This is an interesting question! Some of the variables that you mention are combinations of the fundamental variables (operators), such as the angular momentum, and thus obey commutation relations that derive from the r and p operators. Spin also obeys analogous commutator relations. Field variables such as the E and B fields follow bosonic commutation relations that come from second quantization. Some of the variables that you mention are not defined in QM as operators, such as T, p, S and thus have no conjugate variables.

It appears that every physical value related to space-time has such a conjugate variable. Every pair you have mentioned consists of at least one dependency in space or time.

Georg, this is not quite correct. For example the spin operator has no spatial dependence. The essential feature is whether the physical variable in question can be defined as a (Hermitian) operator in QM or not.

So it seems to me that this question and the one by Lev Goldfarb are related (as pointed out by Georg Gesek on the other question). Unfortunately I am too inept to link them, so I will just answer here, and hopefully answer both questions to an extent.

The uncertainty principle comes about fundamentally from the commutator of any two variables. This is known as the generalised uncertainty principle. So for any two observables, A and B, then the generalised uncertainty principle states that

\geq (1/4)||^2

where the denote expectation values, the \Delta_Q denotes the variance in the operator, and \Delta_Q = Q - , and [A,B] = AB - BA is the usual commutator.

Now this definition shows you that there are really only two choices, either the observables commute or they don't. If they commute ([A,B] = 0) then there are no restrictions on how accurately we may determine them, if they don't commute then the generalised uncertainty principle holds. Now my understanding is that to be conjugate variables, we require slightly more of the observables, namely that they should be a Fourier pair, as for example with position and momentum. Clearly the above definition shows that it is possible (and in fact relatively trivial) to create operators that are not conjugate, but where they do not commute.

Lev Goldfarb's question specifically asked about physical insight. In general the presence of the commutator makes it fairly clear that time ordering is important in the uncertainty relation. If ordering is unimportant then the commutator must be zero, however if time ordering is important then the commutator is non-zero, and hence the uncertainty principle limits what can be achieved in a physical setting. The example I use when teaching quantum mechanics to undergraduates is to ask if there is a difference between the following (classical) processes in reference to a duel: Take 10 paces, turn and fire; turn, fire, take 10 paces.

Lastly, although I'm sure that you're aware of this, Derek, it is important to stress that the energy-time relationship is qualitatively different from the other relationships you have discussed. This is simply because time is not represented by an operator in standard quantum mechanics and \Delta_t has a fundamentally different interpretation from the expectation values I mentioned above. A nice discussion of this can be found in Griffith.

Andrew, it seems that the reason you gave is basically the standard *mathematical* one, while, as you noted, in my question, I asked for a physical insight (if there is one ???)

Dear Lev,

The argument regarding the physical insight comes from the time-ordering that is implied by the commutation properties of the two observables. Every time you perform a measurement you must project the system into an eigenstate of the measurement apparatus. If you perform sequential measurements using two different observables, if the measurements commute, i.e. are in some sense on different properties of the system, then the order is unimportant and no generalised uncertainty relationship holds. Conversely, if the observables do not commute, then the measurements are, again in a very real sense, interfering with each other.

In the case of position and momentum, a position measurement forces the particle into a definite position eigenstate. But to define such a position eigenstate requires a summation over all momentum eigenstates, and vice versa. This follows immediately from Fourier theory.

This to me is the physical insight behind the relationship.

Thanks Andrew!

I pasted your answer also as an answer to my question and will answer there.

@Andy. That's a nice answer. But the question wants you to go a step further and list some more pairs :-) Once we've got an agreed complete list of pairs, I believe it will generate further interesting questions that dig deeper.

Would anyone like to offer some more suggestions of pairs?

