Canonical quantization of fields Flashcards
Classical field theory
How do we switch from point mechanics to classical field theory? What will be the e. o. m. for real scalar fields?
We switch from the Lagrangian and canonical momentum to the Lagrangian density and canonical momentem density (also the Hamiltonian density).
E. o. m. for real scalar fields: Klein-Gordon eq.
- implies L = (∂φ/∂t)^2/2 - (∇φ)^2/2 - m^2 φ^2/2
- H = (∂φ/∂t)^2/2 + (∇φ)^2/2 + m^2 φ^2/2 (movement in time + sheaving in spacetime + exsistance of field)
The first two are the kinetic terms that will lead to a propagator due to relativistic invariance.
Classical field theory
What’s the consequence of Noether’s theorem in terms of local and global conserved quantities?
Local: conservation of current density
Global: conservation of charge (integral of j^0, the charge density)
The local one is stronger: dQ/dt seems 0 only if a charge and its opposite pop up simultaneously but that’s frame dependent, so Lorentz invariance forces us to use local conservation laws.
Classical field theory
Why should we switch to a complex scalar field instead of a real one and how?
Because for real scalar fields there are no internal symmetries. The Lagrangian density and the current density will be different but they will give a global charge.
- L = [∂(μ)φ] [∂^μφ] (complex) – m^2 φ(complex )φ
- j^μ = i[(∂^μφ)(complex)φ – φ(complex)(∂(μ)φ)]
Classical field theory
What is the energy-momentum tensor? What are its components?
It’s the conserved current for translations in spacetime: T^μ(ν) The formula can be constructed based on the Taylor expansion of φ(x + a) and the infinitesimal change in the Lagrangian density.
Interpretation:
- ν = 0: energy conservation
- ν = i: momentum conservation
- μ = 0: charge density components
- μ = i: current density components
Important components:
- T^0(0) = H (density)
- T^(0i) = –T^0(i) = total momentum density(?)
Constructing the scalar quantum field
What are the steps in constructing the scalar quantum field?
Procedure by analogy to the harmonic oscillator.
- Taking a real, time-indep. scalar field: φ(x,t) = φ(x), φ(conj.) = φ
- Diff. eq. of motion: KG eq.
- Fourier transform to momentum space to get rid of gradient, new eq.: φ” + ω(p)^2 φ = 0, ω(p) = √(p^2 + m^2)
- Difference from the HO: φ(conj.)(p,t) = φ(–p, t)
- Field to operator + ladder ops.: φ = 1/√(2ω(p)) (a(p) + a+(-p))
- Going back to coordinate space: φ(x) = ∫ d^3p/(2π)^3 [1/√(2ω(p))] [a(p) exp(ipx) + a+(p) exp(–ipx)]
Constructing the scalar quantum field
What’s the commutation relation between the creation and annihilation operators?
[a(p), a+(q)] = (2π)^3 δ^3(p–q)
[a(p), a(q)] = 0
[a+(p), a+(q)] = 0
Constructing the scalar quantum field
What does the term normal ordering refer to?
Shifting (redifining) the zero point energy in the Hamiltonian so that the infinite vacuum energy disappears.
Original Hamiltonian: H = ∫d^3/(2π)^3 [a+(p)a(p) + (2π)^3 δ^3(0)] + const.
Normal ordered Hamiltonian: :H: = ∫d^3/(2π)^3 [a+(p)a(p)]
[H, a(p)] = –ω(p)a(p), [H, a+(p)] = ω(p)a+(p), which are consistent w/ the Planck hypothesis .
Constructing the scalar quantum field
How can we define states and particle content for the scalar fields?
In Fock space (Hilbert space is not enough here), where each state is characterized by the amount of particles in them. Fock space can be broken up into different sectors like this: 1-particle, 2-particle, …, n-particle sectors.
- two-particle states: the creation operators commute so interchanging two particles makes no difference —» Bose-Einstein statistics
- the entire theory here describes bosonic particles
- normalization: ⟨p|q⟩ = (2π)^3 2ω(p) δ^3(p–q)
Constructing the scalar quantum field
How can we bring in time dependence for a scalar field?
We have to switch to the Heisenberg picture which is more convenient for relativistic treatment: φ(H)(x,t) = exp(iHt)φ(S)(x) exp(–iHt).
The time-dependent quantum scalar field in the Heisenberg picture:
φ(x) = ∫ d^3p/(2π)^3 [1/√(2ω(p))] [a(p) exp(–ipx) + a+(p) exp(ipx)],
where px = p0x0 – p(vec)x(vec), p0 = ω(p) = √(p(vec)^2 + m^2).
Warning: φ(x) is in the Heisenberg picture (time-dependent) but a(p) and a+(p) are still in the Schrödinger picture (time-independent).
Constructing the scalar quantum field
What’s the interpretation of the form of a scalar quantum field?
- Particle content by creation & annihilation ops
- plane wave solution to KG eq: exp(ipx), exp(–ipx) are just the things that multiply the ops
- both exp(ip0t) and exp(–ip0t) are there: a+(p) creates and a(p) annihilates a particle
- H is bounded from below because the eigenvalues of the number op. N(p) are positive integers
Constructing the scalar quantum field
How is causality satisfied for real scalar fields?
