Introduction to the Second Quantization

Tien-Tsan Shieh

Institute of Mathematics Academic Sinica

July 27, 2011

Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 1 / 35 Identical particles Examples of quantum many body problems: I Photons in a block-body cavity I Electrons in a solid I Nucleons in a nucleus I Neutrons in a neutron star I Helium atoms in a superﬂuid The concept of identical particles: I Intrinsic: Those do not depend on the dynamical state of the particle, such as the rest mass, the charge, or the total spin. I Extrinsic: Those depend on the state of the particle or on its preparation: the position, and velocity, spin orientation and other internal parameters. Here two particles are considered identical if they possess the same intrinsic properties. More mathematically speaking, we say n particles are identical if all O(x1, p1; ..., ; xn, pn) are invariant under the exchange of their coordinates (xi , pi ), i = 1,..., n. n 2 X |pi | X H(n) = H (n) + V (n) = + V (|x − x |). 0 2m i j i=1 i

with the inner product Z hϕ|ψi = dxϕ∗(x)ψ(x).

If a particle posses a spin s, its wavefunction will have 2s + 1 components ϕ(x, σ), σ = −s, −s + 1,..., s − 1, s + 1. i.e.  ϕ(x, −s)   ϕ(x, −s + 1)     .  2s+1 ϕ(x) =  .  ∈ H ⊗ C .    ϕ(x, s − 1)  ϕ(x, s) Now, I will focus on the case s = 0. Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 3 / 35 One-particle system: Observables Position q and momentum p are linear operators on H :

(qϕ)(x) = xϕ(x) ∂ ∂ ∂ (pϕ)(x) = −i ∇ϕ(x), ∇ = , , ~ ∂x1 ∂x2 ∂x3 The canonical commutation relations

α β [q , p ] = i~δα,β, α, β = 1, 2, 3. In the Fourier representation 1 Z ϕ(k) = dx e−ik·xϕ(x) e (2π)3/2 the position operator q and the momentum operator q becomes

(qϕe)(k) = −i∇kϕe(k) (pϕe)(k) = ~kϕ(k)

Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 4 / 35 One-particle system: Evolution equations

The evolution of the particle is governed by Schrodinger equation ∂ i ϕ(x, t) = (Hϕ)(x, t) ~∂t where H is the given Hamiltonian. The formal solution is given by

ϕ(x, t) = (U(t)ϕ)(x, 0), U(0) = I

where U(t) is the evolution operator (semi-group on H). For example, if the external ﬁelds are independent of time, then

itH U(t) = exp − ~

∗ −1 U (t) = U (t), U(t1)U(t2) = U(t1 + t2).

Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 5 / 35 n-particle system: States The collection of the states of n non-relativistic spinless particles form a Hilbert space Z n 2 3n 2 H = L (R ) = {Φ(x1,..., xn); dx1 ... dxn|Φ(x1,..., xn)| < ∞}.

If the particles carry a spin s, the Hilbert space of states is n 2s+1 H ⊗ C . Now we are going to construct the n-particle states in Hn from the one-particle states.

hx1,..., xn|ϕ1 ⊗ ..., ⊗ϕni = ϕ1(x1) . . . ϕn(xn)

Here |ϕi i, i = 1,... n are the single-particle state in H. We interpret it as the joint probability of the state |ϕ1 ⊗ ..., ⊗ϕni to ﬁnd particle 1 in x1, ... , and particle n in xn. The inner product of |ϕ1 ⊗ ..., ⊗ϕni and |ψ1 ⊗ ..., ⊗ψni is

hψ1 ⊗ · · · ⊗ ψn|ϕ1 ⊗ ..., ⊗ϕni = hϕ1|ψ1ihϕ2|ψ2i ... hϕn|ψni

Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 6 / 35 n-particle system: Basis

Suppose {|ϕi i, i = 1, 2,... } form a basis in the Hilbert space of a single-particle system. We know

{|ϕj1 ⊗ ..., ⊗ϕjn i; j1, j2, ··· = 1, 2,... } form a basis of the space H⊗n = H ⊗ · · · ⊗ H. For all wavefunction Φ(x1,..., xn), we can express as X Φ(x1,..., xn) = cj1...jn ϕj1 (x1) . . . ϕjn (xn). j1...jn We could also write as X hx1,..., xn|Φi = cj1...jn hx1,..., xn|ϕj1 ⊗ · · · ⊗ ϕjn i. j1...jn X |Φi = cj1...jn |ϕj1 ⊗ · · · ⊗ ϕjn i. j1...jn This means we can identify n-particle space Hn with the tensor product H⊗n.

Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 7 / 35 n-particle system: One-body observables

Observables of a n-identical-particle system can be constructed from observalbes in a single-particle system. If A is a single-particle observable in H, we deﬁne the action A on the jth factor of the tensor product H⊗n as

(I ⊗· · ·⊗A⊗· · ·⊗I )|ϕ1⊗· · ·⊗ϕj ⊗· · ·⊗ϕni = |ϕ1⊗· · ·⊗Aϕj ⊗· · ·⊗ϕni

and denote it as Aj . The corresponding extensive observable for the n-particle system is deﬁned by n X A(n) = Aj . j=1

Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 8 / 35 n-particle system: Two-body observables

Let V be an observable in two-particle space H ⊗ H, such as interaction potential. ⊗n ⊗n Deﬁne Vij on H as the operator acts on ith and jth factors of H and trivially on the others. The corresponding observable for a n-particle system is

n X V (n) = Vij . i

If V is acting on two identical particles, we have Vij = Vji . Therefore we have n 1 X V (n) = V . 2 ij i6=j

Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 9 / 35 Symmetrization: Permutation operator

⊗n Permutation operator Pπ actin on H is

Pπ|ϕ1 ⊗ · · · ⊗ ϕni = |ϕπ(1) ⊗ · · · ⊗ ϕπ(n)i

1 2 ... n where π = . π(1) π(2) . . . π(n) Some properties:

∗ −1 Pπ1π2 = Pπ1 Pπ2 , Pπ = Pπ

Let A be an observable on a single-particle space H. We have

[A(n), Pπ] = 0 for all permutation π

Pn where A(n) = j=1 Aj .

Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 10 / 35 Symmetrization: Exchange degeneracy

Suppose |Φi is an eigenstate of a n-identical-particle system. i.e.

H(n)|Φi = E|Φi

We will ﬁnd

H(n)Pπ|Φi = EPπ|Φi for all permutation π

In a n-identical-particle system, Pπ|Φi and their linear combinations describe the same physical state. Principle of symmetrization: the state of n identical particles are either completely symmetric or completely antisymmetic. Bosons are particles in the symmetric case. Fermions are particles in the antisymmetic case.

Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 11 / 35 Symmetrization: Symmetric and Antisymmetric States For any |Φi in H⊗n, we can construct corresponding one which is totally symmetric or totally antisymmetric:

|Φi+ = S+|Φi

|Φi− = S−|Φi

where two operators S+ and S− deﬁned by 1 X 1 X S = P , S = (−1)πP . + n! π − n! π π∈Pn π∈Pn Properties: 2 2 ∗ S+ = S+, S− = S− = S−, S+S− = S−S+ = 0. π Pπ = S+Pπ = S+, PπS− = S−Pπ = (−1) S−. According to the principle of symmetrization, the space of state of a system of n identical particle particles is not entire H⊗n. ⊗n ⊗n I Space for bosons (symmetric): H+ ≡ S+H . ⊗n ⊗n I Space for fermions (anti-symmetric): H− ≡ S−H .

Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 12 / 35 Hilbert spaces for bosons or fermions ⊗n ⊗n Denote the symmetric or antisymmetric state of H+ or H− by

|ϕ1, ϕ2, . . . , ϕniν ≡ Sν|ϕ1 ⊗ ϕ2 ⊗ · · · ⊗ ϕni where ν = 1 for bosons and ν = −1 for fermions. Exchanging the ith and jth components, we ﬁnd

|ϕ1, . . . , ϕi , . . . , ϕj , . . . , ϕni+ = |ϕ1, . . . , ϕj , . . . , ϕi , . . . , ϕni+

|ϕ1, . . . , ϕi , . . . , ϕj , . . . , ϕni− = −|ϕ1, . . . , ϕj , . . . , ϕi , . . . , ϕni− The inner product is 1 X hϕ , . . . , ϕ |ψ , . . . , ψ i = (ν)πhϕ |ψ i ... hϕ |ψ i. ν 1 n 1 n ν n! 1 π(1) n π(n) π∈Pn In particular for the fermions case, 1 hϕ , . . . , ϕ |ψ , . . . , ψ i = det{hϕ |ψ i}. − 1 n 1 n − n! i j

Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 13 / 35 Occupation Number Representation

Let {|ψr i} be a basis of a single-particle space H. The occupation number normalized to 1 are deﬁned by √ n! |n1, n2,..., nr ,... i+ = √ S+|ψr1 ⊗ · · · ⊗ ψrn i n1! ... nr ! ... √ |n1, n2,..., nr ,... i− = n! S−|ψr1 ⊗ · · · ⊗ ψrn i

where nr is the number of indices r1,..., rn which are equal to r.