It is additionally worth adding that the non commutativity of self-adjoint operators entails more than just their behavior towards making consecutive measurements, that do not commute time-wise. This can be seen easily, since the mathematical expression of non commutativity does not contain any reference to time in itself. What is really means from a physical point of view is that 'no super measurements exists such that the two considered measurements can be seen as equivalent to sub measurements of this super measurement'. If the two self-adjoint operators do commute, such a super measurement is possible, it consists of the self-adjoint operator made up of the spectral decomposition that combines the two spectral decompositions of the two commuting self-adjoint operators. Hence, commutativity (and non commutativity) have in depth a physical meaning that is independent of considering time. There is also an effect related to time consecutive applications of the two measurements, but that is rather a secundary effect. This can be seen also very well in reference to classical physics. There all observables commute, since they are represented by functions on state space, and not by operators in Hilbert space. However, consecutive measurements also in classical physics often do not commute, in the sense of 'giving rise to the same outcomes when executed in reverse order'.

@Tapio, maybe a misunderstanding; I did not make the statement for operators, but for physical values. I know QM operators are mathematically described together with their canonical commutation relation, but QM operators are just corresponding (according the heuristic principle) with physical variables. Thus regarding your example the spin operator does not have a spatial dependence, but the spin as a physical value has a spatial direction and anyhow one would like to measure it, there have to be an interaction in space-time.

@Derek, At the end the product of two conjugate variables result in an action. The action is defined as the integral of the Lagrangian between two points in time, which results in the above mentioned mathematical description by Andy.

In my prospect the physical fundamental principle for these relations is the quantization of space-time through interactions between quantum systems (please see my answer to the linked question). Every interaction means an action on both partners, which are quantum systems. Thus these quantum systems do have a smallest delta in time (dt) and accordingly in space (ds). These smallest differences between two interactions define the norm in space-time, which is the constant speed of light c. Thus c equals to the ratio of ds/dt. In that way conjugate physical variables are bound to each other by the norm of space-time and do not commute.

With this in mind one can easily find the conjugate to each space-time dependent variable by complementing it with a physical value which results in an action. E.g. the magnetic flux has the SI-unit Volt-second which results, for its conjugate variable regarding the norm of space-time, to the electrical charge of a current in Ampere-second.

Certainly there are other pairs of conjugate variables where one or both of them are not easily physically interpretable.

https://www.researchgate.net/post/What_are_some_of_the_most_promising_theories_concerning_the_underlying_cause_of_quantum_entanglement

Dear Derek,

Consider any abstract variable x. Let's now suppose that the domain of this variable is X. What one must now do is define some notion of a measure on X. The idea is that we would like to build a space of measurable functions over X. In particular, a Hilbert space H of square-integrable/summable functions over X. Now comes he question: does x form a conjugate pair with some other variable y in a domain Y? In order to answer this question one needs to first define a notion of differentiation in H. The final thing that is required is as follows. Suppose you have a well defined linera first-order differential operator in H, then in order to have a well defined variable y conjugate to x you will need to be able to define a unitary operaton U:H--> K, where K is another Hilbert space corresponding to the diagonal representation of the derivative. If this can be done, then y can be constructed. But how does one build U? This is the hard part. You need to look at the harmonic decompositions of the functions f in H, and here lies the importance of Fourier analysis, spherical harmonics, ... and more strange and abstract things. What is often found is that the domain X must be extended in order to use a known 'harmonic-deconstruction'.

So, roughly speaking, you should be able to form a conjugate pair (y,x) given x but you may have to extend the domain of x first. Also, note that a Fourier transform is a unitary operator from the 'position-space' into the 'momentum space' and these two Hilbert space are the same. In the case of spherical harmonics we have the example of a unitary transform form the Hilbert space of square-integrable functions over the circle into a Hilbert space of square integrable functions over the integers. The basic rule here is that the Hilbert spaces for the conjugate pairs must have the same dimension. In the circular case above note that the integers is an infinite dimensional domain, and so is the circle. So know that you can construct unitary operations between discrete and continuous domains.

Sorry to go on, i just wanted to give you a basic idea of what is going on.

From a physical point of view, finding conjugate pairs of variables or observables without its conjugate counerpart is perhaps subordinated to the wider problem of measurement.

