Tuesday, November 29, 2011

Celebrating Grassmann numbers

Off-topic rumor: the LHC will probably only restart at 13 TeV, not 14 TeV, in 2015, after the 2013-2014 break (upgrade).
Hermann Graßmann was born in 1809 as the 3rd child to a math teacher (his father) in Stettin, currently in the Northwestern corner of Poland (Czech: Štětín; much of the city was controlled by Czechoslovakia in 1919-1956, despite the distance, because we cleverly defeated Germany in World War I): his father needed to train integers which are neither powers of primes nor products of two primes so he had to produce at least 12 children.

Hermann became kind of well-known as a linguist. However, this German polymath was also a physicist, neohumanist, general scholar, publisher, and especially mathematician who wasn't appreciated during his life. Even though this man was born more than 200 years ago, some people still misunderstand the nature of his contributions. In particular, the Grassmann numbers were shown to be a part of Nature about 100 years after their discovery in mathematics.

And because they are intrinsically mathematical in character, their relevance for physics
is actually enough to show that the world cannot be classical.




We're not giving Grassmann enough credit if we only mention the Grassmann algebras; he may be viewed as the father of linear algebra and many of his original presentations (including the texts about concepts as simple as linear independence) are remarkably similar to the most modern linear algebra textbooks.

However, I will not even talk about the most general Grassmann algebras which were meant to be "new number systems" or "generalizations of quaternions" in which some multiple products or higher powers have to vanish (and where elements typically anticommute). I will only talk about the more special objects that are called "Grassmann numbers" by most particle physicists, namely "classical anticommuting numbers".

Defining Grassmann numbers

Grassmann variables which will be denoted \(\theta_i\) are supposed to be completely analogous to real or complex variables such as \(x_i\). The main difference is that the multiplication isn't commuting as it is for real or complex numbers; instead, it is anticommuting:
\[ x_i x_j = x_j x_i; \qquad \theta_i \theta_j = - \theta_j \theta_i \] The overall sign changes when you permute the factors in the product. Whenever you need it,
\[ x_i \theta_j = \theta_j x_i \] is commuting because \(x_i\) may be thought of as "something similar to a product of two \(\theta\) variables" and the two minus signs from the two transpositions cancel. One may also define the differentiation with respect to the Grassmann variables. In analogy with the commuting derivatives which obey
\[ \left[ \frac{\partial}{\partial x_i}, x_j \right] = \delta_{ij}, \] we have to define
\[ \left\{ \frac{\partial}{\partial \theta_i}, \theta_j \right\} = \delta_{ij}. \] Instead of the commutator \([A,B]=AB-BA\), we had to use the anticommutator \( \{A,B\}=AB+BA \). Finally, if we add some 20th century insights, we may also integrate over the Grassmann variables. The formulae
\[ \int {\rm d}\theta_i\, 1 = 0, \qquad \int {\rm d} \theta_i \,\theta_i = 1 \] should be viewed as the analogy of the definite integral over a commuting variable
\[ \int_{-\infty}^{+\infty} {\rm d} x\,f(x). \] The integral of the identity had to vanish because the integral of a total derivative should be zero,
\[ \int {\rm d}\theta\,\,\frac{\partial}{\partial \theta} f(\theta) = 0 \] That's expected even for commuting variables \(x_i\) if the function \(f(x)\) drops to zero at infinity. The integral of \(\theta\) is the only one that may be nonzero and we normalized it to one.

However, the Grassmann numbers can't have particular real or complex values, not even infinite values, as I will discuss momentarily. Instead, they may be viewed as intrinsically "infinitesimal" in character. That's why we don't write the "plus minus infinite" limits on the definite integral. Also, the commuting integrals with the integrand equal to \(1\) or \(x_i\) would diverge or be ill-defined. They're nicely defined for the Grassmann numbers. In fact, you may see that the integration of the Grassmann numbers is pretty much the same thing as the differentiation, followed by the substitution \(\theta_i=0\). That's the only consistent operation you may do with the Grassmann numbers, so it is employed both as a derivative and as an integral.

