Grassmann Variables, Supersymmetry and Supersymmetric ...

Fermionic Path Integral 

SUSY 

References 

The end 

Grassmann Variables, Supersymmetry and 

Supersymmetric Harmonic Oscillator 

Anna Pachol 

Institute of Theoretical Physics 

University of Wroclaw 

Wroclaw, 17.01.2008 

Anna Pachol GV,SUSY ans SHO

Grassmann Variables 


SUSY 

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Path Integral 

Let suppose η, ξ are two grassmann variables: 

ηξ = −ξη ⇐⇒ {η, ξ} = 0 




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Path Integral 


ηξ = −ξη ⇐⇒ {η, ξ} = 0 

so η 2 = 0. 




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Path Integral 


ηξ = −ξη ⇐⇒ {η, ξ} = 0 

so η 2 = 0. 

General function: 

f (η) = a + bη 

(there is no more terms because η 2 = 0). 


Integration 


SUSY 

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Path Integral 

In analogy to integral with ordinary variables: 

� +∞ � +∞ 

dxf (x) = dxf (x + c) 

−∞ 

−∞ 


Integration 


SUSY 

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Path Integral 


� +∞ � +∞ 


−∞ 

−∞ 

we can derive rules for integrals with anticommuting variables: 


Integration 


SUSY 

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Path Integral 


� +∞ � +∞ 


−∞ 

−∞ 

we can derive rules for integrals with anticommuting variables: 

� 

� 

� 

� 

dηf (η) = dηf (η+ξ) = dη(a+b(η+ξ)) = 

Anna Pachol GV,SUSY ans SHO 

� 

dη(a+bη)+ 

dηbξ

Integration 

� 

� 

dη(a + bη) + 


SUSY 

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� 

dηbξ = 


Path Integral 

� 

dηf (η) + 


dηbξ . � 

= dηf (η)

Integration 

� 

� 

dη(a + bη) + 


SUSY 

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� 

dηbξ = 


Path Integral 

� 

dηf (η) + 

dηbξ . � 

= dηf (η) 

Rules for integrals with anticommuting variables are following: 


Integration 

� 

� 

dη(a + bη) + 


SUSY 

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� 

dηbξ = 


Path Integral 

� 

dηf (η) + 

dηbξ . � 

= dηf (η) 


� 

dηbξ = 0, ∀b − ordinary, ξ − grassmanian 


Integration 

� 

� 

dη(a + bη) + 


SUSY 

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� 

dηbξ = 


Path Integral 

� 

dηf (η) + 

dηbξ . � 

= dηf (η) 


� 


� 

dη = 0 


Integration 

� 

� 

dη(a + bη) + 


SUSY 

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� 

dηbξ = 


Path Integral 

� 

dηf (η) + 

dηbξ . � 

= dηf (η) 


� 


� 

� 

dηη = 1 and 

dη = 0 

� 

� 

dθ 


dηηθ = 1


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Integration and Differentiation 

Integration 


Path Integral 

� 

� 

� 

dη(a + bη) = dηbη, for b − ordinary, = dηηb = b 



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Integration 


Path Integral 

� 

� 

� 


� 

� 

� 

dη(a+bη) = dηbη, for b−grassmanian, = dηη(−b) = −b 



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Integration 


Path Integral 

� 

� 

� 


� 

� 

� 


Differentiation 



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Integration 


Path Integral 

� 

� 

� 


� 

� 

� 


Differentiation 

b for b-ordinary 

d 

d 

dη f (η) = dη (a + bη) ↗ 

↘ −b, for b-grassmanian 



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Complex Grassmann variables 


Path Integral 

Since Dirac field is complex-valued, we have to introduce complex 

Grassmann numbers: η, θ and define the complex conjugation to 

reverse the order of products: (θη) ≡ ηθ = −θη 



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Complex Grassmann variables 


Path Integral 

Since Dirac field is complex-valued, we have to introduce complex 

Grassmann numbers: η, θ and define the complex conjugation to 

reverse the order of products: (θη) ≡ ηθ = −θη 

To integrate over complex Grassmann numbers, let us define: 

