Download INTRODUCTION TO MECHANICS Introduction On the face of it

INTRODUCTION TO MECHANICS
SEAN POHORENCE
Introduction
On the face of it, particle mechanics is the study of the simplest physical systems. However, this seemingly trivial task leads to some surprising consequences
and some deep mathematics. At the quantum level, there are still large gaps in
our mathematical understanding of these systems. In this talk, I hope to give a
small taste of these areas and why one should be interested in studying them. For a
comprehensive reference to the material covered, see Quantum Mechanics for Mathematicians by Leon Takhtajan. If you want to go from zero knowledge to solving
problems quickly (and are okay with neglecting some of the mathematical details),
I recommend Introduction to Quantum Mechanics by David Griffiths.
1. Classical Mechanics
First, we give a brief introduction to the theory of classical mechanics. This will
motivate our approach to quantum mechanics in the next section.
1.1. Principal of Least Action. We will study the motion of a point particle
through space. This is described by a (smooth) map
x : [0, 1] → M,
[0, 1] 3 t 7→ x(t) ∈ M
where M is a manifold which represents the space in which the particle moves. The
image of this map is a path traced out in space. The graph of this map is a path in
spacetime. In order to describe the dynamics of the system, i.e. the shape of this
graph, we first specify a functional
S : C ∞ ([0, 1], M ) → R
which assigns to every path a real number. We require that
R 1 this functional is
“local”1 which we can think of as requiring S is of the form 0 L dt where L is a
function depending only on the first k derivatives of x(t) at each time t. In this talk,
we will only look at the case when k = 1. We think of this action as determining
the “cost” of each path, so physical systems will seek to minimize this action. Thus,
we may say that allowable classical paths x(t) must be critical points of the action.
This formalism is conceptually nice because it is relatively easy to explain. However, it is not especially easy to actually compute the critical
points of an arbitrary
R1
action from this definition. As an example, let S = 0 21 ẋ2 dt. Here, we can see
that the global minimum of the action is clearly achieved when ẋ = 0, i.e. when
x(t) = p ∈ M is the contant path. It is not as obvious that any path with constant
velocity will be a critical point of this action. It turns out that the local nature of
1An example of a non-local functional could be something that counted the total number of
times the path x(t) crosses itself, or some other global behavior.
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SEAN POHORENCE
the action allows us to describe the dynamics of a classical particle in a different
manner in which the above example becomes very easy.
1.2. States, Observables, Measurement, and Dynamics. We begin with a
question: given a path x(t) in Rn , how much information do we need to specify
the motion of our particle starting from any point on the path, x(t0 )? Thinking
of an example where the path crosses itself, it is clear that we must know both
the position x(t0 ) and the velocity ẋ(t0 ) in order to uniquely specify the motion
at x(t0 ). But is this enough? We could similarly dream up a situation in which
the path passes tangent to itself at x(t0 ), so we would need to additionally specify
the second time derivative at t0 to uniquely determine the motion. We could also
continue this process, producing curves whose first kth order derivates all agreed at
the point x(t0 ), meaning we would need to specify the first k + 1 time derivatives of
x(t) at t0 . This situation seems quite grim, since for very badly behaved paths we
would need to provide an infinite amount of initial data to know how our particle
would move from an arbitrary point x(t0 ). Luckily, the locality of the action ensures
that for physical paths (those which are critical points of the action), the situation
R1
will never be so bad. In fact, if the action is of the form 0 L dt where L depends on
the first k derivates of x(t), then it will be possible to uniquely specify the motion
of the particle by supplying initial conditions consisting of the position x(t0 ) and
the first k time derivatives of x(t) at t0 for any t0 ∈ [0, 1]. In particular, when
k = 1, we may completely determine the behavior of the particle from any starting
point x(t0 ) by supplying initial position and velocity conditions. In other words,
for any physical path x(t) ∈ Rn , the path (x(t), ẋ(t)) ∈ R2n , will never cross itself.2
R1
As a summary of the above, given an action S = 0 L(x, ẋ)dt and the initial
conditions (x(0), ẋ(0)) ∈ R2n , we may specify the motion of our particle for all
t ∈ [0, 1] uniquely. Using calculus of variations, we can justify this statement
by showing that the classical paths x(t) must satisfy n second order differential
equations which are called the Euler-Lagrange equations of the system. The initial conditions (x(0), ẋ(0)) then uniquely specify a solution to the Euler-Lagrange
equations. This is known as the Lagrangian description of classical mechanics.
