+ h.c.

+ h.c. is an abbreviation for “plus the Hermitian conjugate”; it means is that there are additional terms which are the Hermitian conjugates of all of the preceding terms, and is a convenient shorthand to omit half the terms actually present.^[1]

Context and use

The notation convention “+ h.c.” is common in particle physics in the context of writing out formulas for Lagrangians and Hamiltonians, which conventionally are required to be a Hermitian operators.^[1] The mathematics of quantum mechanics is based on complex numbers, whereas almost all observations (measurements) are only real numbers. Adding its own conjugate to an operator guarantees that the combination is Hermitian, which in turn guarantees that the combined operator's eigenvalues will be real numbers, suitable for predicting values of observations/measurements of particle behavior.

Hence, interpret the expression

A+B+C+{\text{h.c.}}

as meaning^[1]

A+A^{\dagger }+B+B^{\dagger }+C+C^{\dagger }

In the expressions above, $A^{\dagger }$ is used as the symbol for the Hermitian conjugate of operator $A$ , which is an older notation in mathematics, but still widespread in quantum-mechanics; it is commonly written as $A^{\ast }$ in mathematics (particularly linear algebra) but that notation is sometimes used in quantum mechanics for the complex conjugate only, and not the conjugate transpose (Hermitian conjugate).

References

1 2 3 "Meaning of h.c. in Lagrangians (& elsewhere?)". Physics Forums. 2010-12-05. Retrieved 2018-05-23.

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[physfrms2010-1] 1 2 3 "Meaning of h.c. in Lagrangians (& elsewhere?)". Physics Forums. 2010-12-05. Retrieved 2018-05-23.