So far, I haven't discussed one trivial implication of the anticommuting rule. It also holds for \(i=j\) which means
\[ \theta_i \theta_i = -\theta_i \theta_i = \dots = 0 \] The square of a Grassmann variable has to vanish because it's equal to minus itself and zero is the only number that has this property. The vanishing of the square, \(\theta^2=0\), is also satisfied by "infinitesimal numbers" because we neglect \(({\rm d}x)^2\) if we only calculate at the accuracy of \({\rm d}x\). That's why the Grassmann numbers are sometimes said to be "infinitesimal".

At any rate, the vanishing of the second and higher powers means that all functions of the Grassmann variables are at most linear. That's also why I didn't have to define the integral of any higher powers: they're zero because the integrand vanishes by itself. Only if the exponents in the powers of \(\theta\) variables are \(0\) or \(1\), we may get nonzero terms. That will also be the reason why the number of fermions in a single state will be \(0\) or \(1\) even though for bosons, it (and the exponent) may be any non-negative integer.

Let's stop with the basic maths at this point. What are these crazy rules good for?

Bosons and fermions: the second, previously invisible, group of numbers

For centuries, physicists have known the concept of a field. In the real world, there apparently exist functions of space such as \(E_x(t,x,y,z)\), the \(x\)-component of the electric field, which have particular values at each point of space and each moment of time. People would first use such fields for the velocity, pressure, or temperature of a gas, liquid, or solid. However, the electric and magnetic field – that I used as my example – turned out to be more fundamental.

However, around 1930, people realized that fields were composed of particles. In particular, they suddenly understood why the energy of the electromagnetic waves at frequency \(\omega\) had to be quantized,
\[ E = N \hbar \omega, \] as originally assumed by Max Planck in 1900 when he managed to explain the black-body curve. The quantization of \(N\), i.e. the condition \( N\in{\mathbb N} \), follows from the equivalence of the electromagnetic field and an infinite-dimensional harmonic oscillator (each mode with a given \(\vec k\) and a given polarization vector \(\vec \epsilon\) represents one independent direction in which the oscillator may oscillate). And in quantum mechanics, the energy carried by a harmonic oscillator has to be integer-valued (above the ground-level energy). The integer may be interpreted as the number of particles (in this case, photons) in the given mode.

People soon realized that particles may be either bosons or fermions. The wave functions of the particles are either symmetric or antisymmetric under the permutations of any pair of identical particles:
\[ \psi(\vec y, \vec x) = \pm \psi (\vec x, \vec y) \] The plus sign holds for the bosons which is why they love to be in the same state. That's also why the Bose-Einstein condensation and coherent lasers may exist. The minus sign is chosen for fermions which is why they obey the Pauli exclusion principle that implies, among similar conditions, \(\psi(x,x)=0\). Wolfgang Pauli was the first one who proved the spin-statistics theorem: the symmetric wave function exists for particles with integer spin \(j\) while the antisymmetric wave function applies to the particles with a half-integer spin \(j\).

If you added a more general coefficient \(\varepsilon\) instead of \(\pm 1\), you would still have the condition that \(\varepsilon^2=1\) because if you permute the pair of particles twice, you surely get to the same state. That implies that \(\varepsilon=\pm 1\), anyway. So there can only be two possibilities of what the wave function does if you exchange two identical particles.

(In 2+1 dimensions, one particle orbiting around the other one creates a "winding" that can't be continuously undone. That's why the condition \(\varepsilon^2=1\) may be relaxed and one may end with more general kinds of statistics for identical particles. Such particles are known as anyons.)

Fine. We have seen that fermions have a minus sign somewhere. How is it exactly related to the Grassmann numbers? Well, the state vector for a pair of photons can be written as
\[ |\psi\rangle \sim \int {\rm d}^3 x\,{\rm d}^3 y\,\,\psi(\vec x, \vec y) \,a^\dagger(\vec x) a^\dagger(\vec y) |0\rangle. \] Normally, we would write the creation operators in the momentum basis but I chose the position basis to pump a little bit of populism to this essay: positions are more familiar to the people than momenta and in some sense, the operation above could be done, anyway.

Because it should be only possible to create photons by the creation operators \(a^\dagger\) into symmetric wave functions \(\psi\), it follows that
\[ [a^\dagger_i, a^\dagger_j] = 0. \] And indeed, this is true because similar vanishing (or small, \(\hbar\)-like) commutators exist for the fields \(\vec E, \vec B\) and the creation operators are linear combinations of the electric and magnetic fields.

