Supersymmetry Formalism from Commutation Relations
Soumyadip Nandi
July 3, 2026
Contents
1 Introduction 2
2 Commutation relations 2
3 GrassMann Variables 4
3.1 GrassMann Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.2 SUSY Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4 Superfields 7
5 References 9
1
1 Introduction
Quantum mechanics is formulated in terms of operators acting on a Hilbert space. The algebra satisfied
by these operators determines the kinematics and statistics of quantum systems. Bosonic operators obey
commutation relations, while fermionic operators obey anticommutation relations. Supersymmetry
arises by extending these two distinct algebraic structures into a single graded algebra capable of
transforming bosonic and fermionic degrees of freedom into one another
2 Commutation relations
We are going to start from commutation relations. Now, in order to dictate whether or not two physical
observables can be measured simultaneously, we use commutation relations. An important relation in
Quantum mechanics is the commutator between two operators.
ˆ
𝐴
and
ˆ
𝐵
. It is written as
[
ˆ
𝐴,
ˆ
𝐵]
and is
defined as:-
[
ˆ
𝐴,
ˆ
𝐵] =
ˆ
𝐴
ˆ
𝐵 −
ˆ
𝐵
ˆ
𝐴
If the commutator is zero, they share common eigenfunctions and are comptabile. If the commutator
is non-zero, they follow the Heisenberg’s Uncertainty Principle, defining fundamental limits on
measurement, polarization etc.
If
[
ˆ
𝐴,
ˆ
𝐵] = 0
, the two operators are said to commute.
ˆ
𝐴
and
ˆ
𝐵
are compatible.
ˆ
𝐴,
ˆ
𝐵 =
ˆ
𝐵
ˆ
𝐴
. If the
commutator is non-zero, they follow the uncertainty principle.
One of the key implications of commutation relations is simultaneous measurability. If
[
ˆ
𝐴,
ˆ
𝐵] = 0
, the observables
𝐴
and
𝐵
can be measured simultaneously with precision. If
[
ˆ
𝐴,
ˆ
𝐵] ≠ 0
, they are
incompatible and measuring one disturbs the other. The Canonical Commutation Relation is one
such instance where the commutation relation fails. Or a better way to phrase it would be:- Bosons are
quantized using canonical commutation relations, whereas fermions are quantized using canonical
anticommutation relations. This distinction is required by the spin-statistics theorem. There are a
few reasons for this. When apllied to finite dimensional Hilbert spaces, the trace of the operator
𝑇𝑟 (𝐴𝐵 −𝐵𝐴)
must be zero. This contradiction results in the failure of CCR. It becomes mathematically
invalid. The trace of the identity operator
𝐼
is non-zero because it represents a sum of eigen values of
an identity operator over an infinite dimensional Hilbert space. Broadly speaking, this is the bosonic
story.
But first let’s discuss the ladder operators.
Considering the Simple Harmonic Oscillator,
𝐻 =
1
2
𝑝
2
+ 𝜔
2
𝑝
2
, with the canonical commutation
relations
[𝑞, 𝑝] = 𝑖
where
𝑞
𝑎
are the generalized co-ordinates and
𝑝
𝑎
is the conjugate momenta. We
have promoted them to operators. The Poisson bracket structure of classical mechanics morphs into
the structure of commutation relations between operators.
[𝑞
𝑎
, 𝑞
𝑏
] = [𝑝
𝑎
, 𝑝
𝑏
] = 0
2
[𝑞
𝑎
, 𝑝
𝑏
] = 𝑖𝛿
𝑏
𝑎
where ,
𝛿
𝑏
𝑎
is the Kronecker Delta Function. To find the spectrum, we have to define the creation and
annihilation operators (also known as ladder operators).
