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'''+ h.c.''' is an abbreviation for "plus the ''H'' ermitian ''c'' onjugate"; 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.<ref name=Özkan-HueyWen-2019-03-16/><ref name="physfrms2010"/>

==Context and use==
The [[Language of mathematics|notation convention]] "+ h.c." is common in [[Physics:Quantum mechanics|quantum mechanics]] in the context of writing out formulas for [[Lagrangian mechanics#Quantum mechanics|Lagrangian]]s and [[Physics:Hamiltonian (quantum mechanics)|Hamiltonians]], which conventionally are both required to be Hermitian operators.<ref name=physfrms2010>{{cite web |title=Meaning of h.c. in Lagrangians (& elsewhere?) |url=https://www.physicsforums.com/threads/meaning-of-h-c-in-lagrangians-elsewhere.454071/ |date=2010-12-05 |website=Physics Forums |access-date=2018-05-23 |df=dmy-all}}</ref>

The expression
:<math>\mathcal{L} = A + B + C + ~ \text{h.c.} ~</math>

means
:<math>\mathcal{L} = A + B + C + A^\dagger + B^\dagger + C^\dagger ~.</math><ref name=physfrms2010/>

The mathematics of quantum mechanics is based on [[Complex number|complex number]]s, whereas almost all observations (measurements) are only [[Real number|real number]]s. 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.<ref name=Özkan-HueyWen-2019-03-16>{{cite book |last1=Özkan |first1=Tristan |last2=Lin |first2=Huey-Wen |chapter=Quantum3: Learning QCD through Intuitive Play |title=Proceedings of the 36th Annual International Symposium on Lattice Field Theory (LATTICE2018) |series=Proceedings of Science |volume=334 |date=29 May 2019 |pages=326 |arxiv=1901.00022 |doi=10.22323/1.334.0326 |s2cid=119383350 |oclc=1082145757 |doi-access=free }}</ref>

===Dagger and asterisk notation===
In the expressions above, <math>A^\dagger</math> is used as the symbol for the Hermitian conjugate (also called the ''[[Conjugate transpose|conjugate transpose]]'') of <math>A</math>, defined as applying both the [[Complex conjugate|complex conjugate]] and the [[Transpose|transpose]] transformations to the operator <math>A</math>, in any order.

The dagger (<math>\dagger</math>) is an old notation in mathematics, but is still widespread in quantum-mechanics. In mathematics (particularly [[Linear algebra|linear algebra]]) the Hermitian conjugate of <math>A</math> is commonly written as <math>A^\ast</math>, but in quantum mechanics the asterisk (<math>\ast</math>) notation is sometimes used for the [[Complex conjugate|complex conjugate]] only, and not the combined [[Conjugate transpose|conjugate transpose]] (Hermitian conjugate).

==References==
{{reflist}}

[[Category:Hamiltonian mechanics]]
[[Category:Quantum mechanics]]
[[Category:Lagrangian mechanics]]
[[Category:Operator theory]]

{{Sourceattribution|+ h.c.}}

Physics:+ h.c. - Revision history

WikiHarold: change