η = η1 + iη2 

√ 2 

and η = η1 − iη2 

√ 2 

now we can treat η; η as independent Grassmann numbers and 

adopt the convention � dη � dηηη = 1 


Gaussian integral 


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Path Integral 

η, η − grassman variables; a − ordinary 

� � 

dη dηe ηaη 




SUSY 

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Path Integral 


� � 


� 

= 

� 

dη 

dη(1 + ηaη) 



� 

= 


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Path Integral 


� � 


� 

dη 

� 

= 

� 

dη 

� 

dη ηaη = 

dη(1 + ηaη) 

� 

dηaη = a 


dηη = a = e lna

Generalisation 


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Path Integral 

In n-dimensions: η = (η1, η2, ..., ηN) and η T = (η 1, η 2, ..., η N) 




SUSY 

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Path Integral 


Notation: 

� � � � 

dη1 dη1 dη2 dη2... � � 

dηN dηN = �N i=1 


� � 

dηi dηi



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Path Integral 


Notation: 

� � � � 

dη1 dη1 dη2 dη2... � � 


� Ni=1 η iAijηj = ηAη 


� � 

dηi dηi



SUSY 

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Path Integral 


Notation: 

� � � � 

dη1 dη1 dη2 dη2... � � 


� Ni=1 η iAijηj = ηAη 

i=1 

Now we can prove: 

N� 

� � 

dηi dηie ηAη = detA 


� � 

dηi dηi

Proof 


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Path Integral 

Let assume A can be diagonalized by U-unitary: 

A = U † DU 

where D is diagonal. 


Proof 


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Path Integral 


i=1 

A = U † DU 


N� 

� � 

dηi dηie ηAη N� 

� � 

= dηi dηie ηU† DUη 

i=1 


Proof 


SUSY 

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Path Integral 


i=1 

A = U † DU 


N� 

� � 


� � 

= dηi dηie ηU† DUη 

i=1 

ξ = ηU † ; ξ = Uη ⇔ η = ξU; η = U † ξ 


Proof 

i=1 


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i=1 


Path Integral 

N� 

� � 


� � 

= dξi dξ ie ξDξ N� 

� � 

= dξi 


i=1 

� 

dξ i ie 

λi ξi ξi =

Proof 


SUSY 

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Path Integral 

N� 

� � 

dηi dηie i=1 

ηAη N� 

� 

= 

� 

dξi dξ ie 

i=1 

ξDξ N� 

� 

= 

� 

dξi 

� 

dξ i ie 

i=1 

λi ξi ξi = 

N� 

� � 

dξi 

� 

dξ i e 

i=1 

i 

λi 

N� 

� 

ξi ξi = 

� 

dξi 

� 

dξ i (1 + λiξ iξi) = 

i=1 

i 


Proof 


SUSY 

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Path Integral 

N� 

� � 


ηAη N� 

� 

= 

� 

dξi dξ ie 

i=1 

ξDξ N� 

� 

= 

� 

dξi 

� 

dξ i ie 

i=1 

λi ξi ξi = 

N� 

� � 

dξi 

� 

dξ i e 

i=1 

i 

λi 

N� 

� 

ξi ξi = 

� 

dξi 

� 


i=1 

i 

N� 

� 

λi 

� 

dξi 

N� 

dξ i ξiξi = λi = detA 

i=1 

i=1 


Proof 


SUSY 

References 

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Path Integral 

N� 

� � 


ηAη N� 

� 

= 

� 

dξi dξ ie 

i=1 

ξDξ N� 

� 

= 

� 

dξi 

� 

dξ i ie 

i=1 

λi ξi ξi = 

N� 

� � 

dξi 

� 

dξ i e 

i=1 

i 

λi 

N� 

� 

ξi ξi = 

� 

dξi 

� 


i=1 

i 

N� 

� 

λi 

� 

dξi 

N� 

dξ i ξiξi = λi = detA 

i=1 

i=1 

Result � 

N� 

� 

i=1 

� 

dηi dηie ηAη = detA = e TrlnA 



SUSY 

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Path Integral 

One missing step - transformation of measure 

when we do transformation of variables: 