The result above contains all the key ideas you will need to know about classical
mechanics for this talk. Here, we make some superficial changes which will allow
us to present those ideas in a form which parallels the constructions in quantum
mechanics which will follow. The details are not important for our purposes, so
will be omitted for the most part.
In the above, we considered the function
L : R2n → R,
R2n 3 (x, ẋ) 7→ L(x, ẋ) ∈ R.
Applying the Legendre transform τL associated with L to R2n gives us a new
function H(x, p) on the space R2n , now taken with coordinates (x, p).3 In particular,
the Legendre transform acts by
∂L
(x, ẋ) 7→ x,
∂ ẋ
2For an arbitrary manifold M , this will be a path in the tangent bundle T M .
3For a general M the Legendre transformation is a bundle morphism T M → T ∗ M .
INTRODUCTION TO MECHANICS
3
and
∂L
− L(x, ẋ).
∂ ẋ
We call R2n with these coordinates phase-space and think of x as the position of
the particle and p as its momentum. Similarly to in the Lagrangian description,
the coordinates (x, p) uniquely determine classical paths in the sense that these
paths do not cross themselves in phase space. For this reason, we call the pair
(x, p), the state of the particle. We may think of smooth functions on phase-space
as observables and define the concept of measurement as a map
H ◦ τL = ẋ
States × Observables = R2n × C ∞ (R2n ) → R
which simply sends ((x, p), f ) 7→ f (x, p). Two easy examples are the function x̂
defined by (x, p) 7→ x and the function p̂ defined by (x, p) 7→ p. In this case,
measurement of the observable x̂ just gives the position of the particle, and measurement of the observable p̂ gives the momentum of the particle. For this reason,
we typically call x̂ the position and p̂ the momentum (and usually omit the hat).
As in the Lagrangian description, dynamics are controlled by a set of differential
equations. In particular, we require that
d
f (t) = {H, f (t)}
dt
∂f
∂H ∂f
for any observable f , where {H, f } := ∂H
∂x ∂p − ∂p ∂x is the Poisson bracket (it
is clearly a skew symmetric map { , } : C ∞ (R2n ) × C ∞ (R2n ) → C ∞ (R2n ), and it
turns out that the functions C ∞ (R2n ) equipped with the Poisson bracket form a
Lie algebra). In the special cases when f is the position or momentum, we obtain
ẋ =
∂H
,
∂p
ṗ = −
∂H
∂x
which go by the name of Hamilton’s equations of motion. This formulation is
called the Hamiltonian description of classical mechanics.4 It is worth noting that
the above notation implies that the function H is an observable of the system. H
is called the Hamiltonian of the system, and the observable quantity it measures is
the energy of the system.
1.3. Examples. We start with the example given above, which, in the Hamiltonian
2
description, is given by H = p2 . The equations of motion are then ṗ = 0 and ẋ = p,
so this describes a particle moving with constant velocity p.
An essential example, which is equally important in the quantum case, is the
harmonic oscillator in one dimension (so phase-space is just R2 ), which is given
by Hamiltonian H = 21 (p2 + x2 ). It has equations of motion given by ẋ = p and
ṗ = −x, i.e. the orbits in phase-space are circles. These equations describe any
system with quadratic potential, for example a particle attached to a spring.
4One may note that the proper setting for the Hamiltonian formalism is in symplectic geometry.
In this case, we take space to be a smooth manifold M and phase space to be T ∗ M equipped
with the canonical symplectic form ω = −dη. Then, to each observable f ∈ C ∞ (T ∗ M ), we may
associate a Hamiltonian vector field Xf which satisfies df = ιXf ω (note that the non-degeneracy of
ω ensures this is well defined). Then, we may define the Poisson bracket by {f, g} := ω(Xf , Xg ).
f, defining M
f to be the
In fact, we may apply this construction to any symplectic manifold M
phase-space of the classical system.
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2. Quantum Mechanics
2.1. Motivation. Now, let’s try to turn the ideas above into a quantum theory.
We will need this theory to agree with experimental results, in particular, the way
we describe measurements must agree with experimental observations. This gives
us two goals for the theory:
(1) The order of measurements matters.
