis vector and \mathbf{r} the position; the flow lines form closed loops around the axis, with a nonzero curl that quantifies the local rotational tendency.^[36] In real materials, these continuous symmetries are often broken. Translational symmetry becomes discrete in crystals due to the periodic lattice structure, limiting invariance to specific shifts matching the unit cell dimensions.^[30] Similarly, rotational symmetry is broken in anisotropic materials, such as certain crystals or composites, where properties like electrical conductivity or elasticity vary with direction, lacking full SO(3) invariance—for instance, in iron-based superconductors where orbital anisotropy emerges above the transition temperature.^[37]

Gauge Symmetries

Gauge symmetries represent a class of continuous internal symmetries that are local, meaning the transformation parameters can vary independently at different points in spacetime.^[38] In quantum mechanics and quantum field theory, a prototypical example is the U(1) gauge symmetry under which a charged fermion field \psi transforms as \psi \to e^{i\alpha(x)} \psi, where \alpha(x) is a spacetime-dependent phase.^[39] This locality distinguishes gauge symmetries from global symmetries, as the transformation must leave the action invariant only when accompanied by appropriate adjustments to other fields.^[38] A concrete illustration arises in electromagnetism, where the gauge symmetry acts on the vector potential A_\mu via the transformation A_\mu \to A_\mu + \partial_\mu \Lambda, with \Lambda(x) an arbitrary scalar function.^[40] To maintain invariance under this local U(1) transformation in the presence of matter fields, the ordinary derivative \partial_\mu is replaced by the covariant derivative D_\mu = \partial_\mu - i e A_\mu, where e is the coupling constant.^[38] This substitution ensures that the kinetic term for the fermion, \bar{\psi} i \gamma^\mu D_\mu \psi, remains unchanged under the combined field transformations.^[39] The requirement of gauge invariance necessitates the introduction of gauge fields, such as the photon in electromagnetism, which mediate interactions and propagate the degrees of freedom associated with the symmetry.^[38] In non-Abelian gauge theories, this extends to more complex structures; for instance, quantum chromodynamics (QCD) is based on the SU(3) gauge symmetry, where gluons serve as the gauge bosons that bind quarks through the strong force, enabling the unification of fundamental interactions within the Standard Model.^[41] These gauge fields acquire self-interactions in the non-Abelian case, leading to phenomena like asymptotic freedom in QCD.^[41] Applying Noether's theorem to local gauge symmetries yields identities known as Ward identities, rather than conserved currents in the usual sense, due to the spacetime dependence of the transformations.^[42] These identities constrain scattering amplitudes and correlation functions, ensuring consistency with the underlying gauge invariance, as derived from Noether's second theorem for systems with redundant degrees of freedom.^[43] The concept of gauge symmetry originated with Hermann Weyl's 1918 attempt to unify gravity and electromagnetism through a local scaling invariance in a generalized Riemannian geometry, though this initial formulation faced challenges with observational predictions.^[44] It was later refined in the 1954 work of Chen Ning Yang and Robert Mills, who developed the general framework of non-Abelian gauge theories invariant under local isotopic spin rotations, laying the groundwork for modern particle physics.^[40]

References

[1]
DOE Explains...Symmetry in Physics - Department of Energy
In physics, symmetry refers to how particles behave when space, time, or quantum numbers are reversed. We're used to seeing simple types of symmetry in ...
[2]
[PDF] Lie groups and Lie algebras (Winter 2024)
Lie groups, by contrast, are about so-called continuous symmetries (more aptly called smooth sym- metries), such as translational or rotational symmetries of a ...
[3]
[PDF] Emmy Noether and Symmetry
Noether's (First) Theorem states that to each continuous symmetry group of the action functional there is a corresponding conservation law of the physical ...
[4]
[PDF] 3 Classical Symmetries and Conservation Laws
Noether's theorem: For every continuous global symmetry there exists a global conservation law. Before we prove this statement, let us discuss the connection ...
[5]
Symmetry and Symmetry Breaking
Jul 24, 2003 · The definition of symmetry as “invariance under a specified group of transformations” allowed the concept to be applied much more widely, not ...
