From Special Relativity to Feynman Diagrams.pdf

Recalling the definition given in sect72 a linear

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Recalling the definition given in Sect.7.2 , a linear operator ˆ F on the vector space V ( c ) n is defined as a (not necessarily invertible) mapping from V ( c ) n into itself: ˆ F : | v V ( c ) n → | F v ˆ F | v V ( c ) n , (9.6) satisfying the linearity condition ( 7.3 ) which, in our new notations, reads: ˆ F | v + β | w ) = α ˆ F | v + β ˆ F | w , α, β C . (9.7) Linear transformations are invertible operators on V ( c ) n . Of particular physical rele- vance in quantum mechanics is the notion of expectation value ˆ F of an operator ˆ F on a state | a : ˆ F a | ˆ F | a a | a . Let | u i , i = 1 , . . . , n , be a basis of ket vectors in V ( c ) n , and let u i | be the dual basis of bra vectors. The basis | u i is said to be orthonormal if u i | u j = δ i j . (9.8) With respect to this basis ˆ F can be represented by a matrix F ( F i j ), see ( 7.6 ) and footnote 2 of Chap.7 : | u i ˆ F −→ | Fu i ˆ F | u i = F j i | u j , (9.9) that is, using ( 9.8 ) F i j = u i | ˆ F | u j . Clearly if ˆ F is not invertible, and thus is not a transfromation, the matrix F is singular and the vectors | Fu i do not form a new basis. The identity operator ˆ I on V ( c ) n can be written in the form ˆ I = n i = 1 | u i u i | , (9.10) since it can be easily verified, using the orthonormality of the basis, that ˆ I | u i = | u i , for all i = 1 , . . . , n . The corresponding matrix representation is the n × n identity matrix 1 i j ) .
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266 9 Quantum Mechanics Formalism In quantum mechanics there are two classes of operators which play a special role: The hermitian and the unitary operators. Both of them can be characterized by their properties with respect to their hermitian conjugate operators. The hermitian conjugate ˆ F of ˆ F , is defined as the operator such that ∀| a , | b V ( c ) n , a | ˆ F | b = b | ˆ F | a , or, equivalently, a | Fb = F a | b . This definition implies that Fb | ≡ b | ˆ F is the bra of ˆ F | b . ˆ F is a hermitian operator iff ˆ F = ˆ F , or, equivalently F i j = u i | ˆ F | u j = u j | ˆ F | u i = u j | ˆ F | u i = ( F j i ) , that is, the matrix representing it coincides with the conjugate of its transposed (hermitian conjugate): F = F ( F T ) , and is therefore a hermitian matrix . In other words an operator is hermitian if and only if its matrix representation with respect to an orthonormal basis is hermitian. From this it clearly follows that the expectation value of a hermitian operator on any state is a real number since a | ˆ F | a = a | ˆ F | a = a | ˆ F | a . A unitary operator U is defined by the condition UU = U U = ˆ I . (9.11) From the above definition we derive the corresponding unitarity property of the matrix U = ( U i j ) representing U : δ i j = u i | u j = u i | U U | u j = n k = 1 u i | U | u k u k | U | u j = n k = 1 ( U k i ) U k j , i.e. U U = 1 . While a hermitian operator is not necessarily invertible, a unitary one is, being the corresponding matrix U non-singular: | det U | 2 = 1. Unitary operators are therefore linear transformations and their geometrical and physical meaning will be discussed in the next section. It can be shown, from the properties of the
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