(2) In the “classical limit” the theory agrees with the classical theory.
Many “paradoxes” in experimental results can be boiled down to this first property. For instance, in the double slit experiment, we see that making a preliminary
measurement by checking which slit particles pass through changes the subsequent
measurement of the distribution of particles on the detector.5 We may say that a
potentially viable quantum theory will consist of a map between the classical observables introduced above to some new quantum observables, along with a (possibly new) concept of states. Moreover, we should include a method for making
measurements. Crucially, we should require that these measurements depend on
a parameter that controls the “quantum-ness” of the system and reproduces the
above results as this parameter tends to zero.
2.2. States, Observables, Measurement, and Dynamics. Before giving an
actual construction of quantum mechanics, we provide an axiomatic definition of
how such a theory behaves. The first step is to define the building blocks of the
theory.
States. While classical states were simply points in an even dimensional Euclidean
space (or, more generally, a symplectic manifold), we define states in the quantum
theory to be elements of a complex Hilbert space H with norm 1 (i.e. elements of
the projectivization PH).
Observables. Classically, observables were functions on the state space. Here, we
define the quantum observables to be self-adjoint linear operators on H. That is,
elements O ∈ End(H) satisfying hOϕ, ψi = hϕ, Oψi where h , i is the inner product
on H. We denote the (real) vector space of self-adjoint linear operators on H by
BH . Note that these operators will not, in general, be compact or even bounded.
Measurement. The first attempt to define measurement proceeds as follows. Pretend that any observable
in H. Then, we see that for any state,
P∞O is diagonalizable
P∞
ϕ, we have Oϕ = O ( i=1 ci ei ) = i=1 λi ci ei where ci = hϕ, ei i and the λi are
the eigenvalues of O in the {ei } basis, where we take this basis to be orthonormal.
Then, for the special case where ϕ = ei for some i, we may say that the measurement of the observable O of ϕ, denoted (O, ϕ), will be λi (since O is self-adjoint,
this is a real number). For general ϕ, we define measurement in a non-deterministic
way. Generalizing the special case above, we say that the measurement (O, ϕ) will
2
result
P∞ in 2a value of λi with probability |ci | (note that since ||ϕ|| =P1,∞ we have
2
i=1 λi |ci | .
i=1 |ci | = 1). The expected value of the measurement is given by
This says that, a priori, the state of the system does not completely determine the
results of measurements as it did in the classical case. However, once a measurement is made, it must result in a given value, so we say that making a measurement
resulting in value λi also changes the state of the system to the corresponding ei .
5See http://en.wikipedia.org/wiki/Double-slit_experiment.
INTRODUCTION TO MECHANICS
5
This is where the possibility for the ordering of measurements to matter is built in,
since the act of making a measurement may alter the state of the system.
However, the assumption that O is diagonalizable in H will not always be correct.
Two of the most important operators in the model of quantum mechanics we will
introduce below do not have any eigenstates in the corresponding Hilbert space.
Thus, we need to modify our definition of measurement to accomodate for these
types of operators. The key to this will be to relax how precise the result of a
measurement is. For any measurement (O, ϕ) instead of assigning probabilities to
any one given value, we will assign probabilities to all ranges of values [a, b] that
the measurement may take (we see the above is a specific case of this where we only
worry about intervals [a, a]). In other words, to any measurement, we
R associate a
measure µ on R. The expected value of the measurement is given by R x dµ(x). In
the description above, we said that the state of a system “collapses” after making
a measurement in such a way that subsequent measurements will result in the
same value with probability one.6 In the case where measurements are no longer
exact, we say that if the result of a measurement (O, ϕ) is an interval [a, b] that the
state will collapse in such a way that subsequent measurements will lie within that
range with probability one. Exactly how this manifests itself in terms of a general
model of quantum mechanics is more technical than our previous construction, so
we instead postpone these details until we have a concrete model to work with.