Underlying this problem in quantum mechanics (by the way, nice recent discussions to understand the time ordered measurements can be found in http://arxiv.org/abs/1208.3203 and http://arxiv.org/abs/1206.6224), we have the question of integrability. Integrability in classical mechanics mean having enough conserved quantities in involution (in phase space), and quantum integrability, although the concept may have some sutil diferences respect to the classical one, demands from us to find enough commuting observables to find all the quantum numbers to categorize and understand the solutions of a dynamical system.

So, from this point of view, one should pose Derek's question in the perspective of (quantum) integrability. For example: components of coordinates form a set of commuting variables (which are also observables), but this has nothing to do with integrability.

To conclude I would pose the question: how integrability and time ordered product of operators are related?

Dear Derek, the real question is more like which observable are conjugate to each other, as any pair of observables chosen at random will not be. Also, you should really be asking which independent pairs are canonical. After all, I can tale Q,P and perform infinitely many canonical transformations, each producing a new set of canonical variables.

As for uncertainty, the pair need not even be canonical as long as they do not commute. This is the content of the Schrodinger-Robertson generalisation.

You may be interested in our recent work on uncertainty arXiv:1108.1106v2.

I hope I have at least addressed your concern in part. If not feel free to get back!

Hari Dass

As I believe, you forgot the square of the angular momentum (spin), which commute, as far as I know, with angular momentum (spin).

I mean of course the z-component of the angular momentum (spin).

Yes, Kay, thats why there is no uncertainty restriction on the total angular momentum and any one of the components.

It is worth noting that the energy-time uncertainty relation differs significantly from the others, in that it is NOT based on non-commuting operators. There is no time operator in QM and it is even proven that there cannot possibly be one. QM is a "background dependent" theory; time and space are assumed from the beginning to be independent variables. Although space can be represented by position operators, there is no equivalent representation for time.

Good point, Howard!

For me also a good point Howard, but since you are absolutely right, isn't it a minus point of QM when the theory does not show the dependency of space and time from the outset? From where comes the "background" on which QM "depends"?

@Georg: The prototypical background-independent theory is General Relativity, where space and time are not assumed beforehand but are rather generated by the theory itself. So if you could unify GR and QM, you'd probably have a background-independent quantum theory.

The Dirac equation isn't enough; it unifies QM and Special Relativity, but SR assumes flat Minkowski spacetime (i.e. no gravity). You need a theory that generates its own "curvature", i.e. a theory of quantum gravity. I think string theory and loop quantum gravity are aimed in that direction; too bad they (so far) make no experimentally testable predictions.

@Sanjay: Yes, there are several approaches to constructing "time-like" operators in QM, but they all have limitations since each one needs to violate at least one of the (quite reasonable) assumptions of the impossibility proof.

Answer to Derek's question: any set of variables that commute.

@Sanjay: I don't think that's true. Plane wave basis sets are very common; they are eigenfunctions of momentum and satisfy orthogonality relations.

... but if it's quantum mechanics that you want to understand then learn from the master http://arxiv.org/pdf/math-ph/0702079.pdf His notations have very deep significance, so bear this in mind before discarding as unnecessary. I do appreciate, none the less, that this kind of mathematical physics is not to everyone's taste.

Conjugate pairing arises from the Hamiltonian formulation of classical mechanics, where the conjugate momentum is defined as the partial derivative of the Lagrangian with respect to the time derivative of coordinate, for any generalized coordinate. In field theory, it is the derivative of the Lagrange density with respect to the time derivative of the field. Therefore, there will be a conjugate momentum to any dynamical variable or field that appears in the action, if and only if its time derivative appears in the action as well.

P.S. The conjugacy is unchanged by the change to quantum theory, since Poisson bracket go over to commutators. The fundamental Poisson bracket becomes a fundamental commutator.

Pair of conjugate variables arise from the Hamiltonian formulation of a classical theory. The corresponding canonical quantization leads to commutation relations once Poisson brackets are substituted by the commutators. Time-energy uncertainty relation, in particular, has different origins and interpretations, see for example, http://arxiv.org/abs/quant-ph/0105049v3 (it was already treated by Aharonov and Bohm 2 years after their famous paper of 1959). Although it is not possible to derive the expected uncertainty relation, the meaning of "conjugate variables" for time and energy may be seen with the introduction of reparametrization invariance in a classical theory (the quantization leads naturally to the Schrödinger equation), see http://arxiv.org/abs/1105.0313.