[6]
[PDF] Introduction to Lie Groups and Lie Algebras Alexander Kirillov, Jr.
classes of Lie groups (namely compact Lie groups and semisimple Lie groups, to be defined later) ... Bump, Lie Groups, Graduate Texts in Mathematics, vol. 225, ...
[7]
https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Electro-Optics/Direct_Energy_(Mitofsky)/14%3A_Lie_Analysis/14.03%3A_Continuous_Symmetries_and_Infinitesimal_Generators
[8]
[PDF] Lie Group Actions
A manifold M endowed with a continuous G-action is called a (left or right). G-space. • If M is a smooth manifold and the action is smooth, M is called a smooth.
[9]
[PDF] Lecture Notes on Group Theory in Physics (A Work in Progress)
... Lie groups . . . . . . . . . . . . . . . . . . . . . . . 29. 1.5.3 ... continuous symmetries. Group theory – big subject! Our concern here lies in its ...
[10]
[PDF] and its applications in physics - Group Theory
Infinite dimensional symmetries. String theory. 85. 2015-01-24 iii ... So far we have considered discrete symmetry groups with at most countable number of.
[11]
Lie Group -- from Wolfram MathWorld
A Lie group is a smooth manifold obeying the group properties and that satisfies the additional condition that the group operations are differentiable.
[12]
[PDF] lie groups and lie algebras womp 2007
Definition. A Lie group is a group G that is also a smooth manifold, such that the multipli- cation G × G → G, (g ...
[13]
What is a Lie group?
A Lie group is a group of symmetries where the symmetries are continuous. A circle has a continuous group of symmetries.
[14]
[PDF] About Lie Groups
Oct 6, 2005 · The tangent space TeG at the identity is a real vector space. Using the three classes of maps inherent in the Lie group structure, we can equip ...
[15]
[PDF] Differential Geometry and Lie Groups A Computational Perspective
This book is written for a wide audience ranging from upper undergraduate to advanced graduate students in mathematics, physics, and more broadly ...
[16]
Introduction to Smooth Manifolds, Second Edition
This book is an introductory graduate-level textbook on the theory of smooth manifolds. Its goal is to familiarize students with the tools they will needMissing: definition | Show results with:definition
[17]
[PDF] Theory of Angular Momentum and Spin
Show that in this case holds ρ0(AB) = ρ0(B)ρ0(A). Generators. One can assume then that R(~ϑ) should also form a Lie group, in fact, one isomorphic to SO(3).
[18]
[PDF] Chapter 7 Continuous Groups, Lie Groups, and Lie Algebras
This Lie group is called the general linear group in two dimensions and is denoted by GL(2,R), where the 'R' signifies that the entries are real; the ...
[19]
[PDF] The SO(3) and SE(3) Lie Algebras of Rigid Body Rotations ... - arXiv
This document discusses how to modify these methods so they can be applied to non-Euclidean state vectors, such as those containing rotations and full motions ...
[20]
[PDF] One parameter subgroups
In a compact connected Lie group G, every element lies on some. 1-p subgroup. This is not true in a non-compact G, i.e. there are elements in G which do not ...
[21]
[PDF] Fall, 2022 Lecture III The Exponential Map, Local Lie Groups, and ...
Sep 26, 2022 · Definition 1.4. We define the exponential map, expG : g → G by sending. A ∈ g to γA(1) where γA is the one-parameter subgroup whose tangent.
[22]
[PDF] Colloquium: A Century of Noether's Theorem - arXiv
Jul 11, 2019 · In the summer of 1918, Emmy Noether published the theorem that now bears her name, establishing a profound two-way connection.
[23]
Emmy Noether's Wonderful Theorem (rev ed.). - AIP Publishing
Dec 1, 2018 · In 1918, the mathematician Emmy Noether published two wonderful theorems that had a tremendous impact in physics, mathematics, and beyond.<|control11|><|separator|>
[24]
[PDF] Noether's theorem - Physics Department, Oxford University
Feb 27, 2025 · In order to understand the Euler-Lagrange equations correctly, it is important to note what is held constant in each partial derivative, and ...