However, Nature also contains fermions with \(\varepsilon=-1\): our bodies are composed of quarks and electrons (and there are other leptons). Those obey the Pauli exclusion principle so the creation operators for these fermions have to anticommute:
\[ \{a^\dagger_i, a^\dagger_j\} = 0. \] By a similar "dictionary" as in the previous paragraph, it may also be seen that the Dirac fields and other quantum fields describing the fermions have to be anticommuting fields. So they have to anticommute with each other (at various points) or, to be more precise, they have to have an anticommutator that is mostly zero up to some \(\hbar\)-weighted delta functions associated with the "quantum deformation". We have equations like
\[ \{\Psi^a(x,y,z,t),\bar\Psi^b (x',y',z',t) \} = \hbar\delta^{(3)}(\vec x-\vec x') \delta^{ab} \] I added the explicit \(\hbar\) factor to the right hand side to emphasize that the nonzero right hand side is fully analogous to the nonzero (but small) commutators \([x,p]=i\hbar\) etc. But now it's the anticommutator that's slightly nonzero (but small).

Feynman path integrals for fermions

Is there some way to get rid of this nonzero anticommutator? Well, we should recall what we did in the case of bosons. We may neglect the nonzero commutator between the electromagnetic fields if we switch from quantum field theory to its most straightforward classical limit, the classical field theory. According to Maxwell's classical theory, the components of \(\vec E\) and \(\vec B\) are commuting functions of the spacetime coordinates.

Because the fermions only differ by a sign, it should be the same thing. So one may also define "classical Dirac fields" which are Grassmann-valued. They should be taking values in the realm of the Grassmann numbers. Except, as I tried to suggest from the very beginning, there exist no particular Grassmann numbers! ;-) In particular, Grassmann variables can never be equal to particular nonzero real or complex numbers because the latter commute with each other while the Grassmann numbers must anticommute. It can't ever occur that one Grassmann number is \(\theta_1=5\) and another one is \(\theta_2=7\) because if this were the case, you could prove that \(35=-35\).

So there really exists nothing like a "set of Grassmann numbers" that you could enumerate or whose elements you could describe "separately from others" (with a possible uninteresting exception of zero which may be considered a Grassmann number, too: in some sense, setting all Grassmann numbers equal to zero is the only kind of a straightforward fully classical limit you may generate).

But there exists a system of concepts where it's totally mudane for mathematical objects to be nothing more than uncertain, ill-defined, abstract, mathematical, semi-finished, ready-to-cook meals that only become physically meaningful if you sum them, square them, cook them, and interpret the results as probabilities. The system of concepts is known as quantum mechanics and it's the framework underlying all modern physics from 1925 on. So can the Grassmann numbers that have no particular nonzero values describe something in quantum mechanics?

Why Grassmann numbers can't describe the real state of a system

You bet. As I hinted, the ordinary classical Grassmann numbers that obey an algebra but don't belong to any particular set describe "classical fermion fields" such as the Dirac field for the electron in the classical limit. How do we get from those abstract concepts to the physically testable predictions, namely the probabilities?

In this context, the most natural approach to quantum mechanics is Feynman's path-integral formulation of quantum mechanics. In this formulation, one first has to compute the probability amplitude by integrating over all histories connecting the initial and final state:
\[ {\mathcal A}_{f\leftarrow i} = \int {\mathcal D} \phi \,\,e^{iS/\hbar} \] Then the probability amplitude is used to calculate the probability using the rule that is shared with any approach to quantum mechanics,
\[ {\rm Prob} (i\to f) = |{\mathcal A}_{f\leftarrow i}|^2 \] What's funny is that these formulae involving the action \(S\) that depends on the Dirac fields \(\Psi\) don't require us to talk about any particular values of \(\Psi\). It's enough to know how to formally integrate over \(\Psi\). And we know how to integrate over the Grassmann variables; as we mentioned at the beginning, the integration is almost the same thing as the differentiation (followed by setting \(\Psi=0\)).

