Anti-symmetric operator

In quantum mechanics, a raising or lowering operator (collectively known as ladder operators) is an operator that increases or decreases the eigenvalue of another operator. In quantum mechanics, the raising operator is sometimes called the creation operator, and the lowering operator the annihilation operator. Well-known applications of ladder operators in quantum mechanics are in the formalisms of the quantum harmonic oscillator and angular momentum.

Introduction

Discovered as of late, there is another type of operator in quantum field theory known as anti-symmetric operator. This operator, similar to spin in non-relativistic quantum mechanics is a ladder operator that can create two fermions of opposite spin out of a boson or a boson out of two fermions. A Fermion, named after Enrico Fermi, is a particle with a half-integer spin, such as protons and electrons. This is a matter particle. A force carrying particle, a boson, is distinguished by their integer spin.

pin

First, we will review spin for non-relativistic quantum mechanics. Spin, an intrinsic property similar to angular momentum, is defined by a spin operator S that plays a role on a system similar to the operator L for orbital angular momentum. The operators $$S^2 and $$S_z whose eigenvalues are $$S^2|s,m>=s(s+1)hbar^2|s,m> and $$S_z|s,m>=mhbar|s,m> respectively. These formalisms also obey the usual commutation relations for angular momentum $$ [S_x,S_y] =ihbar S_z, $$ [S_y,S_z] =ihbar S_x, and $$ [S_z,S_x] =ihbar S_y. The raising and lowering operators, S_+ and S_-, are defined as $$S_+=1/2(S_x+S_y) and $$S_-=1/2i(S_x-S_y) respectively. These ladder operators act on the state in the following $$S_+|s,m>=hbar sqrt{s(s+1)-m(m+1)}|s,m+1> and $$S_-|s,m>=hbar sqrt{s(s+1)-m(m-1)}|s,m-1> respectively.

The operators S_x and S_y can be determined using the ladder method. In the case of the spin 1/2 case (fermion), the operator $$S_+ acting on a state produces $$S_+|+>=0 and $$S_+|->=hbar|+>. Likewise, the operator $$S_- acting on a state produces $$S_-|->=0 and $$S_-|+>=hbar|->. The matrix representations of these operators are constructed as follows:

: $$ [S_+] = egin{bmatrix}<+|S_+|+> & <+|S_+|-> \<-|S_+|+> & <-|S_+|-> end{bmatrix}=hbar cdotegin{bmatrix}0 & 1 \0 & 0 end{bmatrix}

: $$ [S_-] = egin{bmatrix}<+|S_-|+> & <+|S_-|-> \<-|S_-|+> & <-|S_-|-> end{bmatrix}=hbar cdotegin{bmatrix}0 & 0 \1 & 0 end{bmatrix}

Therefore $$S_x and $$S_y can be represented by the matrix representations:

:$$ [S_x] = frac{ hbar}{2} cdotegin{bmatrix}0 & 1 \1 & 0 end{bmatrix}

:$$ [S_y] = frac{ hbar}{2} cdotegin{bmatrix}0 & -i \i & 0 end{bmatrix}

Recalling the generalized uncertainty relation for two operators A and B, $$Delta_{psi} A , Delta_{psi} B ge frac{1}{2} left|leftlangleleft [{A},{B} ight] ight angle_psi ight
, we can immediately see that the uncertainty relation of the operators $$S_x and $$S_y are as follows:

:$$Delta_{psi} S_x , Delta_{psi} S_y ge frac{1}{2} left|leftlangleleft [{S_x},{S_y} ight] ight angle_psi ight
=frac{1}{2} (i hbar S_z)=frac{ hbar}{2} S_z

Therefore, like orbital angular momentum, we can only specify one coordinate at a time. We specify the operators $$S^2 and $$S_z.

Application in Quantum Field Theory

The creation of a particle and anti-particle from a boson is defined similarly but for infinite dimesions. Therefore the Levi-Civita symbol for infinite dimensions is introduced.

:$$varepsilon_{ijkelldots} =left{egin{matrix}+1 & mbox{if }(i,j,k,ell,dots) mbox{ is an even permutation of } (1,2,3,4,dots) \-1 & mbox{if }(i,j,k,ell,dots) mbox{ is an odd permutation of } (1,2,3,4,dots) \0 & mbox{if any two labels are the same}end{matrix} ight.