𝑎 =
𝜔
2
𝑞 +
𝑖
√
2𝜔
𝑝, 𝑎
†
=
𝜔
2
𝑞 −
𝑖
√
2𝜔
𝑝
which can easily be inverted to give,
𝑞 =
𝑖
√
2𝜔
(𝑎 + 𝑞
†
), 𝑝 = −
𝑖
√
2𝜔
(𝑎 − 𝑞
†
)
Now, by substituting it to the above equation, we get
[𝑎, 𝑎
†
] = 1
While the Hamiltonian is given by:-
𝐻 =
1
2
𝜔(𝑎𝑎
†
+ 𝑎
†
𝑎) = 𝜔(𝑎
†
𝑎 +
1
2
)
These relations ensure that
𝑎
and
𝑎
†
takes us between energy eigen states. For bosons, swapping the
order of the creation operators (
𝑎
†
) and annihilation operators (
𝑎
) results in a non-zero commutator.
Now, because both the creation and annihilation operators commute with each other, the bosonic states
are completely symmetric under the exchange of identical particles.
The canonical commutation relations fails for fermions because their wave function must be anti-
symmetric under particle exchange. As mentioned previously, if
[
ˆ
𝐴,
ˆ
𝐵] ≠ 0
, they are incomptabile and
measuring one can cause disturbance in the other. it can impose limits like
Δ𝑥Δ𝑝 ≥ ℏ/2
. Non-zero
commutator implies that there are just no eigen-states. In classical mechanics, one can assign a value
to
𝑥
and
𝑝
. So, simultaneous measurability exists at the classical level. When we quantize, we replace
the variables with operators.
Now, let’s understand why commutation fails entirely for fermions.
Part of the reason for the failure is because there is no commutative classical phase space for fermions.
While classical mechanics , models particles as point-like entities with definite trajectories, fermions
as mentioned possess intrinsic anti-symmetric wave functions , meaning that they cannot occupy the
same quantum state (Pauli’s Exclusion Principle). The reason why this incompatibility exists is
because a ”commutative” (traditional) phase-space has no mechanisms to prevent fermions from piling
up on the same point in phase-space, which violates the quantum behaviour of fermions.
To put this into a logical format, if we consider the fermionic annihilation operators
{𝐶, 𝐶
†
} = 1
.
There is no classical variable
𝑐 ∈ C
such that
{𝐶, 𝐶} = 0
. This is because ordinary numbers commute
𝑐𝑐 = +𝑐𝑐 ≠ 0. So, we can say that for fermions, the obstruction exists even before quantization.
3
So, a clear description on why bosons have a commutative classical phase space meanwhile the
fermions do not can be:-
” Bosons admit a commutative classical phase space because their quantum statistics arise
only after quantization, whereas fermions do not because their statistics are enforced from
the start. For the bosonic system, one can begin with ordinary classical variables
𝑥, 𝑝 ∈ R
that commute and represent simultaneously well defined configurations; quantization
then promotes these variables to operators whose non-commutativity encodes quantum
uncertainty , and the classical limit
ℏ → 0
is smooth. For fermions, by contrast are
constrained by the Pauli-Exclusion principle and the spin-statistics theorem, which require
their fundamental degrees of freedom anti-commute. No set of classical numbers can
reproduce this algebra since ordinary numbers cannot square to zero or encode exclusion.
As a result, fermions possess no underlying commutative classical phase space at all -
their ”classical” description must already be formulated in terms of anti-commutating
Grassmann variables , and only bilinear combinations corresspond to physical observables.
”
So, bosons are [𝑎, 𝑎
†
] = 1, fermions = {𝐶, 𝐶
†
} = 1, {𝑐, 𝑐} = 0, 𝐶
2
= 0
Let’s try to understand, how to use Grassmann variables to sove this issue.
3 GrassMann Variables
So, how do we fix this issue ?. Remember that we need to relate the bosonic field
𝜙
with the fermionic
field
𝜓
We arrive at something known as Grassmann variables. A grassmann variable,
𝜃
is a number
like object with one defining feature,
𝜃
2
= 0
and
𝜃
1
𝜃
2
= −𝜃
2
𝜃
1
.