This is an important problem in quantum measurement theory going under the name of "parameter-based uncertainty relations", i.e. observables that may not have corresponding operators in QM such as time . It has been addressed in great detail by Braunstein & Caves as well as several others. For a nice review, see e.g. Braunstein et al, Ann. Phys. 247, 135 (1996). I think there is an arxiv version as well if you hunt for it. Obviously, general relativity is not included in this theory yet, but for most "earthly physics" you may not care too much !!

Some uncertainty relations can be derived directly from Fisher Information theory. For example, in simple 1-D particle-in-a-box QM problems, the F.I. about position I is exactly proportional to the momentum squared. (This probably generalizes to many more situations, but I've only personally cranked through the math for that case.) The Cramer-Rao inequality tells us that the variance of position is >= 1 / I, so that gives a position-momentum uncertainty relation similar to (but not identical to!) the Heisenberg one. The difference is that this one involves the momentum squared rather than the variance of the momentum. It implies that a low-root-mean-square-momentum particle must be uncertain in space.

B. Roy Frieden has written a couple of books about this sort of thing; the latest and most complete is Science From Fisher Information. Some people have dismissed it as "if all you have is a hammer, everything looks like a nail"; but I think if we are ever to succeed in J. A. Wheeler's dream of "it from bit", deriving all of physics from information theory, then Fisher Information is likely to be a major contributor.

@Gurudev: Thanks for the Braunstein reference, that's an interesting approach. I note that they make heavy use of Fisher Information, starting at equation 9. :-)

@howard: that book by Friden sounds interesting. thanks for the ref, I will try to look it up sometime. and indeed they do use the Fisher info heavily :)

Spin is a good example of a QM variable with no canonically conjugate variable associated with it.

The conjugate variable to a Dirac spinor is the conjugate spinor times the Dirac matrix, gamma^0.

Here the conjugate pair and two quantities obeying or not uncertainly principle are independent two concepts. First how to define conjugate pair? x and p called the conjugate pair is because they obey Hamilton equation instead of uncertainly principle. Mathematically, uncertainly principle seems not related to conjugate pairs that obey Hamilton equation. The question is that the generalzed coordination and its canonical momentum obeyed Hamilton equation whether must be non- communicated such that they violate uncertainly principle. Or Do non-communicate pairs must obey Hamilton equation?

In relativistic quantum mechanics operators of the components of energy-momentum are canonically conjugate to operators of corresponding coordinates of four-dimensional space-time. Therefore the energy-time relationship isn't dramatically different from the other momentum-coordinate relationships.

Pierre Amiot, retired, Université Laval

I agree with Rizzuti about the origin in canonical classical mechanics. Any pair of canonically conjugate variables will have a Poisson's bracket equal to 1 and that will translate into a pair of corresponding quantum operators to have a commutator equal to ihbar, that generates an Heisenberg's dispersion relation (I do not like the word uncertainty). Any pair of angle-action variables (Hamilton-Jacobi) will do, because they are canonically conjugate. In relativity, the fouth coordinate is time and the fourth momentum is energy. They are conjugate variables, hence the Heisenberg relation between them. In the relativistic world, the difficulty comes from the fact that the quantum Hamiltonian is a "square root" of a classical one, but Wheeler gives us the road to achieve it.

Can somebody have a go at listing physical variables as conjugate pairs? Then list the ones that don't exist in pairs? It is not that easy!

I think such a list would make for a good discussion. Why are some variables in the conjugate list and some not?

The conjugate pair obey Hamilton equation ( as discussed by S. D. Liang) then how we can ascertain that E & t and phase & particle number satisfy it.