If we measure quantity A around point x and B at point y, for (x–y)² < 0 (so spacelike distances) their commutator has to be: [A, B] = 0 because two events can only be causally connected if they’re not spacelike.
- physical content: there’s no info between x and y, so they’re causally disconnected
- generally: [A, B] ≠ 0 becasue the measurement of one will influence the other
- if A = φ(x) and B = φ(y), [φ(x), φ(y)] = 0 can be proven by applying a Lorentz trafo such that (x – y)^μ —» –(x – y)^μ
The L-trafo can only be done for spacelike distances because for timelike ones we’d reflect the parabola into the past cone (insert ábra).
Complex scalar fields
How can we construct a complex scalar field? Why is such a field needed?
We need a complex scalar field because it has an internal structure unlike a real one; these allow an extra degree of freedom to be able to go beyond just the energies of the particles: φ(x) —» exp(iα)φ(x).
- L = (∂(μ)φ)(∂^(μ)φ)(conj.) – m²φ(conj.)φ
- canonical momentum: ∂(0)φ = Π —» [φ(x,t), Π(y,t)] = iδ^3(x – y)
- the field: φ(x) = ∫ d^3p/(2π)^3 [1/√(2ω(p))] [a(p) exp(–ipx) + b+(p) exp(ipx)] —» different types of particles
- Hamiltonian: :H: = ∫d^3/(2π)^3 ω(p) [a+(p)a(p) + b+(p)b(p)] ≥ 0
- states: double as many —» |0 ⟩, |p⟩(a), |p⟩(b)
- relationship between the types: charge conjugation: φ —» φ+, a «—» b, particle «—» antiparticle
- the charge here: :Q: = ∫d^3/(2π)^3 [a+(p)a(p) – b+(p)b(p)], can be > or < 0 —» electric charge
- The term containing the a operators in the field correspond to the real part and the b term to the imaginary part.
- a type particles: charge +1, b type particles: charge –1
Fermionic fields
How can we construct a fermionic field? Why is this useful?
With fermionic fields, ψ(α), we can account for the existence of nonzero spin in the states as well.
- ψ —» operator —» L = ψ¯(x) (iγ(μ)∂^(μ) − m) ψ(x) —» Dirac eq.: (iγ(μ)∂^(μ) − m)ψ(x) = 0
- 2 times 2 solution to the Dirac eq. respectively: ψ(x) = u(p)exp(ipx), ψ(x) = v(p)exp(-ipx), corresponding to spin up and down
- the fermionic field quantized: ψ(x) = Σ(s)∫ d^3p/(2π)^3 [1/√(2ω(p))] [a^s(p)u(s)(p) exp(–ipx) + b+^(s)(p)v(s)(p) exp(ipx)]
- Hamiltonian: H(density) = iψ+∂(0)iψ —» H = ∫d^3/(2π)^3 Σ(s) ω(p) [a+(s)(p)a(s)(p) – b(s)(p)b+(s)(p)]
For each spin, the coeffiecients and the operators are different because they are independent from each other.
Fermionic fields
What are the spin sums and orthogonality relations of u(s)(p) and v(s)(p)?
Spin sums:
Σ(s) u(s)(p)(α) u¯(s)(p)(β) = (p(slash) + m)(αβ),
Σ(s) v(s)(p)(α) v¯(s)(p)(β) = (p(slash) – m)(αβ)
Orthogonality:
u+(s)(p)u(r)(p) = 2ω(p)δ(rs),
v+(s)(p)v(r)(p) = 2ω(p)δ(rs)
Fermionic fields
According to what rule do we quantize fermionic fields? How does the normal ordered Hamiltonian look like?
We quantize with anticommutators: {a(s)(p), a+(r)(q)} = {b(s)(p), b+(r)(q)} = (2π)^3δ^3(p – q)δ(rs). All other commutators vanish. This is necessary to keep causality and H bounded from below.
- :H: = ∫d^3/(2π)^3 Σ(s) ω(p) [a+(s)(p)a(s)(p) + b+(s)(p)b(s)(p)]
Spin-statistics theorem: commutators for scalars and anticommutators for fermions.
For commutation relations the necessary conditions above wouldn’t be met.
Fermionic fields
What about the particle content and causality for fermionic fields?
Particle content:
- both a and b op.s annihilate the vacuum
- Pauli exclusion principle: a+(s)(p)a+(s)(p)|0⟩ = 0
- Fermi-Dirac statistics: |p(a),s; q(a), r ⟩ = –|q(a),r; p(a), s ⟩, exchanging particles lead to a factor of –1
- :Q: = ∫d^3/(2π)^3 Σ(s) [a+(s)(p)a(s)(p) – b+(s)(p)b(s)(p)] —» a type: charge +1, b type: charge –1
Causality:
- {ψ(x,t), ψ¯(y,t)} = {ψ(x), ψ¯(y)} = 0 for (x – y) ² < 0
