rotational invariance
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
defined on an
inner product space In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often den ...
is said to have rotational invariance if its value does not change when arbitrary
rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
s are applied to its argument.


Mathematics


Functions

For example, the function :f(x,y) = x^2 + y^2 is invariant under rotations of the plane around the origin, because for a rotated set of coordinates through any
angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle. Angles formed by two ...
''θ'' :x' = x \cos \theta - y \sin \theta :y' = x \sin \theta + y \cos \theta the function, after some cancellation of terms, takes exactly the same form :f(x',y') = ^2 + ^2 The rotation of coordinates can be expressed using
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
form using the
rotation matrix In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix :R = \begin \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \en ...
, :\begin x' \\ y' \\ \end = \begin \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \\ \end\begin x \\ y \\ \end. or symbolically x′ = Rx. Symbolically, the rotation invariance of a real-valued function of two real variables is :f(\mathbf') = f(\mathbf) = f(\mathbf) In words, the function of the rotated coordinates takes exactly the same form as it did with the initial coordinates, the only difference is the rotated coordinates replace the initial ones. For a real-valued function of three or more real variables, this expression extends easily using appropriate rotation matrices. The concept also extends to a
vector-valued function A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors. The input of a vector-valued function could ...
f of one or more variables; :\mathbf(\mathbf') = \mathbf(\mathbf) = \mathbf(\mathbf) In all the above cases, the arguments (here called "coordinates" for concreteness) are rotated, not the function itself.


Operators

For a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
:f : X \rightarrow X which maps elements from a
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
''X'' of the
real line In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...
ℝ to itself, rotational invariance may also mean that the function
commute Commute, commutation or commutative may refer to: * Commuting, the process of travelling between a place of residence and a place of work Mathematics * Commutative property, a property of a mathematical operation whose result is insensitive to th ...
s with rotations of elements in ''X''. This also applies for an operator that acts on such functions. An example is the two-dimensional Laplace operator :\nabla^2 = \frac + \frac which acts on a function ''f'' to obtain another function ∇2''f''. This operator is invariant under rotations. If ''g'' is the function ''g''(''p'') = ''f''(''R''(''p'')), where ''R'' is any rotation, then (∇2''g'')(''p'') = (∇2''f'' )(''R''(''p'')); that is, rotating a function merely rotates its Laplacian.


Physics

In
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, if a system behaves the same regardless of how it is oriented in space, then its
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
is rotationally invariant. According to
Noether's theorem Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether ...
, if the
action Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 fil ...
(the integral over time of its Lagrangian) of a physical system is invariant under rotation, then angular momentum is conserved.


Application to quantum mechanics

In
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, rotational invariance is the property that after a
rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
the new system still obeys Schrödinger's equation. That is : ,E-H= 0 for any rotation ''R''. Since the rotation does not depend explicitly on time, it commutes with the energy operator. Thus for rotational invariance we must have 'R'', ''H''= 0. For
infinitesimal rotation In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix :R = \begin \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \ ...
s (in the ''xy''-plane for this example; it may be done likewise for any plane) by an angle ''dθ'' the (infinitesimal) rotation operator is :R = 1 + J_z d\theta \,, then :\left + J_z d\theta , \frac \right= 0 \,, thus :\frac{dt}J_z = 0\,, in other words
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
is conserved.


See also

*
Axial symmetry Axial symmetry is symmetry around an axis; an object is axially symmetric if its appearance is unchanged if rotated around an axis.
*
Invariant measure In mathematics, an invariant measure is a measure that is preserved by some function. The function may be a geometric transformation. For examples, circular angle is invariant under rotation, hyperbolic angle is invariant under squeeze mapping, ...
*
Isotropy Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...
*
Maxwell's theorem In probability theory, Maxwell's theorem, named in honor of James Clerk Maxwell, states that if the probability distribution of a vector-valued random variable ''X'' = ( ''X''1, ..., ''X'n'' )''T'' is the same as the distribution of ''GX'' for ...
*
Rotational symmetry Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which i ...


References

*Stenger, Victor J. (2000). ''Timeless Reality''. Prometheus Books. Especially chpt. 12. Nontechnical. Rotational symmetry Conservation laws