From Special Relativity to Feynman Diagrams.pdf

Side of the above equation we have performed the

Info icon This preview shows pages 351–353. Sign up to view the full content.

View Full Document Right Arrow Icon
side of the above equation we have performed the following change in the integration variable p → − p , gaining a minus sign. Note that the last two terms in the integral sum up to an anti-hermitian operator, which cancels against its hermitian conjugate. The first two terms instead are hermitian, so that we end up with: ˆ P = d 3 p ( 2 π ) 3 V p a ( p ) a ( p ) + b ( p ) b ( p ) = p p a ( p ) a ( p ) + b ( p ) b ( p ) , (11.45) where the last equality refers to the case of a finite volume V and discrete momenta. Note that in this case no normal ordering is necessary since, when writing bb in terms of b b in ( 11.45 ) we have p p b ( p ) b ( p ) = p p b ( p ) b ( p ) + 1 = p p b ( p ) b ( p ), (11.46) due to the cancelation of p against p when summing the constant term over all possible values of p . Putting together the results obtained for ˆ H and ˆ P i , we may define the four- momentum quantum operator ˆ p μ = d 3 p ( 2 π ) 3 V p μ a ( p ) a ( p ) + b ( p ) b ( p ) . (11.47) So far we have defined the quantum operator associated with a Klein–Gordon field. We still need to define the Hilbert space of quantum states on which such operator acts. This will allow us to give a particle interpretation of our results. Our discussion so far paralleled the one for the electromagnetic field in Chap.6 . When we wrote the field operators ˆ φ( x ), ˆ φ ( x ) in terms of the a , b operators and of their hermitian conjugates satisfying the commutation relations ( 11.32 ) (or ( 11.33 )), we have described the quantum system as a collection of infinitely many decoupled quantum harmonic oscillators of two kinds: The “ ( a ) ” and the “ ( b ) ” oscillators, asso- ciated with the positive and negative energy solutions to the Klein–Gordon equation.
Image of page 351

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
11.2 Quantization of the Klein–Gordon Field 371 Each value of p defines a corresponding oscillator of type ( a ) and ( b ), the operators a ( p ), a ( p ) and b ( p ), b ( p ) being the corresponding destruction and creation oper- ators, respectively. For the two kinds of oscillators we define the (hermitian) number operators : ˆ N ( a ) p = a ( p ) a ( p ) ; ˆ N ( b ) p = b ( p ) b ( p ), (11.48) We see that both the energy ( 11.43 ) and the momentum ( 11.45 ) are expressed as infinite sums over such operators. In particular the Hamiltonian operator ˆ H is the sum over the Hamiltonian operators ˆ H ( a ) p , ˆ H ( b ) p of the various oscillators (we use here, for the sake of simplicity, the finite volume notation): ˆ H = p ˆ H ( a ) p + ˆ H ( b ) p , ˆ H ( a ) p E p ˆ N ( a ) p = E p a ( p ) a ( p ) ; ˆ H ( b ) p E p ˆ N ( b ) p = E p b ( p ) b ( p ). (11.49) Since these harmonic oscillators correspond to independent, decoupled degrees of freedom of the scalar field, operators associated with different oscillators commute, as it is apparent from ( 11.32 ). In particular the hermitian operators ˆ N ( a ) p , ˆ N ( b ) p form a commuting system 5 and thus can be diagonalized simultaneously. As a consequence of this the quantum states of the field can be expressed as products of the infinite states pertaining to the constituent quantum oscillators, each constructed as an eigenstate of the corresponding number operator.
Image of page 352
Image of page 353
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern