Identical Particles

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Identical particles, or '''indistinguishable particles''', are particles that cannot be distinguished from one another, even in principle. Species of identical particles include Elementary Particle s such as Electron s, as well as composite microscopic particles such as Atom s and Molecule s. There are two main categories of identical particles: Boson s, which can share Quantum State s, and Fermion s, which are forbidden from sharing quantum states (this property of fermions is known as the Pauli Exclusion Principle .) Examples of bosons are Photon s, Gluon s, Phonon s, and Helium-4 atoms. Examples of fermions are Electron s, Neutrino s, Quark s, Proton s and Neutron s, and Helium-3 atoms. The fact that particles can be identical has important consequences in Statistical Mechanics . Calculations in statistical mechanics rely on probabilistic arguments, which are sensitive to whether or not the objects being studied are identical. As a result, identical particles exhibit markedly different statistical behavior from distinguishable particles. For example, the indistinguishability of particles has been proposed as a solution to Gibb's Mixing Paradox . DISTINGUISHING BETWEEN PARTICLES There are two ways in which one might distinguish between particles. The first method relies on differences in the particles' intrinsic physical properties, such as Mass , Electric Charge , and Spin . If differences exist, we can distinguish between the particles by measuring the relevant properties. However, it is an empirical fact that microscopic particles of the same species have completely equivalent physical properties. For instance, every electron in the universe has exactly the same electric charge; this is why we can speak of such a thing as " The Charge Of The Electron ". Even if the particles have equivalent physical properties, there remains a second method for distinguishing between particles, which is to track the trajectory of each particle. As long as we can measure the position of each particle with infinite precision (even when the particles collide), there would be no ambiguity about which particle is which. The problem with this approach is that it contradicts the principles of Quantum Mechanics . According to quantum theory, the particles do not possess definite positions during the periods between measurements. Instead, they are governed by Wavefunction s that give the probability of finding a particle at each position. As time passes, the wavefunctions tend to spread out and overlap. Once this happens, it becomes impossible to determine, in a subsequent measurement, which of the particle positions correspond to those measured earlier. The particles are then said to be ''indistinguishable''. QUANTUM MECHANICAL DESCRIPTION OF IDENTICAL PARTICLES Symmetrical and antisymmetrical states We will now make the above discussion concrete, using the formalism developed in the article on the Mathematical Formulation Of Quantum Mechanics . For simplicity, consider a system composed of two identical particles. As the particles possess equivalent physical properties, their state vectors occupy mathematically identical Hilbert Space s. If we denote the Hilbert space of a single particle as ''H'', then the Hilbert space of the combined system is formed by the Tensor Product ''H×H''. Let ''n'' denote a complete set of (discrete) quantum numbers for specifying single-particle states (for example, for the Particle In A Box problem we can take ''n'' to be the quantized Wave Vector of the wavefunction.) Suppose that one particle is in the state ''n''1, and another is in the state ''n''2. What is the quantum state of the system? We might guess that it is
	:<math> n 1, N 2 Ang	\mbox{constant} imes \bigg( n_1 ang n_2 ang + i n_2 ang n_1 ang \bigg) </math>
	:<math>Pn 1, N 2 S Ang	+ n_1, n_2 S ang</math>
	:<math>Pn 1, N 2 A Ang	- n_1, n_2 A ang</math>
	:<math>H	rac{p_1^2}{2m} + rac{p_2^2}{2m} + U(x_1 - x_2) + V(x_1) + V(x_2) </math>

  :<math> \lang N 1 N 2 \cdots N N S N 1 N 2 \cdots N N S Ang 1, \qquad \lang n_1 n_2 \cdots n_N A n_1 n_2 \cdots n_N A ang = 1 </math>
  As A Result, The Continuous Eigenstates ''x''> Are Normalized To The "http://wwwinformationdelightinfo/encyclopedia/entry/delta_function" class="copylinks">Delta Function instead of unity:
  :<math> \lang X X' Ang \delta^3 (x - x') </math>
  :<math>x 1 X 2 \cdots X N S Ang rac{\prod_j N_j!}{N!} \sum_p x_{p(1)} ang x_{p(2)} ang \cdots x_{p(N)} ang </math>
  :<math>x 1 X 2 \cdots X N A Ang rac{1}{N!} \sum_p \mathrm{sgn}(p) x_{p(1)} ang x_{p(2)} ang \cdots x_{p(N)} ang </math>







where the single-particle wavefunctions are defined, as usual, by

  :<math> \int\!\int\!\cdots\!\int\ \left\Psi^{(S/A)} {n 1 N 2 \cdots N N} (x 1, X 2, \cdots X N) Ight^2 D^3\!x 1 D^3\!x 2 \cdots D^3\!x N 1 </math>