𝜃
2
1
= 0
,
𝜃
2
2
= 0
This makes them
anti-commute and any power higher than
1
vanishes. This makes Grassmann Variables useful for
fermionic degrees of freedom. They encode the Pauli Exclusion principle at the algebraic level.
The appearance of Grassmann variables is not merely a mathematical trick. Their introduction changes
the geometry itself.
A couple of points that are essential here:-
• Two identical fermions cannot occupy the same quantum state
•
Algebraically squaring a fermionic co-ordinate gives zero. Basically, we have Bosonic
−→
Commuting numbers and fermionic −→ Anticommuting numbers.
This is a structure that is forced by spin statistics. The reason why path integrals are necessary is
because spin 1/2 particles obey Fermi-Dirac statistics with the quantum state picking up a minus sign
upon the interchange of any two particles. This fact, as mentioned above is baked into the structure of
relavistic QFT. So, to have states that obey fermionic statistics, we need anti-commutation relations.
{
𝐴, 𝐵
}
= 𝐴𝐵 + 𝐵 𝐴. In this case, the spinor fields should satisfy.
4
𝜓
𝛼
(
−→
𝑥 ), 𝜓
𝛽
(
−→
𝑦 )
=
𝜓
†
𝛼
(
−→
𝑥 ), 𝜓
†
𝛽
(
−→
𝑦 )
= 0
𝜓
𝛼
(
−→
𝑥 ), 𝜓
†
𝛽
(
−→
𝑦 )
= 𝛿
𝛼𝛽
𝛿
(3)
(
−→
𝑥 −
−→
𝑦 )
Now, for commutation relations to work properly, we need the path integrals. To properly represent
fermionic fields in a functional integral(sum over histories), the variables must obey the same anti-
commutating algebra, thus necessitating the use of Grassmann numbers. The reason why Grassmann
numbers are required for anti-commutation to work are as follows.
•
In the path integral formulation, integrating over fermionic grassmann variables produces a
determinant ,
𝑑𝑒𝑡(𝐷)
, rather than the inverse determinant
(𝑑𝑒𝑡𝐷)
−1
produced by bosonic fields.
This correctly calculates the degrees of freedom for fermions.
•
Grassmann Integration rules:- The rules for integration specifically ,
𝑑𝜃 = 0
and
𝜃𝑑𝜃 = 1
might seem bizarre at first , until you realize that the integration actually acts as differentiation,
and only linear terms survive.
•
Path integrals using Grassmann variables allow for the construction of fermionic coherent states
and the correct evaluation of transition amplitudes.
In summary, the path integral for fermions is essentially a formal construction using Grassmann variables
to ensure that the ”sum over all paths” is anti-symmetric , which is essentially capturing the spin-statistics
of fermionic systems. To make the path integral work correctly, we need 𝜓(𝑥)𝜓(𝑦) = −𝜓 (𝑥)𝜓(𝑦).
This is the algebraic foundation for boson-fermion cancellation in SUSY.
Now, we know that we have to relate both the bosonic as well as the fermionic fields for Supersymmetry.
The logical path is to start from the Grassmann Harmonic Oscillator (GHO), just like we had done for
the Simple Harmonic Oscillator and build our intuition towards SUSY fields.
3.1 GrassMann Harmonic Oscillator
Considering, we have the fermionic creation and annhilation operators:-
𝑓 , 𝑓
†
= 1
,
𝑓
2
= ( 𝑓
†
)
2
= 0
.
