Quantum Field TheoryPHYS-P 621

Radovan Dermisek, Indiana University

Notes based on: M. Srednicki, Quantum Field Theory

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Attempts at relativistic QMbased on S-1

The result of this effort is relativistic quantum field theory - consistent description of particle physics.

A proper description of particle physics should incorporate bothquantum mechanics and special relativity.

Today we review some of these attempts.

However historically combining quantum mechanics and relativity was highly non-trivial.

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Quantum mechanicsTime evolution of the state of the system is described by Schrödinger equation:

For a free, spinless, nonrelativistic particle we have:where H is the hamiltonian operator representing the total energy.

in the position basis,

where is the position-space wave function.

and S.E. is:

H =1

2mP2

P = !ih!

!(x, t) = !x|!, t"

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Relativistic generalization?Obvious guess is to use relativistic energy-momentum relation:

Schrödinger equation becomes:

☹Not symmetric in time and space derivatives

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Relativistic generalization?Obvious guess is to use relativistic energy-momentum relation:

Schrödinger equation becomes:

☺

Applying

Obvious guess is to use relativistic energy-momentum relation:Obvious guess is to use relativistic energy-momentum relation:

on both sides and using S.E. we get

Klein-Gordon equation looks symmetric

i! !

!t

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Special relativity(physics is the same in all inertial frames)

Space-time coordinate system: xµ = (ct,x)define: xµ = (!ct,x)

or

is the Minkowski metric tensor. Its inverse is:

where

that allows us to write:

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Interval between two points in space-time can be written as:

ds2 = (x! x!)2 ! c2(t! t!)2

= gµ!(x! x!)µ(x! x!)!

= (x! x!)µ(x! x!)µ

General rules for indices:repeated indices, one superscript and one subscript are summed; these indices are said to be contracted

any unrepeated indices (not summed) must match in both name and height on left and right side of any valid equation

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Two coordinate systems (representing inertial frames) are related by

Lorentz transformation matrixtranslation vector

Interval between two different space-time points is the same in all inertial frames:

which requires:

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Notation for space-time derivatives:

matching-index-height rule works:

For two coordinate systems related by

derivatives transform as:

which follows from

not the same as

will prove later!

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Is K-G equation consistent with relativity?

Physics is the same in all inertial frames: the value of the wave function at a particular space-time point measured in two inertial frames is the same:

This should be true for any point in the space-time and thusa consistent equation of motion should have the same form in any inertial frame.

Is that the case for Klein-Gordon equation?

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Klein-Gordon equation: in 4-vector notation:

in 4-vector notation: ☐ =

Is it equivalent to:

?Since

K-G eq. is manifestly consistent with relativity!

d’Alambertian operator

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Is K-G consistent with quantum mechanics?

Schrödinger equation (first order in time derivative) leaves the norm of a state time independent. Probability is conserved:

!"

!t= #! !#

!t+ #

!#!

!t

=i!2m

#!!2# " i!2m

#!2#!

=i!2m!.(#!!# " #!#!)

# "!. j

=!

d3x!(x)

!

!t!", t|", t" =

!d3x

!#

!t= #

!d3x$. j = #

!

Sj.dS = 0

Gauss’s law

j(x) = 0 at infinity

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Is K-G consistent with quantum mechanics?

Schrödinger equation (first order in time derivative) leaves the norm of a state time independent. Probability is conserved:

=!

d3x!(x)

!

!t!", t|", t" =

!d3x

!#

!t= #

!d3x$. j = #

!

Sj.dS = 0

Klein-Gordon equation is second order in time derivative and the norm of a state is NOT in general time independent. Probability is not conserved.

Klein-Gordon equation is consistent with relativity but not with quantum mechanics.

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Dirac attemptDirac suggested the following equation for spin-one-half particles (a state carries a spin label , a = 1,2):

!(x, t) = !x|!, t"

|!, a, t!

Consistent with Schrödinger equation for the Hamiltonian:

Dirac equation is linear in both time and space derivatives and so it might be consistent with both QM and relativity.

Squaring the Hamiltonian yields:

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!(x, t) = !x|!, t"Eigenvalues of should satisfy the correct relativistic energy-momentum relation:

can be written as anticommutator

and also can be written as

and so we choose matrices that satisfy following conditions:

H2

it can be proved (later) that the Dirac equation is fully consistent with relativity.

we have a relativistic quantum mechanical theory!

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Discussion of Dirac equationto account for the spin of electron, the matrices should be 2x2but the minimum size satisfying above conditions is 4x4 - two extra spin states

H is traceless, and so 4 eigenvalues are: E(p), E(p), -E(p), -E(p)

negative energy states no ground state

Dirac’s interpretation: due to Pauli exclusion principle each quantum state can be occupied by one electron and we simply live in a universe with all negative energy states already occupied.Negative energy electrons can be excited into a positive energy state (by a photon) leaving behind a hole in the sea of negative energy electrons. The hole has a positive charge and positive energy antiparticle (the same mass, opposite charge) called positron (1927)

also a problem for K.-G. equation

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Quantum mechanics as a quantum field theory

Consider Schrödinger equation for n particles with mass m, moving in an external potential U(x), with interparticle potential

It is equivalent to:

is the position-space wave function.

if and only if satisfies S.E. for wave function!17

and

is a quantum field and its hermitian conjugate;

they satisfy commutation relations:

state with one particle at

state with one particles at and another at

Total number of particles is counted by the operator:

commutes with H!

is the vacuum state

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creation operators commute with each other, and so without loss of generality we can consider only completely symmetric functions:

we have a theory of bosons (obey Bose-Einstein statistics).

For a theory of fermions (obey Fermi-Dirac statistics) we impose

and we can restrict our attention to completely antisymmetric functions:

We have nonrelativistic quantum field theory for spin zero particles that can be either bosons or fermions. This will change for relativistic QFT.

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