η ′ i = Uijηj 



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Path Integral 




the product of new (grassmann) variables is following: 

n� 

i=1 

η ′ i = 1 

n! ɛij...lη ′ iη ′ j...η ′ l = 1 

n! ɛij...lUii ′ηi ′Ujj ′ηj ′...Ull ′ηl ′ = 



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Path Integral 





n� 

i=1 

η ′ i = 1 

n! ɛij...lη ′ iη ′ j...η ′ l = 1 


= 1 

n! ɛij...lUii ′Ujj ′...Ull ′ηi ′ηj ′...ηl ′ 



SUSY 

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Path Integral 





n� 

i=1 

η ′ i = 1 

n! ɛij...lη ′ iη ′ j...η ′ l = 1 


= 1 

n! ɛij...lUii ′Ujj ′...Ull ′ηi ′ηj ′...ηl ′ 

but ηi ′ηj ′...ηl ′ = ɛi ′ j ′ ...l ′ � n i=1 ηi 

and detU = 1 

n! ɛij...l Uii ′Ujj ′...Ull ′ɛi ′ j ′ ...l ′ 



SUSY 

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transformation of measure 

so 


Path Integral 

n� 

η 

i=1 

′ n� 

i = detU ηi 

i=1 



SUSY 

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η ′ i = η i(U † )ij =⇒ 

so 


Path Integral 

n� 

η 

i=1 

′ n� 

i = detU ηi 

i=1 

and for 

n� 

η 

i=1 

′ i = detU † 

n� 

ηi i=1 



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η ′ i = η i(U † )ij =⇒ 

so 


Path Integral 

n� 

η 

i=1 

′ n� 

i = detU ηi 

i=1 

and for 

n� 

η 

i=1 

′ i = detU † 

n� 

ηi i=1 

n� 

η 

i=1 

′ n� 

i η 

j=1 

′ j = (detU † n� n� n� n� 

)(detU) ηi ηj = ηi ηj 

i=1 j=1 i=1 j=1 



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Path Integral 

In a general integral: 

( � 

� 

dηidηi)f (η) 

i=1 

the only term of f (η) that survives has exactly one factor of each 

η i and ηi 

and it is proportional to ( � η i)( � ηi). 



SUSY 

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Path Integral 

In a general integral: 

( � 

� 

dηidηi)f (η) 

i=1 

the only term of f (η) that survives has exactly one factor of each 

η i and ηi 

and it is proportional to ( � η i)( � ηi). 

If we replace η by Uη, this term acquires a factor of 

(detU)(detU † ) = 1, so the 

integral is unchanged under the unitary transformation. 


Dirac field 


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Path Integral 

In similar way we can prove: 

� dη � dη ηiηje ηAη = (detA)(A −1 )ji 


Dirac field 


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Path Integral 



A Grassmann field is a function of spacetime whose values are 

anticommuting numbers. We can define a Grassmann field ψ(x) in 

terms of any set of orthonormal basis functions: 

ψ(x) = � 

ψiφi(x) 

The basis functions φi(x) are ordinary complex valued functions, 

while the coefficients ψi are Grassmann numbers. 

i 


Dirac field 


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Path Integral 



A Grassmann field is a function of spacetime whose values are 

anticommuting numbers. We can define a Grassmann field ψ(x) in 

terms of any set of orthonormal basis functions: 

ψ(x) = � 

ψiφi(x) 

The basis functions φi(x) are ordinary complex valued functions, 

while the coefficients ψi are Grassmann numbers. 

To describe the Dirac field we take the φi to be a basis of 

four-component spinors. 

i 


Partition function 

� 

Z = 


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Path Integral 

DψDψe i � d 4 xψ(iγ µ ∂µ−m)ψ = C det(iγ µ ∂µ−m) = C e Trln(iγµ ∂µ−m) 



� 

Z = 


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Path Integral 


G (2) (x1, x2) = 

Two point Green function: 

� DψDψe i � d 4 xψ(iγ µ ∂µ−m)ψ ψ(x1)ψ(x2) 

� DψDψe i � d 4 xψ(iγ µ ∂µ−m)ψ 


=


� 

Z = 


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Path Integral 


G (2) (x1, x2) = 




= C det(iγµ ∂µ − m)[−i(iγ µ ∂µ − m)] −1 

C det(iγ µ ∂µ − m) 


⇒ 

=


� 

Z = 


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Path Integral 


G (2) (x1, x2) = 




= C det(iγµ ∂µ − m)[−i(iγ µ ∂µ − m)] −1 

C det(iγ µ ∂µ − m) 

⇒ (in Fourier space): 

G (2) � 

(x1, x2) = S(x1 − x2) = 


⇒ 

d 4p (2π) 4 

ie−ip(x1−x2) p − m + iɛ 

=


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Basic idea of Supersymmetry 

Supersymmetry 

Supersymmetric harmonic oscillator 

Hilbert space contain Fermionic and Bosonic Hilbert spaces: 

H = HB ⊕ HF 



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H = HB ⊕ HF 

Coordinates space : (xµ, θµ) and Momentum space : (pα, ηα) 

where xµ and pα are bosonic variables i.e. commuting : 

[xµ, xν] = 0; [pα, pβ] = 0 

and θµ, ηα are fermionic variables i.e. anticommuting (grassmanian 

numbers) : {θµ, θν} = 0; {ηµ, ην} = 0 



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H = HB ⊕ HF 

Coordinates space : (xµ, θµ) and Momentum space : (pα, ηα) 

where xµ and pα are bosonic variables i.e. commuting : 

[xµ, xν] = 0; [pα, pβ] = 0 

and θµ, ηα are fermionic variables i.e. anticommuting (grassmanian 

numbers) : {θµ, θν} = 0; {ηµ, ην} = 0 

A supersymmetry transformation turns a bosonic state into a 

fermionic state, and vice versa. The operator Q that generates 

such transformations must be an anticommuting spinor, with 

Q|Boson >= |Fermion >, Q † |Fermion >= |Boson > 



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The generators Q and Q † must satisfy an algebra of 

anticommutation and commutation relations with the schematic 

form: 

{Q, Q † } = Pµ, {Q, Q} = {Q † , Q † } = 0, [Pµ, Q] = [Pµ, Q † ] = 0, 

where Pµ is the four-momentum generator of spacetime 

translations. 



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form: 



translations. 

The single-particle states of a supersymmetric theory fall into 

irreducible representations of the supersymmetry algebra, called 

supermultiplets. 



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form: 



translations. 




Each supermultiplet contains both fermion and boson states, 

which are commonly known as superpartners of each other. 



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form: 



translations. 




Each supermultiplet contains both fermion and boson states, 

which are commonly known as superpartners of each other. 

By definition, if |Ω > and |Ω ′ > are members of the same supermultiplet, 

then |Ω ′ > is proportional to some combination of Q and Q † operators acting 

on |Ω >, up to a spacetime translation or rotation. 



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1. Harmonic bosonic oscillator. 





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Hamiltonian:HB = 1 

2p2 + 1 

2ω2 Bq2 where p and q are the 

momentum and coordinate operators respectively and [p, q] = 1 



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Hamiltonian:HB = 1 

2p2 + 1 

2ω2 Bq2 where p and q are the 

momentum and coordinate operators respectively and [p, q] = 1 

Hamiltonian can be presented in terms of bosonic 

creation-annihilation operators: b † , b, as follows: 

HB = 1 

2 ωB{b † , b} = 1 

2 ωB(b † b + bb † ) 

where � = 1; m = 1; b = 1 

√ 2ωB (ip + ωBq) and 

[b, b † ] = 1; [b, b] = 0; [b † , b † ] = 0; [NB, b] = −b; [NB, b † ] = b † 

and NB = b † b is bosonic number operator. 



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Harmonic bosonic oscillator (2). 

After simple calculation Hamiltonian can be presented as: 

HB = ωB(NB + 1 

2 ) = ωBNB + E0 

where E0 = 1 

2 �ωB is energy of the ground state (vacuum energy). 



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Harmonic bosonic oscillator (2). 

After simple calculation Hamiltonian can be presented as: 

HB = ωB(NB + 1 

2 ) = ωBNB + E0 

where E0 = 1 

2 �ωB is energy of the ground state (vacuum energy). 