Therefore, the Hamiltonian would be:-
𝐻
𝐹
= ℏ𝜔( 𝑓
†
𝑓 −
1
2
)
There are only two states here. The degrees of freedom are intrinsically fermionic. We need to keep in
mind that this algebra is not representable by commuting numbers. So, if we want a classical looking
formulation (for eg:- a path integral) we must introduce new objects. Its necessary because path
integrals manifest symmetry and they can generalize easily to fields and spacetime. As mentioned
previously, a Dirac spinor
𝜓
is constructed from two independent grassmann variables
𝜓
𝛾
(𝑥)
and
¯
𝜓
𝛾
(𝑥), where the components are anti-commuting. (𝜓
𝑖
𝜓
𝑗
− 𝜓
𝑗
𝜓
𝑖
).
5
So, the grassmann variables
𝜓(𝑡)
,
¯
𝜓(𝑡)
with
𝜓
2
=
¯
𝜓
2
= 0
,
𝜓
¯
𝜓 = −
¯
𝜓𝜓
. The GrassMann Harmonic
Oscillator action is:-
𝑆 =
𝑑𝑡
¯
𝜓(𝑖𝜕
𝑡
− 𝜔)𝜓
From here, the equations of motion can be derived. The equations of motion will therefore be:-
(𝑖𝜕
𝑡
− 𝜔)𝜓 = 0. Hence, the solution :- 𝜓 (𝑡) = 𝜓
0
𝑒
−𝑖𝜔𝑡
.
Here,
𝜓
0
is Grassmann and we cannot assign a numerical value to it. Only bilinears like
−
¯
𝜓𝜓
are
physical.
For the bosonic system, we know that [𝑎, 𝑎
†
] = 1, 𝐻
𝐵
= ℏ𝜔(𝑎
†
𝑎 +
1
2
).
Now, if we were to consider the combined system, 𝐻 = 𝐻
𝐵
+ 𝐻
𝐹
, we are going to get,
𝐻 = ℏ𝜔( 𝑓
†
𝑓 + 𝑎
†
𝑎)
If you notice, bosonic = zero-point energy +
1
2
ℏ𝜔 and Fermionic = zero-point energy -
1
2
ℏ𝜔.
This gets cancelled and is the reason behind why SUSY cancels divergences. Now, we need an operator
that can transform fermions into bosons and vice-versa . We can define them as Supercharges as
fermionic operators.
So, let’s define these operators. 𝑄 = 𝑎 𝑓
†
, 𝑄
†
= 𝑎
†
𝑓 .
These are fermionic operators . For now, all we need to know is that supercharges are described by
the Super-Poincare algebra . Supercharge and it’s adjoint commute with the Hamiltonian Operator
𝑄, 𝑄
†
= 𝐻
,
𝑄
2
= (𝑄
†
)
2
= 0
. This is the first glimpse of supersymmetry which is already present in
the oscillator. We haven’t included spacetime yet, just symmetry.
Note:- The condition
[𝑄, 𝐻] = 0
means the supercharge
𝑄
commutes with the Hamiltonian
𝐻
. In
quantum mechanics, if an operator commutes with the Hamiltonian, it is a conserved quantity (constant
of motion). Therefore, the symmetry operator
𝑄
maps eigenstates of
𝐻
to other eigenstates of
𝐻
with
the same energy eigenvalue.
As a conserved quantity, supercharges have another implication:- We know that they transform bosonic
states into fermionic states (and vice versa) without changing the energy of the system. This means that
for every boson, there is a fermion with the exact same energy, leading to degenerate supermultiplets.
3.2 SUSY Transformations
As per our objective, we need to be able to relate particles with different spin-statistics, specifically
mapping bosons to fermions and vice-versa. It is therfore natural to look for an infinitesmal
transformation parameter that represents the transformation between fields of opposite GrassMann
6
parity. We can introduce a GrassMann odd infinitesmal parameter
𝜖
to describe the infinitesmal
supersymmetry transformation. 𝜃 −→ 𝜃 + 𝜖, in the fermionic direction.