So the action may be formally written in terms of the fermionic Grassmann classical fields and the integral may be calculated and seen to yield a nontrivial nonzero result even though none of the fermionic fields can ever take any particular nonzero value!

You may find this operation "unnatural" or "counterintuitive" but the problem is with your intuition rather than with physics. A proper rigorous mathematical analysis of the physics problem speaks a clear language: the Grassmann numbers with their Berezin integration are on par with the ordinary commuting numbers. The democracy is as justified as the statement that \(\varepsilon=-1\) is as good a solution to the equation \(\varepsilon^2=1\) as the solution \(\varepsilon=+1\). You simply can't overlook or discriminate against the negative solution. It is as good and as relevant as the positive one. And Nature has used both solutions (and their corresponding bosonic and fermionic field machineries). In fact, all atomic matter is primarily composed out of the fermions that correspond to the negative solution and that require the Grassmann numbers and Berezin integrals.

When Feynman derived his Feynman diagrams for the first time, he actually did use the path-integral approach and integrated over all histories. For the fermions, one has to integrate over the Grassmann numbers and use the Berezin integration to define the result. If an interacting field theory is expanded around a free field theory, the most important actions are the bilinear ones. Just like we may integrate the "Gaussian" exponential
\[ \int {\rm d}^N x\,\,\exp(-\vec x \cdot M \cdot \vec x) = \frac{\pi^{N/2}}{\sqrt{{\rm det}\,M}}, \] which may be calculated by diagonalizing the symmetric matrix \(M\), we may also calculate the corresponding multi-dimensional "Gaussian" Berezin integral,
\[ \int {\rm d}^N \theta\,\,\exp(-\vec\theta \cdot A \cdot \vec \theta) = \sqrt{{\rm det} \,A} \] which may be calculated by diagonalizing the antisymmetric matrix \(A\): the result is only nonzero for an even \(N\) in this case because the eigenvalues of antisymmetric matrices come in pairs \(\pm\lambda\). The square root of the determinant of an antisymmetric matrix is a polynomial of the matrix entries, the so-called Pfaffian.

Note that there is no Gaussian power of \(\pi\) in the fermionic result. Even more importantly, the square root of the determinant of the matrix appears in the numerator (for fermions) and not in the denominator (as in the case of bosons). This fact boils down to the fact that for bosons, the integration is the opposite of the differentiation but for fermions, they're the same thing. The opposite (numerator vs denominator) location of the determinant in the bosonic and fermionic cases is a part of the reason why various factors cancel between bosons and fermions in supersymmetric theories (where each boson is married to a fermion and vice versa).

Quantum numbers are more elementary than the classical ones

I want to end up with some extra deserved Bohm bashing.

We have seen that in quantum mechanics, the testable physical predictions – the probabilities of various outcomes – are almost "fundamental" but not quite: they're calculated as the squared absolute values of the probability amplitudes. And if we adopt Feynman's path-integral calculational framework, the probability amplitudes may be computed as integrals over all histories.

However, the histories also need to know about the evolution of the fermionic fields (e.g. the Dirac field for the electrons) and these fields have classical values represented as the Grassmann numbers that don't take values from any particular set of "imaginable" possibilities.

So the intermediate results and variables – such as the particular histories – don't even "sharply" exist. You can't enumerate any examples of the histories of the fermionic fields you're integrating over (except for the vanishing fields). Nevertheless, when you do all the operations prescribed by quantum mechanics with this ready-to-cook semi-finished product, you get a real, positive, totally meaningful probability of an outcome.

There exists no physical reason why the auxiliary, intermediate concepts in a calculation should take values one can imagine – and indeed, the possible histories of the fermionic fields can't take any such "visualizable" values. Still, the integral over such histories is totally well-defined.

When people talk about the wave function, a problem is that they may "draw the graph" of such a wave function which is why all the stupid people are imagining that the wave function is a real wave. They're doing so all the time. And in this kindergarten way, they are attributing lots of classical properties to such a "wave function-like classical wave". For example, in the de Broglie-Bohm pilot wave theory attempting to mimic non-relativistic quantum mechanics, this wave exists independently of the "sharp position of the particle". It is a classical wave that literally "pushes" the classical particle by ad hoc laws meant to imitate quantum mechanics.