Quantization and Dynamics. With these building blocks, we may now define what
a model of quantum mechanics actually is. Here, we will only discuss models which
come up as the quantization of a classical model.7 Given such a classical model
consisting of a Lie algebra of observables (A, { , }), we define a quantization to
consist of a choice of Hilbert space H and an bijective map of sets Q~ : A →
BH which depends on a parmeter ~. Note that on BH the operation i[ , ], where
[O1 , O2 ] = O1 O2 − O2 O1 is the commutator of operators, is well defined (i.e. i
times the commutator of two self-adjoint operators is self-adjoint).8 We require
that this map satisfies the following
lim [Q~ (f ), Q~ (g)] = 0,
~→0
and
i
[Q~ (f ), Q~ (g)].
~
The first condition means that as ~ → 0, the quantum observables commute. The
second condition means that as ~ → 0 the map Q~ takes Poisson brackets to ~i
times the commutator (i.e. Q~ becomes an isomorphism of Lie algebras). Thus,
in this limit, we obtain a commutative algebra of observables isomorphic to the
original one, so it reproduces the classical theory. For this reason, this is called the
classical limit of the quantum theory.
We define the dynamics of the quantum system in an analogy with the classical
case. Namely, given a classical Hamiltonian Hcl , we define the quantum Hamiltonian H := Q~ (Hcl ) and specify that the time evolution of a quantum observable O
lim Q~ ({f, g}) = lim
~→0
~→0
6When we say that subsequent measurements will result in the same value with probability
one, we imagine these measurements occurring immediately after the first. If the system is allowed
to evolve for some time after the first measurement, this will no longer be true.
7Not all quantum theories arise in this way.
8In fact, this turns the vector space of self-adjoint operators into a Lie algebra.
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SEAN POHORENCE
is given by9
d
i
O(t) = [H, O(t)].
dt
~
2.3. A Model of Quantum Mechanics. The model we will describe is a quantization of a one dimensional classical system. In particular, we consider a system
with phase space R2 with coordinates (x, p) and Hamiltonian Hcl = 21 p2 + V (x)
where V (x) is a polynomial function in the position. To quantize this system, we
set our Hilbert space H to be L2 (R), the space of square integrable complex-valued
functions on the real line, with inner product given by
Z
ϕψ dx.
hϕ, ψi :=
R
Here, we only provide part of the data of a full quantization, but this is actually
sufficient to obtain many important results about systems of this type. We define
Q~ (x) = mx where mx is the operator defined by multiplication by x and Q~ (p) =
d
i~ dx
. For convenience, we will denote the quantum position operator by X and
the quantum momentum operator by P . Here, you might protest that neither of
these map all of H into itself. There are a few reasons why this does not create
problems. One is that measurements (which will be defined later) will still be well
defined on all of H, and these are the only physical parts of the theory. Another
is that many of the “important” states in H will be well behaved enough (possibly
they will decay exponentially) so that X and P are well defined on them (in fact, X
and P are well defined on a dense subspace D ⊂ H). To check that these quantum
observables do indeed define a quantization of x and p we may write (for ϕ ∈ D)
d
]ϕ = −i~ϕ
dx
so we see that as ~ → 0, we have [X, P ] → 0 and ~i [X, P ] = 1 = {x, p}. The reason
why the quantizations of position and momentum capture so much of this theory
is they allow us to write the quantization of the Hamiltonian. That is, we set
1
Q~ (Hcl ) = P 2 + V (X).
2
With this quantum Hamiltonian, we can write down how the dynamics of position
and momentum behave, or any other observable once we have defined the full
quantization. We should note here that the only reason we could write down H in
terms of X and P is that the classical Hamiltonian had no cross terms between x
and p. For a general classical observable which is a function of x and p, in order to
quantize it we would need to know how to turn a function of commuting variables
into a functions on non-commuting variables.
[X, P ]ϕ = [x, i~
9This formulation of quantum mechanics, in which observables vary with time and states are
fixed is called the Heisenberg picture. Since the only physical elements of the theory are the
results of measurements, which come from a pairing of states and observables, it is also possible
to construct a version of the theory where states vary in time while operators are fixed which
produces the same physical predictions. This view is called the Schrödinger picture. In the
d
Schrödinger picture, the time evolution of a state ψ is given by i~ dt
ψ = Hψ, which is known as
the Schrödinger equation for the state ψ. In particular, if ψ is an eigenvector of H with eigenvalue
i
d
λ, we have i~ dt
ψ = λψ so the solution of the Schrödinger equation is ψ(t) = e− ~ λt ψ(0). When
it is possible to diagonalize H in H, this resulting solution is often easier to work with than the
equations for the time evolution of operators in the Heisenberg picture.