[25]
[physics/0503066] Invariant Variation Problems - arXiv
Mar 8, 2005 · Authors:Emmy Noether, M. A. Tavel. View a PDF of the paper titled Invariant Variation Problems, by Emmy Noether and M. A. Tavel. View PDF.
[26]
[gr-qc/0608096] Noether's theorem, the stress-energy tensor ... - arXiv
Noether's theorem is reviewed with a particular focus on an intermediate step between global and local gauge and coordinate transformations, namely linear ...Missing: field | Show results with:field
[27]
[PDF] Symmetries and conservation laws in quantum me- chanics
In quantum mechanics, a conserved quantity's probability is time-independent, and a symmetry is a unitary transformation that maps states to equivalent states.
[28]
[PDF] Transformations and Symmetries - Rutgers Physics
Finite Transformations. • As noted above, a finite continuous transformation can be built up out of an infinite number of infinitesimal transformations. ˆ. Ux ...
[29]
[PDF] Time Crystals - Stony Brook University
A standard example of symmetry breaking is the existence of crystals: the symmetry breaking of the continuous spatial translation symmetry to the discrete ...
[30]
[PDF] Units, limits, and symmetries When solving physics problems it's ...
... central forces (like gravity) have spherical symmetry because the force depends only on the distance from the origin. In this case, spherical symmetry means ...<|separator|>
[31]
[PDF] Central Forces - LIGO-Labcit Home
Thus rotational symmetry about the z-axis implies the conservation of the z-component of the angular momentum. Full rotational symmetry, about any axis, implies ...
[32]
Kepler Orbits - Galileo and Einstein
The conservation of angular momentum comes from the spherical symmetry of the system: the attraction depends only on distance, not angle. In quantum mechanics, ...
[33]
[PDF] so(4) Symmetry of the Kepler Problem - Columbia Math Department
Then there exists a corresponding conserved quantity. In this problem, we saw conservation of angular momentum, and conservation of energy is also present ...
[34]
[PDF] Curl, Divergence and Laplacian - Purdue Math
The curl operator encodes information about infinitesimal rotations of a vector field. Think of a small plastic ball inside the stream of a river. If the ball ...
[35]
UM Ma215 Examples: 16.5 Curl
Curl of a Vector Field Intuitively, the curl measures the infinitesimal rotation around a point. This is difficult to visualize in three dimensions, but we ...
[36]
Symmetry-breaking orbital anisotropy observed for detwinned Ba ...
Apr 11, 2011 · Nematicity, defined as broken rotational symmetry, has recently been observed in competing phases proximate to the superconducting phase in ...
[37]
[PDF] Gauge Theory - DAMTP
... gauge symmetry which underlies the Maxwell equations, the Standard Model and, in the guise of diffeomorphism invariance, general relativity. Gauge symmetry ...
[38]
[PDF] GAUGE THEORIES - UT Physics
also called a gauge symmetry — the field transformations at. different points x have independent parameters. For example, a local ...
[39]
Conservation of Isotopic Spin and Isotopic Gauge Invariance
The possibility is explored of having invariance under local isotopic spin rotations. This leads to formulating a principle of isotopic gauge invariance.
[40]
[PDF] P528 Notes #3: Non-Abelian Gauge Theories
Jan 31, 2017 · From this point of view, it is completely natural to try to construct field theories with a gauge invariance under non-Abelian transformation ...
[41]
Noether's Second Theorem and Ward Identities for Gauge Symmetries
Oct 23, 2015 · We reintroduce Noether's second theorem and discuss how to work with the physical remnant of gauge symmetry in gauge fixed systems.
[42]
Noether's second theorem and Ward identities for gauge symmetries
Feb 4, 2016 · We reintroduce Noether's second theorem and discuss how to work with the physical remnant of gauge symmetry in gauge fixed systems.
[43]
Gauge theory: Historical origins and some modern developments
Jan 1, 2000 · In this article the authors review the early history of gauge theory, from Einstein's theory of gravitation to the appearance of non-Abelian gauge theories in ...