For bosons, nr = 0, 1, 2, 3,... . For fermions, nr = 0, 1. P The system has exactly n = r nr particles.

Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 14 / 35 Occupation Number Representation: Observables

Let |ψr i are the eigenvector of a single-particle observable A such that

Therefore if [Sν, A(n)] = 0, we have ! X A(n)|n1, n2,..., nr ,... iν = nr αr |n1, n2,..., nr ,... iν. r Orthogonality:

νhm1, m2,..., mr ,... |n1, n2,..., nr ,..., iν = δm1,n1 δm2,n2 . . . δmr ,nr ... ⊗n Basis: {|n1, n2,... iν} form a basis of Hν . Since |Φi = P c S |ψ ⊗ · · · ⊗ ψ i. ν r1,...,rn c1,...,rn ν r1 rn

Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 15 / 35 Systems with Variable Particle Numbers

In many physical systems, the number of particles of a given type is not exactly known or may vary during reactions. A photon gas Phonon in condensed-matter In high-energy physics, particles may be created or annihilated.

Because of these reasons, it is better to construct a Hilbert space of many-particle system which allowing undetermined number of particles. Such construction is called the second quantization.

Note that the second quantization does not introduce any new theory. Conceptually, it is just an useful formulation of quantum mechanics.

Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 16 / 35 Fock Space ⊗n Let H be a Hilbert space of single-particle states and Hν be the corresponding Hilbert space of n identical particles. (ν = 1 for bosons, ν = −1 for fermions) 0 Vacuum state: H = {λ|Φ(0)i; λ ∈ C} where |Φ(0)i = |0i is a unit vector.

0 ⊗1 ⊗n V = H ⊕ Hν ⊕ · · · ⊕ Hν ⊕ .... The elements |Φi in the space V take the form

|Φi = {|Φ(0)i, |Φ(1)i, |Φ(2)i,..., |Φ(n)i,... } = {|Φ(n)}n. Addition and scalar multiplication:

|Φi + |Ψi ≡ {|Φ(n)i + |Ψ(n)i}n

λ|Φi ≡ {λ|Φ(n)}n Inner product on the space V: ∞ X hΦ|Ψi ≡ hΦ(n)|Ψ(n)i n=0 ⊗n where hΦ(n)|Ψ(n)i is an inner product in Hν . Fock space: Fν (H) = {|Φi ∈ V; hΦ|Φi < ∞}

Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 17 / 35 Observables in Fock space

Let A(n) be an observable for n particle. The corresponding observable A acting on Fock space is given by

∞ X A|Φi = A(n)|Φ(n)i. n=0 The vacuum sate is an eigenvector of all extensive observables with eigenvalue zero. P∞ The particle number operator N is deﬁned by N|Φi ≡ n=0 n|Φ(n)i. Expectation of A:

Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 18 / 35 Basis in Fock space

The collection of {|ϕ1, . . . , ϕniν} generates H and the vacuum state |0i generate the Fock space Fν(H) when {|ϕi i} form a basis of a single-particle space H. Occupation number representation: Given a basis {|ϕi } of one-particle space H, the collection of X |n1, n2,..., nr ,... iν, nr < ∞ r form a basis in the Fock space.

I for bosons, nr = 0, 1, 2,... . I for fermions, nr = 0, 1.

Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 19 / 35 Creation and Annihilation Operators

∗ The creation operator a (ϕ) of one particle in the state |ϕiν is deﬁned by √ ∗ a (ϕ)|ϕ1, . . . , ϕniν ≡ n + 1|ϕ, ϕ1, . . . , ϕniν

The annihilation operator a(ϕ) of a particle in the state |ϕi is deﬁned by a(ϕ) ≡ (a∗(ϕ))∗ where the second ∗ means the Hermitian conjugate.

Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 20 / 35 Properties of creation operators and annihilation operators a∗(ϕ) is linear but a(ϕ) is anti-linear: ∗ ∗ ∗ a (λ1|ϕ1i + λ2|ϕ2i) = λ1a (ϕ1) + λ2a (ϕ2) ∗ ∗ a(λ1|ϕ1i + λ2|ϕ2i) = λ1a(ϕ1) + λ2a(ϕ2) Inner product:

ν hψ1, . . . , ψm|a(ϕ)|ϕ1, . . . , ϕniν √ = n + 1ν hϕ, ψ1, . . . , ψm|ϕ1, . . . , ϕniν  0 for m 6= n − 1,    n √1 P i−1 = n i=1(ν) ×  ν hψ1, . . . , ψn−1|ϕ1, . . . , ϕi−1, ϕi+1, . . . , ϕniν hϕ|ϕi i.   for m = n − 1 Therefore, a(ϕ) decreases the number of particles by one unit, while preserving the symmetry of the state n 1 X a(ϕ)|ϕ , . . . , ϕ i = √ (ν)i−1|ϕ , . . . , ϕ , ϕ , . . . , ϕ i hϕ|ϕ i 1 n ν n 1 i−1 i+1 n ν i i=1

Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 21 / 35 Commutation and Anti-commutation Relations

a∗(ϕ)a∗(ψ) − a∗(ψ)a∗(ϕ) = 0 (bosons)

a∗(ϕ)a∗(ψ) + a∗(ψ)a∗(ϕ) = 0 (fermions)

a(ψ)a∗(ϕ) − a∗(ϕ)a(ψ) = hψ|ϕiI (bosons)

a(ψ)a∗(ϕ) + a∗(ϕ)a(ψ) = hψ|ϕiI (fermions)

Here I is the identity operator in Fν(H). Let {|ψr i} be a given ﬁxed basis of one-particle space. Set ∗ ∗ ar = a (ψr ) and ar = a(ψr ). The commutation and anti-commutation relation become ∗ ∗ ∗ ∗ ar as ∓ as ar = 0

ar as ∓ as ar = 0 ∗ ∗ ar as ∓ as ar = δr,s Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 22 / 35 Formation of States from the Vacuum (I) All states of n independent particles can be obtained by successive application of the creation operators on the vacuum. 1 |ϕ , . . . , ϕ i = √ a∗(ϕ ) ... a∗(ϕ )|0i. 1 n ν n 1 n

The state is symmetric or antisymmetric depending on whether ∗ a (ϕi ) is commute or anti-commute. All of states in Fock space is their linear combination of such states or their limit. Occupation number representation relative to the single-particle basis ∗ ∗ {|ψr i} and ar = a (|ϕr i):

∗ n1 ∗ nr (a1) (a1) nr = 0, 1, 2,... (bosons) |n1,..., nr ,... i = √ ... √ ... |0i for n1! nr ! nr = 0, 1 (fermions)

Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 23 / 35 Formation of States from the Vacuum (II)

For bosons,

∗ √ a |n1,..., nr ,... i+ = nr + 1 |n1,..., nr + 1,... i+ r √ ar |n1,..., nr ,... i+ = nr |n1,..., nr − 1,... i+

For fermions,

∗ γr ar |n1,..., nr ,... i− = (1 − nr )(−1) |n1,..., nr + 1,... i− γr ar |n1,..., nr ,... i+ = nr (−1) |n1,..., nr − 1,... i−

Pr−1 where γr = s=1 ns . For both bosons and fermions, the occupation number states are ∗ eigenstates for all of the operations ar ar :

∗ ar ar |n1,..., nr ,... iν = nr |n1,..., nr ,... iν

Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 24 / 35 Coherent States of Bosons

States of type 1 |ϕ , . . . , ϕ i = √ a∗(ϕ ) ... a∗(ϕ )|0i 1 n ν n 1 n have a determined number of particles. This is typical for the case of a scattering process in which the incoming particles are speciﬁed and the particles produced by the reaction are observed. States of type consist of superposition of n-particle state with n = 0, 1, 2,...

A coherent state represents a state of photons (or phonons) associated with a classical electromagnetic (or acoustic) wave.

One has a(ψ) |ϕicoh = (hψ|ϕi) |ϕicoh.

Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 25 / 35 One-Body Operators in Fock Space

Consider a single-particle observable A. The operator Pn A(n) = j=1 Aj is the associated observable of n-particle system. The corresponding observable A in the Fock space is

n X A |ϕ1, . . . , ϕniν = |ϕ1, . . . , ϕj−1, Aϕj , ϕj+1, . . . , ϕniν j=1

Let {|ψr i} be some basis of single-particle space.