I think that the problem of the canonical conjugate is associated withthe action and any two variables that can give the action are a canonical conjugate to each other

As someone has already told you, these are two different concepts. The uncertainty relation is directly related to the commutator between the (quantum mechanical) operators associated to the variables.

Now, the concept of (pairs of) conjugate variables is well introduced in Thermodynamics by means of the First Law in its differential form. This is well explained in Callen's Book.

Examples of this are: Pressure and volume, entropy and temperature, electric potential and charge, electric field and electric dipole moment, magnetic flux density and magnetic dipole moment, surface tension and area, spin and frequency, and pairs of variables which "do work" (in a more general sense).

Now, from the quantum mechanical point of view, not all of these variables is associated to a quantum mechanical operator, so many of these pairs don't have a non-zero commutation relationship, so they don't obey a uncertainty relation. For example, you may or may not define a "charge" operator, a "frequency" operator, and so on.

The challenge is to list out all the pairs and ask if there are any variables that aren't paired.

Force?

If we assume there is a variable say in the entire universe that does not exist as a pair, we would have to come up with the universe itself as that variable; however, for any variable without any pairing, e.g., the universe, semantically that variable, the universe, may be paired with its negation or non-existence, nothing, zero v. one (uni-) if you will, pre-Big Bang, theoretical vacuum. On the other hand, my thinking is before becoming extinct or created, the universe reaches a limit, known as the Planck length (or ultimate 'string') and perhaps albeit a constant that may be regarded as having NO physical, counterfactual variable pairing or uncertainty relation. Furthermore, although h and h-bar are constants, most if not all constants equate with ratios between two variables, in other words, pairs. Long ago, due to symmetry in nature, I have come to and confirm the conclusion that the universe necessarily is therefore fractally a binary universe, two's or pairs of and within the one, uni-verse, whether in curved spacetime (not linearly/causally) infinite or not.

A good starting point for getting a different perspective may be

http://www.mathematik.uni-muenchen.de/~bohmmech/BohmHome/files/PhaseOperators.pdf

and

"Naive Realism about operators" http://link.springer.com/article/10.1007%2FBF00276801#page-1

Why in special theory of Relativity we treat space coordinates normally and convert time coordinate to space like using ' ict'? Why not keep time as it is and normalise space coordinates to time like coordinates. This may help not bringing in the velocity of light c, in the picture!

Any physical quantity, which transforms as a 4-vector under Lorentz transforms becomes a basis for conjugate pairs. For instance, 4-velocity, 4-momentum, 4-wavevector, 4-position etc can be shown to give rise to conjugate pairs.

https://www.researchgate.net/publication/236742214_On_the_Complementary_Wave_Interpretation_of_the_Dirac_Equation

As for the present form of QM, if observables satisfy the canonical-commutation rule, then those observables do form a conjugate pair. However, in QM, time is not considered a quantum mechanical operator.

In view of that, in this paper, it is shown that the result one obtains by using two operators which satisfy the canonical-commutation relation on a single component wave function, is the same when one considers the wave function of a particle in motion to be associated with a pair of complementary waves. Thereby, both the energy-time and momentum-position uncertainty relations are derived from first principles, in accordance with both the QM and the theory of relativity.

So, Derek, in response to your question, the answer boils down to, listing down all the physical properties which can be associated into 4-vectors under Lorentz transforms.

Article On the Complementary Wave Interpretation of the Dirac Equation

Thanks, Dinesh for the comment. However, i am still not happy why one can not reverse the role played by space and time coordinates in QM. Why prefer space as primary and time as secondary and then use 4-dimensional space, including time in space configuration!

In short, i am propagating leaving current biases that got historically introduced in Physics. Have a free, innovative mind!

Narendra, the time and space are inter-connected together through the basis of 4-vectors. Thus, it really doesn't matter. i.e. if one wants to express the relation in terms of time like coordinates, one can derive the same from the 4-vector. On the other hand if one wants to express in terms of space-like coordinates (as 'ict' in your discussion), that also can be obtained from the conventional 4-vector representation.

