Canonical quantization of scalar ‍elds - IU B dermisek/QFT_08/qft-I-2-1p.pdf  Canonical...

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Transcript of Canonical quantization of scalar ‍elds - IU B dermisek/QFT_08/qft-I-2-1p.pdf  Canonical...

  • Canonical quantization of scalar fieldsbased on S-3

    Hamiltonian for free nonrelativistic particles:

    Furier transform:

    a(x) =

    d3p

    (2)3/2eipx a(p)

    we get:

    can go back to x using:

    d3x

    (2)3eipx = 3(p)

    d3p

    (2)3eipx = 3(x)

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  • Canonical quantization of scalar fields

    (Anti)commutation relations:

    [A,B] = AB BAVacuum is annihilated by :

    is a state of momentum , eigenstate of with

    is eigenstate of with energy eigenvalue:

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  • Relativistic generalization

    Hamiltonian for free relativistic particles:

    spin zero, but can be either bosons or fermions

    Is this theory Lorentz invariant?

    Lets prove it from a different direction, direction that we will use for any quantum field theory from now:

    start from a Lorentz invariant lagrangian or action

    derive equation of motion (for scalar fields it is K.-G. equation)

    find solutions of equation of motion

    show the Hamiltonian is the same as the one above

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  • A theory is described by an action:

    where is the lagrangian.

    Equations of motion should be local, and so

    Thus:

    where is the lagrangian density.

    is Lorentz invariant:

    For the action to be invariant we need:

    the lagrangian density must be a Lorentz scalar!

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  • Lets consider:

    Any polynomial of a scalar field is a Lorentz scalar and so are products of derivatives with all indices contracted.

    and lets find the equation of motion, Euler-Lagrange equation:

    (we find eq. of motion from variation of an action: making an infinitesimal variation in and requiring the variation of the action to vanish)

    arbitrary constant

    integration by parts, and at infinity in any direction (including time)

    (x) = 0

    is arbitrary function of x and so the equation of motion is Klein-Gordon equation

    ! = 1, c = 1

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  • Solutions of the Klein-Gordon equation:

    one classical solution is a plane wave:

    is arbitrary real wave vector and

    The general classical solution of K-G equation:

    where and are arbitrary functions of , and

    where and are arbitrary functions of , and

    is a function of (introduced for later convenience) |k|

    if we tried to interpret as a quantum wave function, the second term would represent contributions with negative energy to the wave function!

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  • real solutions:

    k k

    thus we get:

    (such a is said to be on the mass shell)k

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  • Finally lets choose so that is Lorentz invariant:

    manifestly invariant under orthochronous Lorentz transformations

    on the other hand

    sum over zeros of g, in our case the only zero is k0 = for any the differential is Lorentz invariant

    it is convenient to take for which the Lorentz invariant differential is:

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  • Finally we have a real classical solution of the K.-G. equation:

    where again: , ,

    For later use we can express in terms of :

    where and we will call .

    Note, is time independent.

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  • Constructing the hamiltonian:

    Recall, in classical mechanics, starting with lagrangian as a function of coordinates and their time derivatives we define conjugate momenta and the hamiltonian is then given as:

    In field theory:

    hamiltonian density

    and the hamiltonian is given as:

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  • In our case:

    Inserting we get:

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  • d3x

    (2)3eipx = 3(p)

    39