# Mathematical-physics dictionary

The interplay between theoretical physics and mathematics is a beautiful topic. The two contrasting backgrounds may and do, however, lead to a somewhat different scientific vocabulary. The language of physicists tends to be more descriptive and intuitive, but less accurate, than that of mathematicians. This difference may at times be confusing or annoying — at least to me. Moreover, unlike in mathematics, where everything is (and actually *can* be) defined to a level of precision that may sometimes obscure what is actually meant, I am not aware of many references for the language used by theoretical physicists.

For lingo related to (quantum) field theory a good starting point is the Glossary in volume 1 of *Quantum Fields and Strings: A Course for Mathematicians*. For notions from various areas in (theoretical) physics discussed from a more mathematical perspective I also like

- Various questions and answers at MathOverflow, Mathematics.SE and Physics.SE,
- The
*n*Lab, as well as some discussions at The*n*-Category Café, often but not always from a (higher) category-theoretic viewpoint, - John Baez’s website, including his column
*This Week’s Finds*, - Terence Tao’s website.

Besides such topic-specific notions, however, physicists tend to use a slightly different language more generally, and there are more widely used phrases that may be confusing as well. To help filling in this gap, the following is a modest collection of potentially confusing terminology used in theoretical physics that I have compiled over the past few years.

*Disclaimer.* The following overview is by no means complete, and I certainly do not claim to have the best possible translation. Do not hesitate to contact me with contributions or corrections!

theoretical physicist’s language |
mathematical meaning |

abstract algebra | algebra (as opposed to a representation thereof, cf. algebra) |

algebra | 1. relations defining an algebra; 2. representation of an algebrathe ⁓ closes: this actually/indeed is a representation |

bosonic quantity | commuting quantity: even element of a supercommutative superalgebra |

c-number |
scalar (cf. q-number) |

continuous group | Lie group |

correspondence | 1. bijection; 2. ‘partial bijection’ (for part of the data/aspect); 3. (striking) relation or similarity |

equivalence | isomorphism |

fermionic quantity | see Grassmann variable |

finite | 1. not infinite; 2. nonzero; 3. finite dimensional |

generic | usually: general, arbitrary |

Grassmann variable | anticommuting quantity: odd element of a supercommutative superalgebra, e.g. element of exterior algebra |

group manifold | Lie group |

group theory | representation theory of Lie groups and Lie algebras (often: of su_{2}) |

linear space | vector space |

nonlinear realization | often: (module for a) group action (cf. realization 1) |

on shell | satisfying certain equations (characterizing the ‘physical’ quantities, e.g. mass-shell condition, equations of motion, Bethe equations) |

q-number |
operator (cf. c-number) |

realization | 1. representation, module; 2. instance, example; 3. ?? |