X ∗ X ∗ A = a (Aψr )a(ψr ) = hr|A|si as ar . r r,s

If |ψr i are chosen as eigenvector of A with eigenvalue αr , then we have X ∗ A = αr ar ar . r

Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 26 / 35 Some typical one-body operators in Fock space (I)

Particle-number operator

X ∗ N = a (ψr )a(ψr ) r If we express the operator N in terms of the basis |xσi of the conﬁguration representation

Z X Z N = dx a∗(x, σ)a(x, σ) = dx n(x) Λ σ Λ P where the particle density in x is n(x) = σ n(x, σ) and the particle density of particles with spin σ is n(x, σ) = a ∗ (x, σ)a(x, σ).

Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 27 / 35 Some typical one-body operators in Fock space (II)

Kinetic energy

I The Hamiltonian of free particles

X ∗ H0 = k akσakσ kσ

where k is the kinetic energy of the particle. I The momentum of free particles

X ∗ P = ~k akσakσ kσ

Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 28 / 35 Some typical one-body operators in Fock space (III)

Consider a system of particles under an external potential V (x) which is independent of spin. In the basis |xσi

0 0 0 hxσ|V |x σ i = δ(x − x )δσ,σ0 V (x)

The representation of the external potential V is Z V = dx V (x)n(x). Λ In the basis of plane wave |kσi

0 0 1 0 hkσ|V |k σ i = δσ,σ0 Ve (k − k ) L3

R −ik·x where Ve (k) = λ dx e V (x).

Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 29 / 35 Some typical one-body operators in Fock space (IV) In the basis of plane wave |kσi

0 0 1 0 hkσ|V |k σ i = δσ,σ0 Ve (k − k ) L3 R −ik·x where Ve (k) = λ dx e V (x).

1 X X 0 ∗ V = Ve (k − k )akσak0σ L3 kk0 σ

1 X X ∗ = Ve (k)a(k+k0)σak0σ L3 kk0 σ

1 X ∗ = n (k)Ve (k) L3 e k where

Z −ik·x X ∗ X ∗ en(k) = dx e a (x, σ)a(x, σ) = ak0σa(k0+k)σ Λ σ k0σ X ∗ ∗ = a(k0−k)σak0σ = en (−k). 0 Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) tok theσ Second Quantization July 27, 2011 30 / 35 Two-Body Operators

Consider an operator V acting on two-particle space H ⊗ H.

n X V|ϕ1, . . . , ϕniν = Vij |ϕ1, . . . , ϕniν. i

We express V in terms of the creation and annihilation operators

1 X V = hr r |V|s s i a∗ a∗ a a . 2 1 2 1 2 r1 r2 s1 s1 r1r2s1s2

Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 31 / 35 Two-Body Operator: Conﬁguration Representation

Consider a two-body potential V (x1 − x2) which is invariant under translations and independent of spin.

0 0 0 0 hx σ , x σ |V |x σ , x σ i = V (x − x )δ(x − x )δ(x − x )δ 0 δ 0 . 1 1 2 2 1 1 2 2 1 2 1 1 2 2 σ1,σ1 σ2,σ2 Therefore we have Z Z ! 1 X ∗ ∗ V = dx1 dx2 V (x1 − x2) × a (x1, σ1)a (x2, σ2)a(x2, σ2)a(x1, σ1). 2 Λ Λ σ1σ2

Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 32 / 35 Two-Body Operator: Momentum Representation Here we express V in the plane-wave basis |kσi with periodic boundary conditions.

0 0 1 X hk1σ1, k2σ2|V |k σ1, k σ2i = δσ ,σ0 δσ ,σ0 VeL(k)δk ,k0 +kδk ,k0 +k. 1 2 1 1 2 2 L3 1 1 2 2 k R −ik·x where VeL = Λ dx e V (x). Therefore we can write

1 X X ∗ ∗ V = VeL(k) a a ak σ ak σ . 2L3 (k1+k)σ1 (k2−k)σ2 2 2 1 1 k1k2k σ1σ2

Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 33 / 35 Reference

Ph. A. Martin and F. Rothen, Many-Body Problems and Quantum Field Theory — An Introduction

Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 34 / 35 Thank you for your attention!

Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 35 / 35