Download Quantum Field Theory

Quantum mechanics
Time evolution of the state of the system is described by
Schrödinger equation:
Quantum Field Theory
PHYS-P 621
where H is the hamiltonian operator representing the total energy.
For a free, spinless, nonrelativistic particle we have:
Radovan Dermisek, Indiana University
H=
1 2
P
2m
in the position basis, P = −ih∂ and S.E. is:
Notes based on: M. Srednicki, Quantum Field Theory
where
ψ(x, t) = !x|ψ, t" is the position-space wave function.
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Relativistic generalization?
Attempts at relativistic QM
Obvious guess is to use relativistic energy-momentum relation:
based on S-1
A proper description of particle physics should incorporate both
quantum mechanics and special relativity.
Schrödinger equation becomes:
However historically combining quantum mechanics and relativity
was highly non-trivial.
Today we review some of these attempts.
Not symmetric in time and space derivatives
The result of this effort is relativistic quantum field theory consistent description of particle physics.
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!
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Interval between two points in space-time can be written as:
Relativistic generalization?
ds2
Obvious guess is to use relativistic energy-momentum relation:
= (x − x! )2 − c2 (t − t! )2
= gµν (x − x! )µ (x − x! )ν
= (x − x! )µ (x − x! )µ
Schrödinger equation becomes:
General rules for indices:
repeated indices, one superscript and one subscript are summed; these indices
are said to be contracted
∂
Applying i!
on both sides and using S.E. we get
∂t
Klein-Gordon equation
looks symmetric
any unrepeated indices (not summed) must match in both name and height on
left and right side of any valid equation
!
Obvious guess is to use relativistic energy-momentum relation:
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Two coordinate systems (representing inertial frames) are related by
Special relativity
(physics is the same in all inertial frames)
Space-time coordinate system:
xµ = (ct, x)
define: xµ = (−ct, x)
or
Lorentz transformation matrix
translation vector
Interval between two different space-time points is the same in
all inertial frames:
where
is the Minkowski metric tensor. Its inverse is:
which requires:
that allows us to write:
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Notation for space-time derivatives:
Klein-Gordon equation: in 4-vector notation:
!!"
d’Alambertian operator
in 4-vector notation:
matching-index-height rule works:
Is it equivalent to:
For two coordinate systems related by
derivatives transform as:
which follows from
not the same as
?
Since
will prove later!
K-G eq. is manifestly consistent with relativity!
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Is K-G equation consistent with relativity?
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:
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:
=
∂
!ψ, t|ψ, t" =
∂t
This should be true for any point in the space-time and thus
a consistent equation of motion should have the same form in
any inertial frame.
∂ρ
∂t
Is that the case for Klein-Gordon equation?
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!
∂ρ
d x
=−
∂t
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∂ψ
∂ψ ∗
+ψ
∂t
∂t
i! ! 2
i!
=
ψ ∇ ψ−
ψ∇2 ψ ∗
2m
2m
i!
=
∇.(ψ ! ∇ψ − ψ∇ψ ∗ )
2m
≡ −∇. j
= ψ!
!
d x∇. j = −
3
!
!
d3 xρ(x)
j.dS = 0
S
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:
=
∂
!ψ, t|ψ, t" =
∂t
!
∂ρ
d x
=−
∂t
3
!
d x∇. j = −
3
!
!
can be written as anticommutator
d3 xρ(x)
and also
can be written as
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Eigenvalues ofψ(x,
satisfy
H t)should
= !x|ψ,
t" the correct relativistic energymomentum relation:
j.dS = 0
S
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.
and so we choose matrices that satisfy following conditions:
it can be proved (later) that the Dirac equation is fully consistent with relativity.
Klein-Gordon equation is consistent with relativity but not with
quantum mechanics.
we have a relativistic quantum mechanical theory!
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Dirac attempt
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Discussion of Dirac equation
to account for the spin of electron, the matrices should be 2x2
Dirac suggested the following equation for spin-one-half particles
(a state carries a spin label |ψ, a, t!, a = 1,2):
but 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)
Consistent with
Schrödinger
for the Hamiltonian:
ψ(x,
t) = !x|ψ,equation
t"
negative energy states
no ground state
also a problem for K.-G. equation
Dirac equation is linear in both time and space derivatives and so
it might be consistent with both QM and relativity.
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)
Squaring the Hamiltonian yields:
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Quantum mechanics as a quantum field theory
creation operators commute with each other, and so without loss of
generality we can consider only completely symmetric functions:
Consider Schrödinger equation for n particles with mass m, moving in an
external potential U(x), with interparticle potential
we have a theory of bosons (obey Bose-Einstein statistics).
For a theory of fermions (obey Fermi-Dirac statistics) we impose
is the position-space wave function.
It is equivalent to:
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.
satisfies S.E. for wave function!
if and only if
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and
is a quantum field and
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Lorentz invariance
its hermitian conjugate;
based on S-2
they satisfy commutation relations:
Lorentz transformation (linear, homogeneous change of coordinates):
that preserves the interval
:
is the vacuum state
All Lorentz transformations form a group:
product of 2 LT is another LT
state with one particle at
state with one particles at
identity transformation:
and another at
inverse:
Total number of particles is counted by the operator:
for inverse can be used to prove:
commutes with H!
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Infinitesimal Lorentz transformation:
How do operators and quantum fields transform?
Lorentz transformation (proper, orthochronous) is represented
by a unitary operator
that must obey the composition rule:
thus there are 6 independent ILTs: 3 rotations and 3 boosts
not all LT can be obtained by compounding ILTs!
infinitesimal transformation can be written as:
+1 proper
are hermitian operators = generators of the Lorentz group
-1 improper
from
proper LTs form a subgroup of Lorentz group; ILTs are proper!
using
and expanding both sides, keeping only linear terms in
Another subgroup - orthochronous LTs,
since
we get:
are arbitrary
ILTs are orthochronous!
general rule: each vector index undergoes its own Lorentz transformation!
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When we say theory is Lorentz invariant we mean it is invariant
under proper orthochronous subgroup only (those that can be
obtained by compounding ILTs)
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using
Transformations that take us out of proper orthochronous
subgroup are parity and time reversal:
and expanding to linear order in
we get:
These comm. relations specify the Lie algebra of the Lorentz group.
We can identify components of the angular momentum and boost
operators:
orthochronous but improper
and find:
nonorthochronous and improper
A quantum field theory doesn’t have to be invariant under P or T.
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in a similar way for the energy-momentum four vector
P µ = (H/c, P i ) we find:
using
and expanding to linear order in
Similarly:
we get:
Derivatives carry vector indices:
or in components:
in addition:
is Lorentz invariant
Comm. relations for J, K, P, H form the Lie algebra of the Poincare group.
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Finally, let’s look at transformation of a quantum scalar field:
Recall time evolution in Heisenberg picture:
this is generalized to:
P x = P µ xµ = P · x − Ht
x is just a label
we can write the same formula for x-a:
e+iP a/! e−iP x/! φ(0)e+iP x/! e−iP a/! = φ(x − a)
we define space-time translation operator:
and obtain:
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