User:Count Truthstein/Transformation (physics)
In physics, a transformation is a change of coordinate system which changes the coordinates which label points of physical space or spacetime, and also the numbers describing physical quantities. If a theory describes the same physical state after a transformation, obeying the same rules, then the transformation is a symmetry of the theory.
Examples of transformations[edit]
Examples of transformations are rotations, translations, and Lorentz transformations. These are often expressed as linear transformations of a vector space; concretely, as a matrix R multiplying a column of Euclidean coordinates v to give the new column of coordinates Rv.
A collection of a particular type of transformation may form a group, in which case the quantities transformed thereby are a representation or projective representation of the said group (for example, transformed objects may be representations of the Lorentz group, or of the covariance group in general relativity).
Infinitesimal transformations[edit]
Co-ordinate systems and associated quantities can be said to transform under infinitesimal transformations (such as infinitesimal rotations). These can be thought of as very small movements of the co-ordinate system which can applied in sequence (integrated) to give the change after some period of time.
Infinitesimal transformations belong to the Lie algebra corresponding to the group of transformations. Transformed quantities belong to a representation of this algebra as well as a representation of the group.
Active and passive[edit]
A transformation can be imagined as either a motion of the coordinate axes, or as a motion of space keeping the axes where they are.
Transformation of physical quantities[edit]
Physical quantities, just like points of physical space, can also be labelled with elements of a vector space. Such vector spaces may be either finite- or infinite-dimensional over the real or complex numbers, depending on the physical quantity. An example of a quantity which is described with an infinite number of numbers is a wavefunction in quantum mechanics.
The transformation rule for a physical quantity is often given as a linear map. However, the linear map may be different from that used for the co-ordinates: in general, given a co-ordinate transform x to Λx, there will be a linear transformation D(Λ) saying how the physical quantity transforms.
Quantities such as tensors may be labelled with several indices (for example, the Riemann curvature tensor Rρσμν has four), in which case there is commonly a linear transformation for each index.
Physical quantities can be categorized by how they transform under (certain) transformations:
- Scalar - Always has the same value, regardless of coordinate system
- Vector - Objects with magnitude and direction which transfer in a straightforward way
- Tensor - A generalization of scalars and vectors
- Spinor - Similar to vectors, but have the property that applying a sequence of transformations which return the coordinates to their original values does not in general return the value of the spinor to the original value
- Pseudoscalar, Pseudovector and Pseudotensor - transform like scalars, vectors and tensors, except that they transform to their negatives under reflections