Anti-symmetric Operator - Application in Quantum Field Theory

Application in Quantum Field Theory

The creation of a particle and anti-particle from a boson is defined similarly but for infinite dimensions. Therefore the Levi-Civita symbol for infinite dimensions is introduced.

\varepsilon_{ijk\ell\dots} = \left\{ \begin{matrix} +1 & \mbox{if }(i,j,k,\ell,\dots) \mbox{ is an even permutation of } (1,2,3,4,\dots) \\ -1 & \mbox{if }(i,j,k,\ell,\dots) \mbox{ is an odd permutation of } (1,2,3,4,\dots) \\ 0 & \mbox{if any two labels are the same} \end{matrix} \right.

The commutation relations are simply carried over to infinite dimensions . is now equal to where n=∞. Its eigenvalue is . Defining the magnetic quantum number, angular momentum projected in the z direction, is more challenging than the simple state of spin. The problem becomes analogous to moment of inertia in classical mechanics and is generalizable to n dimensions. It is this property that allows for the creation and annihilation of bosons.

Read more about this topic:  Anti-symmetric Operator

Famous quotes containing the words application, quantum, field and/or theory:

    “It is known that Whistler when asked how long it took him to paint one of his “nocturnes” answered: “All of my life.” With the same rigor he could have said that all of the centuries that preceded the moment when he painted were necessary. From that correct application of the law of causality it follows that the slightest event presupposes the inconceivable universe and, conversely, that the universe needs even the slightest of events.”
    —Jorge Luis Borges (1899–1986)

    “A personality is an indefinite quantum of traits which is subject to constant flux, change, and growth from the birth of the individual in the world to his death. A character, on the other hand, is a fixed and definite quantum of traits which, though it may be interpreted with slight differences from age to age and actor to actor, is nevertheless in its essentials forever fixed.”
    —Hubert C. Heffner (1901–1985)

    “What though the field be lost?
    All is not lost; the unconquerable Will,
    And study of revenge, immortal hate,
    And courage never to submit or yield:
    And what is else not to be overcome?”
    —John Milton (1608–1674)

    “The things that will destroy America are prosperity-at-any- price, peace-at-any-price, safety-first instead of duty-first, the love of soft living, and the get-rich-quick theory of life.”
    —Theodore Roosevelt (1858–1919)