In reality, the wave function isn't an observable that should be directly interpreted (in particular, it is not a real classical wave): it is a semi-finished product that only yields physically interpretable results once it's squared in the quantum way to calculate the probabilities. Some other auxiliary objects that people may want to consider real – like particular histories of fields – can't even be drawn. That's a pity that people don't start to learn quantum mechanics in Feynman's path-integral way. They would understand that the intermediate calculations leading to the probability amplitudes are purely mathematical and the objects in them don't have to "exist" in the same sense as the classical hammers (approximately) "exist". Grassmann variables can't even take any particular values.

OK, why does it really disprove the Bohmian imitation of quantum mechanics?

If you tried to construct a Bohmian theory for quantum fields, i.e. in other words if you wanted to use this pseudoscientific paradigm to imitate not only some elementary quantum models from 1925 but also the description of physics that's been used since 1930 or so – quantum field theory – you would need to decide what to do with the fields.

In the Bohmian mechanics, one doesn't let the quantum mechanical Hamiltonian to choose the "preferred states" that should emerge as possible outcomes that may be perceived. Instead, like a communist dictator, David Bohm wants to prescribe the "beables" – the observables that are obliged to behave classically. For a non-relativistic particle, the position of the particle "exists" as a "beable". It literally exists in the classical sense.

It is impossible to treat the quantum mechanical spin in this communist manner because if you allowed a "real classical bit" remembering whether the particle's spin is up or down to exist, you would inevitably produce laws of physics that violate the rotational symmetry (because there would only exist one axis with respect to which the spin takes a binary value). If you want to avoid violations of the rotational symmetry, the spinors may only be used for objects that are interpreted as probability amplitudes (or something proportional to them) and the basis vectors of the 2-dimensional space of electron's spin can't correspond to any classical limit because the spin, when it's this small, can't have any nonzero classical limit at all.

The existence of fermionic fields described by the Grassmann numbers is another, although related, problem that also rules out the Bohmian picture of physics.

In electrodynamics, the only conceivable strategy is to imagine that there objectively exists a "preferred value of the classical electromagnetic field" at each point and this classical field is being driven by the "pilot wave" which is given by the functional depending on the classical configurations (a particular infinite-dimensional representation of the state vector). This is of course wrong even for the electromagnetic field because the values of the electromagnetic field aren't the universal quantity we may measure (sometimes we measure the location of a photon which is a totally different, non-commuting, observable that couldn't be predicted from the Bohmian theory).

But because Nature also contains fermions, and they're very important, any Bohmian model of field theory inevitably runs into a problem that's much more serious: we can't even define what the "current value of the classical Dirac field" is simply because this classical field must be anticommuting and the only particular value that such a field could have is therefore zero.

You can't ever create any Bohmian "model" of the fermionic fields much like you can't ever create any Bohmian "model" of the spin. These problems are related but may be viewed as independent from one another. These problems are also potentially independent from the most obvious problem of any Bohmian approach: whenever you consider the wave function to be a real wave, you need physically superluminal influences to guarantee the predicted (and verified) correlations that may violate Bell's inequalities. Therefore, every "realist" theory contradicts locality and consequently the principles of special relativity, too.

I have to emphasize that in proper quantum mechanics, interpreted in the proper quantum mechanical way, no superluminal influence needs to occur to yield the right predictions. In fact, it doesn't occur and in quantum field theories, one may prove this assertion in the full generality and exactly.

The very idea that the quantum probabilities are "composed of" something fundamental (and "real") is misguided and kind of upside down. We've seen that the probabilities are computed from probability amplitudes and/or integrals over histories of Grassmann variables. The latter don't even exist as elements of a well-defined set so they obviously can't be composed out of (or functions of) anything that is "real". The abstract mathematical building blocks such as the Grassmann fields (to be integrated over) or probability amplitudes (waiting to be squared) are the fundamental building blocks; and one may study what can be constructed out of them that takes real values independent of conventions and gauge choices. We find out that the only variables with this property – truly physical predictions – are the probabilities of different outcomes and other expectation values of Hermitian linear operators.

Quantum mechanics has been identified as the fundamental layer of the reality in 1925 and this insight can't ever be undone again.

And that's the memo.

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