The grassmann parameters
𝜖
(Anti-commutating,
𝜖
2
= 0
) are best known as infinitesmal generators of
motion along the fermionic direction in an extended configuration space. So, instead of spacetime
alone,
𝑥
𝜇
, we can enlarge it into superspace.
(𝑥
𝜇
, 𝜃
𝛼
,
¯
𝜃
¤𝛼
)
, where
𝑥
𝜇
are the Minkowski co-ordinates
and 𝜃 and
¤
𝜃 are the grassmann co-ordinates.
The question now is, why does the parameter become co-ordinates?. In ordinary physics, if a symmetry
has a parameter
𝑎
, and that parameter can vary freely , we can interpret it as a co-ordinate. Now, SUSY
gives us a new symmetry with a new parameter
𝜖
, that behaves consistently like a ”direction”. So,
here we do the same thing , Parameter
−→
Co-ordinate ,
𝜖 −→ 𝜃
. So, superspace here is the ordinary
space as well as the Grassmann directions. Hence, we can now describe systems using
(𝑡, 𝜃,
¯
𝜃)
. This is
superspace which means that
𝑡
is how far we are moving in time and
𝜃
is how far in the fermionic
direction we are moving. Once, we introduce superspace, SUSY transformations become simple
translations in 𝜃. Superspace = ordinary space + grassmann.
Bosons and fermions live together in one object ( a superfield). A much clearer mental picture here is
:- Ordinary space tells you where you are, SUSY tells you what kind of particle you are.
4 Superfields
A superfield is just a function that is defined on the superspace, meaning that it is depends on both
ordinary time and Grassmann co-ordinates. So, instead of a field like
𝑥(𝑡)
, we can now consider
Φ(𝑡, 𝜃, 𝜃
¯
)
. Now in Supersymetry Quantum Mechanics , superspace has one-ordinary co-ordinate :
𝑡
and two Grassmann co-ordinates:-
𝜃
and
¯
𝜃
. They satisfy
𝜃
2
=
¯
𝜃
2
= 0
,
𝜃
¯
𝜃 = −
¯
𝜃𝜃
. Where
Φ
is the
chiral superfield.
Because, 𝜃
2
=
¯
𝜃
2
= 0 , the most general expansion is:-
Φ(𝑡, 𝜃,
¯
𝜃) = 𝑥 (𝑡) +
√
2𝜃𝜓(𝑡) + 𝜃
2
𝐹 (𝑡) = 0
This is the superfield. Conceptually, we have upgraded from the Fermionic harmonic Oscillator .
There’s a superpotential after it. The
√
2
is convention.
𝜓(𝑡)
here came as a replacement of
𝑓
by
Grassmann variables. That was just fermionic dynamic. Now,
𝜓(𝑡)
becomes a co-ordinate direction in
superspace. The fermionic oscillator is no longer separate , it becomes part of the geometry.
𝑥(𝑡)
is
the bosonic field here. The expansion stops because we cannot have powers that are higher than
1
.
[Fermionic DOF −→ co-ordinates].
𝜓(𝑡),
¯
𝜓(𝑡)
- Fermionic fields. They are Grassmann valued, they anticommute and they encode the
exclusion principle.
𝐹 (𝑡)
is the bosonic auxilliary field , and it has no dynamics. There are no kinetic
terms for F. If we integrate over the superspace, we can find the action. If we have a single grassmann
variable 𝜃 , then
𝑑𝜃 = 0,
𝑑𝜃𝜃 = 1
7
This means that , if we have a function
𝑓 (𝑥, 𝜃) = 𝑓
0
(𝑥) + 𝜃 𝑓
1
(𝑥)
, then the grassmann integration will
pick out the component multiplying 0.
𝑑𝜃 𝑓 (𝑥, 𝜃) = 𝑓
1
(𝑥)
Here , we need to integrate over the superspace parameterized by 𝜃
𝛼
and and
¯
𝜃
¤𝛼
. We define
𝑑
2
𝜃 =
1
2
𝑑𝜃
1
𝑑𝜃
2
,
𝑑
2
¯
𝜃 = −
1
2
𝑑
¯
𝜃
1
𝑑
¯
𝜃
2
.