Dinesh, you have avoided the existing bias factor mentioned by me. Time to me is connected directly with life and living objects in this universe. Space comes by virtue of need for motion from one to another location, also involving time necessarily. In fact the birth of the Universe happened because of 'instant time ' anomaly, releasing huge amount of energy in the process!

Narendra

You will certainly not throw away everything. So, you have selected a reliable foundation. What is that foundation?

I followed a similar procedure. My HBM project is strictly founded on quantum logic, which I refined to Hilbert logic.

In the HBM universe steps with universe wide steps from one static status quo to the next static status quo. Each of these static sub-models conforms to quantum logic. An ordered sequence of these sub-models forms a dynamic model. The step counter acts as proper time clock.

A selected base of atomic quantum logic propositions concern elementary particles.

They are represented by closed subspaces of a Hilbert space.

A selected base of atomic Hilbert propositions concern place holders (step stones) that represent locations where a given elementary particle can be. These step stones live only during a single progression step. During their lifetime they transmit information about their presence and properties. The message is transmitted via a spherical wave that slightly folds and thus curves the embedding continuum. Together these waves constitute the potential of the particle.

The sub-spaces that represent the particles have a greater persistence than the step stones. The step stones are represented by base vectors in this sub-space. At any progression instance only one of these base vectors of this sub-space is used. The other base vectors of the sub-space exist, but are not used in that progression instance. They represent virtual step stones.

The step stones that belong to a particle (Hilbert sub-space) are generated by a stochastic process that consists of a Poisson process in combination with a binomial process. The binomial process is implemented by a 3D spread function.

The spherical waves that transport messages proceed with light speed.

Since all active step stones emit message waves whose influence diminishes with distance, and the fact that the number of contributing step stones increases with distance, the net result will be a huge background potential at every location in universe. This background potential may act as embedding continuum. Coupling to this medium acts as inertia.

The coherent distributions of step stones that include actual as well as virtual step stones can be described by density distributions. For example a combination of a (scalar) step stone density distribution and a (3D vector) current density distribution describe the static situation of the discrete distribution. These density distributions can be combined in a single quaternionic density distribution. The generated potentials correspond to a scalar potential function and a 3D vector potential function. Also these potential functions can be combined in a single quaternionic potential function. Using quaternionic density distributions and potential functions has the advantage that these functions can also be linked with the discrete symmetry characteristics of the particle. In this way the quaternionic density distribution and the corresponding quaternionic potential function describe all aspects of the concerned particle.

Like I mentioned, the 4-position vector can be written in terms of time-like coordinates. Also, simply by changing the basis, one cannot get rid of the speed of light "c" which connects the space-like coordinates with time.

However, my primary intention was to answer the question Derek raised. In doing so, what I mentioned is that, if one can list down all the 4-vectors associated with particles or fields associated with particles, then those physical quantities associated to them can be shown to give rise to conjugate pairs, which can be shown go give rise to uncertainty relations.

If one follows the procedure of quantising a system according to:

(1) write the Lagrangian for the classical system

(2) Find the momentum from the Lagrangian

(3) Calculate the Hamiltonian in the usual way

(4) replace poisson brackets with commutators

then the variables that do not have conjugate pairs at stage (2) will be those without a conjugate pair momentum but will instead form equations of constraint for your system. Also stage (2) provides a mechanism for listing variables that do not exist in pairs (for any given system).

This is a separate question than the one about which operators have a non-trivial Heisenberg uncertainty relation (HUR). As the HUR is a corollary of the Cauchy-Schwarz inequality it stands for any two operators that do not commute (see the beginning of JJ Sakurai's Modern Quantum Mechanics for a nice discussion.

In replies to this question there is some discussion about the uncertainty relation and measurement. Those interested in this area might like to read this paper "Violation of Heisenberg’s Measurement-Disturbance Relationship by Weak Measurements" by Lee A. Rozema, Ardavan Darabi, Dylan H. Mahler, Alex Hayat, Yasaman Soudagar, and Aephraim M. Steinberg Phys. Rev. Lett. 109, 100404 (2012) http://prl.aps.org/abstract/PRL/v109/i10/e100404

About quantization

1) Think first what quantum means

2) Accept that nature is fundamentally countable and finite.