Operators b † , b act on a state |nB >, and the energy spectrum of 

the hamiltonian is: 

EB = ωB(nB + 1 

2 ) 

where nB = 0, 1, 2..., and nB are the eigenstates of the bosonic 

number operator. 



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2. Harmonic fermionic oscillator. 





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Fermionic creation-annihilation operators: f † , f obey the 

anticommutation relations: 

{f , f † } = 1; {f , f } = 0; {f † , f † } = 0 



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{f , f † } = 1; {f , f } = 0; {f † , f † } = 0 

Hamiltonian of the fermionic oscillator can be constructed by the 

analogy with the bosonic hamiltonian, as: 

HF = 1 

2 ωF [f † , f ] = 1 

2 ωF (f † f − ff † ) 



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{f , f † } = 1; {f , f } = 0; {f † , f † } = 0 

Hamiltonian of the fermionic oscillator can be constructed by the 

analogy with the bosonic hamiltonian, as: 

HF = 1 

2 ωF [f † , f ] = 1 

2 ωF (f † f − ff † ) 

Hamiltonian of the fermionic oscillator can be presented as: 

HF = ωF (NF − 1 

2 ) = ωF NF − E0 

where E0 = − 1 

2 �ωF is energy of the ground state (vacuum energy). 



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Harmonic fermionic oscillator (2). 

Operators f † , f act on a state |nF >, and the energy spectrum of 

the fermionic oscillator can be calculated using of the 


EF = ωF (nF − 1 

2 ) 



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EF = ωF (nF − 1 

2 ) 

where nF = 0, 1 (Pauli principle, due to {f † , f † } = 0 there exist 

only vacuum state |0 > and |1 >= f † |0 >, because (f † ) 2 = 0 ), 

and nF are the eigenvalues of the fermionic number operator 

NF = f † f . 



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EF = ωF (nF − 1 

2 ) 

where nF = 0, 1 (Pauli principle, due to {f † , f † } = 0 there exist 

only vacuum state |0 > and |1 >= f † |0 >, because (f † ) 2 = 0 ), 

and nF are the eigenvalues of the fermionic number operator 

NF = f † f . 

And fermionic harmonic oscillator is built only from two states |0 > 

and |1 > with energy E0 = − 1 

2 �ωF and E1 = 1 

2 �ωF 



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3. Supersymmetric harmonic oscillator. 



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For the composite system of the bosonic and fermionic oscillators, 

ω = ωB = ωF and: 

H = 1 

2 ω({b† , b} + [f † , f ]) = ω(NB + NF ) 

with E = ω(nB + nF ) 



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For the composite system of the bosonic and fermionic oscillators, 

ω = ωB = ωF and: 

H = 1 

2 ω({b† , b} + [f † , f ]) = ω(NB + NF ) 

with E = ω(nB + nF ) 

This formula implies that all energy levels of the system are twice 

degenerate except for the ground state nB = nF = 0, characterised 

by zero energy E = 0. 



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Supersymmetric harmonic oscillator (2). 

Bosonic and fermionic hamiltonian together has additional 

symmetry (i.e.supersymmetry) which mix bosonic and fermionic 

degrees of freedom. 



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Supersymmetric harmonic oscillator (2). 

Bosonic and fermionic hamiltonian together has additional 

symmetry (i.e.supersymmetry) which mix bosonic and fermionic 

degrees of freedom. 

Generators of the symmetry (nB � nB ∓ 1; nF � nF ± 1) 

responsible for the degeneracy are supersymmetry generators: 

Q− = √ 2ωb † f ; Q+ = √ 2ωbf † 


References 


SUSY 

References 

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References 

”An Introduction to Quantum Field Theory”; M.E. 

Peskin, D.V. Schroeder 

”A Supersymmetry Primer”; Stephen P. Martin 

(arXiv:hep-ph/9709356) 

”Supersymmetric mechanical harmonic oscillator”; 

A.K.Aringazin 


The end 


SUSY 

References 

The end 

The end 

Thank You 

�

Grassmann Variables, Supersymmetry and Supersymmetric ...

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