Meaning of Commutation Relation

Newtonian mechanics.Now, the most important question is, is the structure of quantum mechanics the same as the structure of a Lie group? And the answer is, no, it's not the same. The reason is that the Lie group is a structure on the vector space of infinitesimal operations, while the structure of quantum mechanics is the structure of a Lie algebra on the vector space of observables.In summary, the commutation relation between two observables is a key to understanding how the abstract objects are connected with real-world observables. The symmetries of space-time models of classical physics are what allow the structure of a Lie group to be reconstructed from the infinitesimal operations around thef
  • #1

I want an explanation of the commutation relation.
According to what I understand if two operators commute then they can be measured simultaneously. If they do not commute then the measurement of one depends on other as per the value of the commutator..I hope this is correct by far.
In QFT, first of all, how can we use commutation relation for say a Klein Gordan Field? I mean how does it become an operator?
Secondly, considering the equal time commutation relation between a scalar field and its conjugate field, which gives a commutator proportional to the dirac delta function. When the two are at the same point they are infinte...suggesting that the measurement of one destroys the possibility of measurement of the other. And If at two different point in space, the measurements can be made simultaneously. Is this correct? If so, what does this even imply? How does the measurement of one field affect the other..?

I am not sure if I have been able to put my question properly, I hope I have.

Thank you in advance.
  • #2
First of all a general remark about the sloppy language of physicists, used to abbreviate everyday conversation. It's however a sin to use this slang in textbooks for beginners. It's not true that an observable is a self-adjoint operator in Hilbert space. I've not seen one around me or in the physics lab yet. An observable is defined by a measurement procedure of this observable and nothing else. The self-adjoint operators in Hilbert space represent the observables in the formal mathematical description of nature, called quantum theory. All elements of this theory have a clear meaning in terms of operationally defined preparation and measurement protocols. Otherwise you'd loose contact between the abstract mathematical model and observations on real systems, and then you'd have a nice mathematical system of axioms but no physical theory.

Now to your question of the commutation relations, which is of course a key to the understanding, how the abstract objects are connected with real-world observables. The key are symmetries. The most important ones are the space-time symmetries. Quantum theory takes the space-time models of classical physics. It's fully understood for the Galilei-Newton space time, leading to nonrelativistic quantum mechanics, which you usually learn first (and that you should!) and for Einstein-Minkowski space-time of Special Relativity, leading to relativistic quantum field theory.

So the entire business is geometry in the sense of the 19th century: Among the most important developments of pure mathematics in the 2nd half of the 19th century and the beginning of the 20th was the development of group theory and the theory of invariance. It started with Riemann and Felix Klein, who found out that geometries, including Euclidean and non-Euclidean geometry, can be classified and even reconstructed in working out the symmetries of them. E.g., Euclidean geometry can be caracterized as a point manifold that is homogeneous (you can define parallel transport of geometrical objects, and the properties of them don't change under the corresponding translations) and isotropic around one (and due to translation invariance thus any) point, i.e., you can define a rotation of geometrical objects, and this operation leaves the properties of these objects unchanged. Also there are "discrete operations" like reflections on a point or a plane which are also symmetry operations. All so defined symmetry operations form a mathematical structure called a group, and for the continuous operations as are translations and rotations in Euclidean affine space they even form a Lie group, which roughly speaking means you can build up finite translations and rotations from "infinitesimal ones" in the sense of calculus. All operations have an operation inverting them, e.g., you can rotate an object and also make another rotation (the inverse one) to bring the object back to its original position etc. Of course, there's also the trivial operation of doing nothing at all to your objects, which is the neutral element. Doing two operations after the other leads again to a operation among the symmetry operations. All this defines (roughly speaking) a Lie group.

You can learn a great deal about a Lie group by only looking at the infinitesimal operations around the neutral element. It turns out that the infinitesimal operations build a vector space (techinically speaking it's the tangent space at the group's neutral element), which has a kind of antisymmetric product, making it to a socalled Lie-algebra.

Now, the early works of the quantum phycisists brought out the Hilbert-space structure of quantum theory. Now the question is, how to systematically build the concrete quantum theory of, say a point particle within Newtonian mechanics. First of all Newtonian mechanics demands that the physical observables are consistent with the structure of Galilei-Newtonian space-time, and this structure can be reconstructed by analysing the part of the symmetry group which is continuously connected with the neutral element of this group. Each inertial observer describes the geometry of the space of locations of point particles as an Euclidean three-dimensional affine space. So at fixed time you have the above described symmetry group (called ISO(3) by the group theorists).

Then there are further symmetries involving time: first of all, there's no way to distinguish one point in time among all other points in time, leading to time-translation invariance, i.e., the behavior of a point particle doesn't depend on the time when I observe it: Given an initial condition at ##t=0## leads to the same observations as when setting the initial conditions at ##t=t_0 \neq 0##. Further, there's no way to distinguish the physics of an observer moving with constant velocity with respect to another inertial observer (particularly the former observer than is also an inertial one). All these operations together build a 10-dimensional Lie group, built by the spatial translations and rotations (making up ISO(3)), the time translations, and the "Galilei boosts". All together any such operation needs 10 parameters to specify it (3 numbers to specify the spatial translation, a unitvector and an rotation angle for a rotation, giving 3 more parameters, 1 parameter for a time translation, and a velocity for the boost, giving again 3 parameters).