3) Still it can be thought to be embedded in a (reference) continuum.

4) Try to discover what the discrete constituents are.

5) Coherent distributions of discrete objects can be described by continuous density distributions.

6) With appropriate Green's functions the coherent discrete distributions can be described by potential functions.

7) These descriptors characterize higher order building blocks.

8) If quaternionic descriptors are used, then not only the object density distribution, but also the object current density distributions can be described,

9) Quaternionic functions can store discrete symmetries. This can be used to store discrete properties, such as charges.

I think the uncertainty relation is a reflection to more general concept which is related to the differences between space time of elementary particles and space time of classical objects this can be noticed from the relation of position and momentum one is a funtion of position and the other of velocity, and time and energy one is a fuction of time and the other of velocity both of these relations are related to space time.

The uncertainty principle is a property of Fourier transformations and the uncertainty of location x momentum depends on the spread (width) of the quantum state function.

The uncertainty of the time x energy product depends on the spread (width) of the energy density as a function of time.

Both of these can be due to differences in space time of elementary particles and classical objects.

Concepts of space and time and their homogeneity as assumed are the things to ponder over. Entanglement in QM is a concept that demands that appear unrealistic to the brain that can think one thought at a time. What about the human mind that can

'think' more than one thing at the same time. Mind is not van organ of the body! What is it? Let us ponder over.

For all the dynamical variables that appear in a Lagrangian, conjugate pairs

are defined by the standard relation, p=dL/d(qdot). When carried over to quantum

theory, that produces uncertainty relations. So the variables which do not appear

in conjugate pairs would have to be absent from the Lagrangian. Parameters

characterising symmetry principles can be of this type, e.g. P,C,T quantum

numbers of particles. (For continuous symmetries, the parameters and the

generators would become a conjugate pair, but discrete symmetries are different.)

Apoorva, can you list exhaustively list valid pairs, and then list those that don't exist in pairs? It is the ones that are difficult and make us think twice that are interesting. Let's attempt a list so we can discuss any tricky ones.

When two Hermitian operators A and B commute (i.e., [A,B]= AB-BA=0, for a given eigenstate), their uncertainties product will be zero, leading to the ability of measuring both eigenvalues of A and B at the same time. Proposed by the general uncertainty principle, any pair of Hermitian commutating operators (commutators) provide simultaneously measurable observations when their mutual disturbances has no effect. It is this mutual disturbances that lead to the uncertainty of measurement specially for small systems. A famous explanation due to P. M. Dirac: "There is a limit to the fineness of our power of observation, and the smallness of the accompanying disturbances – a limit which is inherent in the nature of things and can never be surpassed by improved technique". This also means that for large systems, the classical picture of measuring is correct.

The rule doesn't always involve operators that are made of commutators. A known example is the total orbital angular momentum operator L^2, it is the sum of Lx^2, Ly^2 and Lz^2; each pair of the components do not commute, that is

[Lx,Ly]= i h /(2 pi) Lz … (cyclic) ; yet [L^2,Lx^2]=0, and similar for Ly, Lz

This means that 1) L^2 and one of its components can be measured at the same time, and 2)there is simultaneous eigenstate of L^2 and any of its one component, e.g., Lz, yet such eigenstates can't be for Lx or Ly. Similar case goes for the Hamiltonian H(=p^2/2m + V) and p, [H,p] is not zero.

The sequence of operations is quite important, as mentioned by Andrew Greentree. After waking up in the morning, for example, I dress up (A) then (B) go to lecture students and life goes fine; reversing the operations (BA) will definitely lead to lose my job, so AB-BA=(my job), and A and B do not commute, not at all.

So, in general, any two operators that work independently of each others will commute; while those that might affect (disturb) each others will not, and must have a smallest uncertainty value. I can, as another final example, have tea (C) before and after going to class; hence, A and C; and independently B and C do commute,.. luckily. :)

So which are they? Can you list all the uncertainty relationships? That is the question.