INTRODUCTION TO MECHANICS
7
In the special case where V (x) = x2 , this system can actually be solved exactly.
In this case, there is an algebraic trick which allows us to diagonalize H in H and
obtain the energy eigenstates. From there, we may use the Schrödinger picture formalism to solve for the time evolution of any state (once we have written it in the
basis of energy eigenstates). This is a very special property of Hamiltonians of this
form (i.e. with kinetic term p2 and quadratic potential V (x)). For most physical
Hamiltonians, it turns out that we can approximate the behavior of the corresponding systems using a Hamiltonian with quadratic potential. This is sometimes called
the semiclassical approximaiton.
We finish our discussion of this model by briefly describing how measurement is
defined. For the position operator, we define the probability of the measurement
Rb
(X, ϕ) belonging to the interval [a, b] is given by the integral a |ϕ|2 dx. That is,
we specify that theRmeasure µ on R corresponding to the measurement (X, ϕ) is
defined by µ(E) = E |ϕ|2 dx. In order to define measurement for the momentum
operator, we first observe that P ϕ = FXϕ where F is the Fourier transform on
L2 (R) (possibly up to a constant term).10 In this spirit, we define the probability of
Rb
a momentum measurement belonging to the interval [a, b] by the integral a |ϕ̂|2 dx
where ϕ̂ is the Fourier transform of ϕ. We may also define the collapse of the state
resulting from the act of measuring it. For (X, ϕ), we say that after measuring the
position in an interval [a, b], the state will take the form Cχ[a,b] ϕ where C is the
necessary normalization constant. The definition for (P, ϕ) is similar.
2.4. The Path Integral. We now change gears and discuss one of the more mysterious elements of quantum theory. When we introduced classical mechanics, we
started out by defining the principle of least action, which, in turn, led us to the
development of Hamiltonian mechanics. When we sought to quantize a classical system, we modeled our approach on the Hamiltonian formulation of classical
mechanics. One could ask whether the action principle has a counterpart in the
quantum theory. The answer is, in some sense, not quite known.
In the formulation of quantum mechanics above, we focused on the Heisenberg
picture, in which all the dynamics are encoded in observables and the states are
constant in time. As hinted at previously, we may also take the viewpoint that
observables should be constant in time while states must evolve according to some
differential equation (the Schrödinger equation). In this case, one may ask the
following question: give an initial state ψt0 at time t0 , what is the probability of
this state evolving to a new state ψt1 and time t1 ? Let U (t) be the time evolution
operator so that U (t1 − t0 )ψ0 is the state of the system at time t1 . In particular,
U (T ) is of the form e−iHt/~ . Then, given two elements of norm one in a Hilbert
space, a natural way to check how similar they are is to take their inner product.
Thus, we define the propagator
K(ψt0 , t0 ; ψt1 , t1 ) := hψt1 , U (t1 − t0 )ψt0 i.
This provides us with a formula, but we still need to do some calculations in order
to get a meaningful answer. In special cases, it is possible to write the propagator
10This is not just a coincidence. The relationship between the position and momentum operators is an interesting topic both classically and in the quantum case. In both models, these
observables are “complimentary” in the sense that any observable whose braket with both p and
x vanishes must be a multiple of the identity. In the quantum regime, this relationship also has
consequences in terms of how accurately one can measure position and momentum simultaneously.
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SEAN POHORENCE
down explicitly. Namely, suppose that ψt0 = ψx and ψt1 = ψx0 where ψx is a
state completely localized at the position x, i.e. an eigenstate of the position
operator with eigenvalue x (of course, these states do not exist, but let’s suspend
any skepticism for now). In the case where H = P 2 , the propagator can be solved
exactly and written in terms of a Gaussian integral. In the case where H = P 2 +
2
V (X), one may hope that we can write eiHt/~ = eiP t/~ eiV (X)t/~ where the second
exponential reduces to just a complex number since ψx is an eigenstate of the
position operator, so also of V (X), and the behaviour of the first exponential is
the same as above (where V (X) = 0). However, since P and X do not commute,
we may not write U (t) in this form. Luckily for us, there is another approach we
can try. Instead of trying to compute the entire propagator at once, we can split
the time interval [t0 , t1 ] into many smaller intervals partitioned by the times t0 =
s0 < s1 < · · · < sn = t1 . Then, we may try to compute each of the intermediate
propagators. In the limit where n → ∞ and the separation ∆tj := sj+1 − sj → 0,
2
the above formula holds: eiH∆ti /~ = eiP ∆ti /~ eiV (X)∆ti /~ (this result is called the
Lie-Kato-Trotter Product Formula). Of course, when we split the larger propagator
into a number of intermediate ones, we have some degrees of freedom added to the
problem coming from the choice of the intermediate states ψxi assigned at times si .