Now according to Noether there's a one-to-one mapping between the infinitesimal group operations and conserved quantities: Each symmetry implies a certain conserved quantity and each conserved quantity defines a symmetry operation. For the space-time symmetries it turns out that the correspondence is as follows

time translations <-> energy
spatial translations <-> momentum
spatial rotations <-> angular momentum
Galilei boots <-> center-mass location

In classical mechanics the group is best described by the Hamilton formalism with Poisson brackets, which build a Lie algebra on the space of phase-space functions. The space-time symmetries are represented by canonical transformations with the above listed conserved quantities as generating functions. There's thus a mathematically very natural connection between these observables and the symmetries of space and time.

Now in quantum theory the observables are represented by self-adjoint operators, and these build (up to a factor ##\mathrm{i}##) a Lie algebra with the commutator of operators. Thus it's very natural to assume the commutation relations (modulo the important factor ##\mathrm{i}##) building the same Lie algebra as the Poisson brackets in classical mechanics, admitting to build the symmetry transformations on the quantum level in the same way as within Hamiltonian mechanics on the classical level. This leads to the equal-time commutation relations like
##[\hat{x}_j,\hat{p}_k]=\mathrm{i} \hbar \delta_{jk}, \quad [\hat{J}_j,\hat{J}_k]=\mathrm{i} \hbar \epsilon_{jkl} \hat{J}_l.##
There are, of course some subtleties when quantizing in this way through group-theoretical arguments which are very important but quite technical. So the Galilei symmetry of Newtonian mechanics is at the end realized by a slightly different group than the Galilei group in classical mechanics, leading, e.g., to the possibility of half-integer representations of rotations in terms of the covering group SU(2) of the classical rotation group SO(3). This is of course very important, because all the particles around us consist of particles of the spin 1/2 (protons, neutrons, and electrons).

It's a bit different in relativistic quantum theory. There it turns out that you need a many-body theory to begin with, because particles interacting at relativistic energies don't stay among themselves, but new particles are created all the time. The most convenient form of quantum many-body theory (also in the non-relativistic case) is quantum field theory. Nonrelativistic quantum field theory leads to the same results as long as there are no creation and annihilation processes involved and particle number is conserved. In relativistic physics there are no known models which don't lead to the creation and annihilation of particles. That's why you need the quantum-field theoretical formulation to begin with. Otherwise it's the very same line of arguments: You analyze the symmetry group of Minkowski space, known as the Poincare group. It's built up by space-time translations and Lorentz transformations (which include both spatial rotations and Lorentz boosts). Then, when restricted, to local field theories this leads to the known field operators. Fortunately on the fundamental level of the Standard Model of elementary particles we have to deal with spin-1/2, spin-1, and spin-0 particles only.

The equal-time commutators (bosons) or anticommutators (fermions) arise from "canonical quantization", i.e., you use the Hamiltonian formulation of classical field theories and assume the corresponding equal-time (anti-)commutation relations for the field operators of the quantum theory. Then you can build the Poincare-group operations from these field operators by using Noether's theorem. It turns out that this shortcut works for all practical purposes very well, and it has the advantage that you can build local quantum field theories from these field operators as building blocks, which guarantee causality (leading to a causal S-matrix and obedience of the linked-cluster principle) and a stable ground state (Hamiltonian that's bounded from below).

As a textbook to learn non-relativistic quantum theory in this approach I recomment

L. Ballentine, Quantum Mechanics, Addison-Wesley

and for relativistic QFT

S. Weinberg, The Quantum Theory of Fields (Vol. 1), Cambridge University Press
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  • #3
I have a simpler layman's concept. I hope it is correct.

If we allow uncertainty in time, then simultaneous measurement of A and B could mean A then B or B then A. If A and B commute, then the answers are the same either way, and the order doesn't matter. But in QM, some pairs do not commute, so that measuring A then B gives different results than B then A.

The actual nonzero value for non commuting pairs, AB-BA, is proportional to Planks constant, which is very cool because it shows that if Planks constant had zero value, there would be no quantum effects.

In his video lectures, Professor Leonard Susskind works out an example calculating the statistics for AB versus BA in a two state spin system. Search for "susskind quantum" on YouTube.
  • #4
<The commutation relations come as a result of the underlying space-time symmetry>, to make the shortest possible summary of vanhees's encyclopedic post. The Klein-Gordon field is the simplest possible quantum field *upon its so-called 'canonical quantization' the classical Poisson brackets inherent to the Hamiltonian description go to 1/ihbar commutator of (Fock space operator-valued) distributions*, for it has 0 spin. This 'quantization scheme' is the standard (=not deep/typical textbook) receipe which actually fully works for the free (non self-interacting) real/complex K-G field only, It needs some adjustment for the constrained classical field theories.
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