The commutation relations are simply carried over to infinite dimensions $$ [S_i,S_j] =ihbar S_kvarepsilon_{ijk}. $$S^2 is now equal to $$S^2=sum_{m=1}^n S_m^2 where n=∞. It's eigenvalue is $$S^2|s,m>=s(s+1)hbar^2|s,m> . Defining the magnetic quantum number, angular momentum projected in the z direction, is more challenging than the simple state of spin. The problem becomes analogous to moment of inertia in classical mechanics and is generalizable to n dimensions. It is this property that allows for the creation and annihilation of bosons.

Bosons

Characterized by their spin, a bosonic field can be scalar fields, vector fields and even tensor fields. To illustrate, the electromagnetic field quantized is the photon field, which can be quantized using conventional methods of canonical or path integral quantization. This has led to the theory of quantum electrodynamics, arguably the most successful theory in Physics. The graviton field is the quantized gravitational field. There is yet to be a theory that quantizes the gravitational field, but theories such as string theory can be thought of the gravitational field quantized. An example of a non-relativistic bosonic field is that describing cold bosonic atoms, such as Helium-4. Free bosonic fields obey commutation relations:

:$$ [a_i,a_j] = [a^dagger_i,a^dagger_j] =0:$$ [a_i,a^dagger_i] =langle f|g angle,

To illustrate, suppose we have a system of N bosons that occupy mutually orthogonal single-particle states $$ |phi_1 ang, |phi_2 ang, |phi_3 ang , etc. Using the usual representation, we demonstrate the system by assigning a state to each particle and then imposing exchange symmetry.

:$$frac{1}{sqrt{3 left [ |phi_1 ang |phi_2 ang
phi_2 ang + |phi_2 ang |phi_1 ang |phi_2 ang + |phi_2 ang
phi_2 ang |phi_1 ang ight] .

We can represent this wave equation using a second quantized approach, known as second quantization. The number of particles in each single-particle state is listed.

:$$ |1, 2, 0, 0, 0, cdots angle,

The creation and annihilation operators, which add and subtract particles from multi-particle states. These creation and annihilation operators are very similar to those defined for the quantum harmonic oscillator, which added and subtracted energy quanta. However, these operators literally create and annihilate particles with a given quantum state. The bosonic annihilation operator $$a_2 and creation operator $$a_2^dagger have the following effects:

:$$a_2 | N_1, N_2, N_3, cdots angle = sqrt{N_2} mid N_1, (N_2 - 1), N_3, cdots angle,:$$a_2^dagger | N_1, N_2, N_3, cdots angle = sqrt{N_2 + 1} mid N_1, (N_2 + 1), N_3, cdots angle.

Like the creation and annihilation operators $$a_i and $$a_i^dagger also found in Quantum Field Theory, the creation and annihilation operators $$S_i^+ and $$S_i^- act on bosons in multi-particle states. While $$a_i and $$a_i^dagger allows us to determine whether a particle was created or destroyed in a system, the spin operators $$S_i^+ and $$S_i^- allow us to determine how. A photon can become both a positron and electron and vice versa. Because of the anti-symmetric statistics, a particle of spin $$frac{1}{2} obeys the Pauli-Exclusion Rule. Two particles can exist in the same state if and only if the spin of the particle is opposite.

Back to our example, the spin state of the particle is spin-1. Symmetric particles, or bosons, need not obey the Pauli-Exclusion Principle so therefore we can represent the spin state of the particle as follows:

:$$ |1, ix, 0, 0, 0, cdots angle, and $$ |1, -ix, 0, 0, 0, cdots angle,

The annihilation spin operator, as its name implies, annihilates a photon into both an electron and positron. Likewise, the creation spin operator creates a photon. The photon can be in either the first state or the second state in this example. If we apply the linear momentum operator

Bosonization

Fermions

Therefore we define the operator $$S_i + and $$S_i -. In the case of the non-relativistic particle, if $$S_+ is applied to a fermion twice, the resulting eigenvalue is 0. Similarly, the eignenvalue is 0 when $$S_- is applied to a fermion twice. This relation satisfies the Pauli Exclusion Principle. However, bosons are symmetric particles, which do not obey the Pauli Exclusion Principle.

References

*cite book | author=Griffiths, David J.|title=Introduction to Quantum Mechanics (2nd ed.) | publisher=Prentice Hall |year=2004 |id=ISBN 0-13-111892-7

*cite book | author=McMahon, David|title=Quantum Mechanics DeMystified: A Self-Teaching Guide |publisher=The McGraw-Hill Companies |year=2006 |id=ISBN 0-07-145546-9