If we integrate over each of these degrees of freedom, then after some manipulation,
the total propagator takes the form
!
! n−1
2
Z
n−1 Y
xi+1 − xi
i X
− V (xi ) ∆ti
K(ψx , t0 ; ψx0 , t1 ) = lim
exp
dxi .
n→∞
~ i=0
∆ti
i=1
Now, for each n, we may choose a smooth path x(t) such that x(si ) = xi . Then,
where n → ∞, we see that the sum in the exponential becomes
Rint1 the limit
2
(
ẋ(t))
−
V
(x(t)) dt. Writing this integral more compactly as S, and also denott0
Qn−1
ing limn→∞ i=0 dxi by Dx, we may write the propagator as
Z
i
e ~ S(x(t)) Dx
{x(t) | x(t0 )=x, x(t1 )=x0 }
where the integral is over all paths x(t) such that start at x and end at x0 . Now,
this looks very familiar. The term S looks like the classical action of a system with
L(x, ẋ) = ẋ2 − V (x). Then, it is tempting to think of this form of the propagator
as an “integral over paths” where each path is weighted by an exponential factor
depending on how large the classical action of that path is. However, we must
emphasize that this is just a rewritten version of the initial formula above, and
that it should not be interpreted as any sort of integral over path space.11 This
definition of the path integral is just a reinterpretation of the theory developed in
the previous section.
Why Path Integrals? This conclusion, combined with the difficulty of using this
formula to actually compute anything, might lead you to think that the path integral interpretation of quantum mechanics is rather useless. This is why it is
very surprising that forgetting the mathematical subtleties and simply assuming
the path integral exists as an integral can actually produce sensical results and is
11This can actually be made rigorous by considering the so-called Wiener measure. There has
been a lot of energy devoted to providing mathematical definitions of the path integral, especially
in quantum field theory where the problem becomes even more difficult.
INTRODUCTION TO MECHANICS
9
more useful than our original approach to quantization in certain situations. For
example, when the system we are describing has a special kind of symmetry called
supersymmetry 12 the ill-defined path integral may become localized and turn into
a finite-dimensional integral that can be done in the normal manner. For physicists, this phenomenon is part of the reason why supersymmetric theories behave so
nicely and why people would hope that they can make predictions about our universe. For mathematicians, research in supersymmetric quantum mechanics and
supersymmetric quantum field theories has actually uncovered links between these
physical theories and many areas of mathematics (in particular, geometry and topology). A very different example which emphasizes the importance of this approach,
in which the notion of path integral is even harder to define than in the quantum
mechanical case, is quantum field theory. Roughly speaking, quantum field theory
is a theory that unifies the predictions of quantum mechanics and special relativity.
In such a theory, one would like to develop a mathematical framework that is inherently covariant, which means invariant under the symmetries of special relativity.
If one simply tries to extend the Hilbert space definition of quantum mechanics to
quantum field theory, it is not possible (or at least not easy) to manifestly preserve
covariance. It turns out that many results that can be computed will be covariant,
but it may seem like a small miracle that this is the case at first glance. The path
integral formulation, on the other hand, is inherently covariant. This provides motivation for mathematicians and physicists to find a rigorous definition of the path
integral in quantum field theory. Such a definition may shed light on a vast number
of questions in theoretical physics. As it currently stands, what such a definition
would be, or even if one exists, is an open problem.
12Supersymmetry is a symmetry between even (commuting, bosonic) and odd (anticommuting, fermionic) particles in the theory. In particular, supersymmetry imposes that the
theory must be invariant under certain transformations between the even and odd particles.