Those strange factors of
1
2
are because
𝜃
2
= 𝜃
𝛼
𝜃
𝛼
= −2𝜃
1
𝜃
2
.
We then have
𝑑
2
𝜃 𝜃
2
= −
𝑑𝜃
1
𝑑𝜃
2
(𝜃
1
𝜃
2
) = 1,
where the minus sign disappears when 𝑑𝜃
2
moves past 𝜃
1
.
Note that the measure 𝑑
2
¯
𝜃 comes with an extra minus sign, but this cancels the corresponding minus
sign in
¯
𝜃
2
=
¯
𝜃
¤𝛼
¯
𝜃
¤𝛼
= +2
¯
𝜃
1
¯
𝜃
2
.
Once again, we have
𝑑
2
¯
𝜃
¯
𝜃
2
= 1.
Finally, we can use the notation
𝑑
4
𝜃 =
𝑑
2
𝜃 𝑑
2
¯
𝜃.
Now, if we were to build an action out of some function of superfields. That function will itself be a
superfield that we can call
𝐾 (𝑥, 𝜃,
¯
𝜃)
. But
𝐾
is a composite superfield whose components are functions
of other superfields. We can construct the action of the form
𝑆 =
𝑑
4
𝑥𝑑
4
𝜃𝐾 (𝑥, 𝜃,
¯
𝜃)
The action is real if 𝐾 is a real superfield., obeying 𝐾 = 𝐾
†
.
So, going back to where we had started from , multiplying a chiral and an anti-chiral superfield together
gives us the real superfield. Φ
†
Φ, which we can integrate over the superspace to get the action
𝑆
𝑐ℎ𝑖𝑟𝑎𝑙
=
𝑑
4
𝑥𝑑
4
𝜃Φ
†
Φ
Now, some calculation and integration by parts shows us that , the action ,
8
𝑆
𝑐ℎ𝑖𝑟𝑎𝑙
=
𝑑
4
𝑥[𝜕
𝜇
𝜙
†
𝜕
𝜇
𝜙 − 𝑖
¯
𝜓𝜎
𝜇
𝜕
𝜇
𝜓 + 𝐹
†
𝐹]
These are just the standard kinetic terms for a complex scalar
𝜙
and Weyl fermion
𝜓
. But now we
see that there is something special about F: it does not have any kinetic terms. Moreover, this will
continue to be true as we write down further supersymmetric interactions. This is what it means
to be an auxiliary field. Because there are no kinetic terms for F, it has no propagating degrees of
freedom and, when quantised, doesn’t give rise to any particle states. The most general renormalizable
supersymmetric theory also contains a holomorphic superpotential which has not been derived here.
The remarkable feature of the superfield formalism is that supersymmetry is manifest. Rather than
verifying invariance under supersymmetry transformation term by term, one constructs the action
directly in superspace. The integral automatically projects out the highest Grassmann component,
producing a Lagrangian that is invariant under supersymmetry by construction.
The logical development of supersymmetry can therefore be viewed as a sequence of algebraic structures.
One begins with bosonic and fermionic operator algebras, introduces Grassmann variables as the
classical analogue of fermionic degrees of freedom, extends spacetime into superspace, constructs
supercharges as generators of translations along Grassmann directions, packages fields into superfields,
and finally writes a supersymmetric action as an integral over superspace. The entire formalism
emerges naturally from the requirement that bosonic and fermionic degrees of freedom be treated
within a single graded geometric framework
5 References
• Quantum Field Theory - Peskin
• QFT notes - David Tong
• Supersymmetric field Theory- David Tong
• Supersymmetry Quantum Mechanics - david Tong
• Dynamics of classical analogs of fermions, bosons and beyond- APS Journal
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