  We Let The Composite System Evolve In Time, Interacting With A Noisy Environment Because The 0> And 1> States Are Energetically Equivalent, Neither State Is Favored, So This Process Has The Effect Of Randomizing The States (This Is Discussed In The Article On "http://wwwinformationdelightinfo/encyclopedia/entry/quantum_entanglement" class="copylinks">Quantum Entanglement ) After some time, the composite system will have an equal probability of occupying each of the states available to it We then measure the particle states



As can be seen, even a system of two particles exhibits different statistical behaviors between distinguishable particles, bosons, and fermions. In the articles on Fermi-Dirac Statistics and Bose-Einstein Statistics , these principles are extended to large number of particles, with qualitatively similar results.


THE HOMOTOPY CLASS


To understand why we have the statistics that we do for particles, we first have to note that particles are point localized excitations and that particles that are spacelike separated do not interact. In a flat d-dimensional space M, at any given time, the configuration of two identical particles can be specified as an element of M × M. If there is no overlap between the particles, so that they do not interact (at the same time, we are not referring to time delayed interactions here, which are mediated at the speed of light or slower), then we are dealing with the space × M /{coincident points}, the subspace with coincident points removed. (x,y) describes the configuration with particle I at x and particle II at y. (y,x) describes the interchanged configuration. With identical particles, the state described by (x,y) ought to be indistinguishable (which ISN'T the same thing as identical!) from the state described by (y,x). Let's look at the Homotopy Class of continuous paths from (x,y) to (y,x). If M is R^d where $d\backslash geq\; 3$, then this homotopy class only has one element. If M is R², then this homotopy class has countably many elements (i.e. a counterclockwise interchange by half a turn, a counterclockwise interchange by one and a half turns, two and a half turns, etc, a clockwise interchange by half a turn, etc). In particular, a counterclockwise interchange by half a turn is NOT Homotopic to a clockwise interchange by half a turn. Lastly, if M is R, then this homotopy class is empty. Obviously, if M is not isomorphic to R^d, we can have more complicated homotopy classes...

What does this all mean?

Let's first look at the case $d\backslash geq\; 3$. The Universal Covering Space of × M /{coincident points}, which is none other than × M /{coincident points} itself, only has two points which are physically indistinguishable from (x,y), namely (x,y) itself and (y,x). So, the only permissible interchange is two swap both particles. Performing this interchange twice gives us (x,y) back again. If this interchange results in a multiplication by +1, then we have Bose statistics and if this interchange results in a multiplication by -1, we have Fermi statistics.

Now how about R²? The universal covering space of × M /{coincident points} has infinitely many points which are physically indistinguishable from (x,y). This is described by the infinite Cyclic Group generated by making a counterclockwise half-turn interchange. Unlike the previous case, performing this interchange twice in a row does not lead us back to the original state. So, such an interchange can generically result in a multiplication by exp(iθ) (its absolute value is 1 because of Unitarity ...). This is called Anyon ic statistics. In fact, even with two DISTINGUISHABLE particles, even though (x,y) is now physically distinguishable from (y,x), if we go over to the universal covering space, we still end up with infinitely many points which are physically indistinguishable from the original point and the interchanges are generated by a counterclockwise rotation by one full turn which results in a multiplication by exp(iφ). This phase factor here is called the Mutual Statistics .

As for R, even if particle I and particle II are identical, we can always distinguish between them by the labels "the particle on the left" and "the particle on the right". There is no interchange symmetry here and such particles are called plektons.

The generalization to n identical particles doesn't give us anything qualitatively new because they are generated from the exchanges of two identical particles.