By "conjugate pair", I think it means the corresponding two operators A and B satisfy [A,B]=none zero constant up to some dimensional coefficient. What important is that the commutator [A,B] does not equals to an operator (for example, Lx and Ly are not conjugate pair).

I agree with Prof. Patel . I also think one should find the conjugate pair of an operator from the Lagrangian formalism. This makes the cananical coordinates plays a special role.Any other operators should be a function of the cananical coordinates (and their time dirivtives) f(q,qdot).And it is less important to find the conjugate pair of f.

I think what matters is to find the Lagrangain of a system take the operators listed by Prof. Abbott.

As long as quantum state functions exist that are no Dirac deltas, uncertainties are unavoidable.

Number of particles in a state vs phase -> does this pair have the same footing as e.g. Position vs momentum (in a given direction) &/or energy vs time? Answers may be different in nonrelativistic and relativistic q m!

Two points to add to this discussion . . .

1. I didn't see a discussion of charge or color-charge or quark flavor, though they are listed in Derek's question. Electric charge is a quantization constant, and can be expressed in terms of Planck's constant, the fine structure constant, the magnetic constant, and the speed of light, see http://en.wikipedia.org/wiki/Elementary_charge . I assume the other charge-like physical variables have or will be found to have similar characters. In other words, they enter the undertainty relation as factors in the uncertainty constant itself, not as variables which are uncertain or in conjugate pairs. There is of course an uncertainty in the value to which physical constants such as Planck's constant and the base unit of electric charge are known.

2. I appreciate Vadym's comment that time is not especially different than other variables. Time and position are both coordinates, and both involved in conjugate pairs. Incidentally, time can be defined in terms of mass, a somewhat complex side issue. Also, like time-energy, the position-momentum pair can be used to define a "field" but one with different qualities than the time-energy fields, as interactions are through mutual measurement rather than mutual momentum exchanges. Whether other conjugate pairs can form the basis of fields, or if so are they of any use, I do not know but I suspect so. For more information see InertiaFirst.com .

Robert: "time can be defined in terms of mass, a somewhat complex side issue. "

Can you give a reference please?

Sure, see http://physicsessays.org/doi/ref/10.4006/1.3637365 (downloadable from http://mc1soft.com/papers ) see equation (2) delta-T' = delta-T*Gamma where Gamma is a reference frame transformation factor presented in this paper without particular cause other than that it is related to gravitation and acceleration. Time and mass appear invariant in the proper reference frame of any observer because of this relation, but we see the differences as time dilation when observing photons from elsewhere, such as in the Pound-Rebka experiment.

Einstein of course postulates that both spatial and time coordinates are a function of mass, and you might take the entire field of General Relativity also to be a "reference" for my assertion, except that in a QM context it is difficult to incorporate GR.

For my own unorthodox view of a quantum justification of the mass-time relation, using the position-momentum field concept that I mentioned, see http://www.scirp.org/journal/jmp/ vol 4 no 1 Jan 2013 On Dynamics in a Quasi-measurement Field (open access http://www.scirp.org/journal/PaperDownload.aspx?paperID=27250 ).

This is a great conversation. Let me try putting in my two cents.

It appears to me that conjugate variables are arguments of a propagator, which are unitary transformations, and thus do not change the norm of the state vector. For instance, Ipsi(t)> = U(t) Ipsi(0)>, where U(t) = exp(-iHt). Here H is the Hamiltonian. So, U(t) pulls the state vector Ipsi(0)> forward in time without changing its norm.

Likewise, U(x) = exp(ipx) is a propagator (a unitary transformation) that generates a displacement in x.

So a unitary transformation can be generated from the conjugate variables of energy and time,momentum and displacement, and angular momentum and rotation. Each of these unitary transformations leave the norm of the state vector unchanged.

Cheers,

John

Measurable physical quantities A and B can be measured simultaneously only if their operators commute. Example: the the x-component of the orbital momentum Lx and the orbital momentum squared L^2 can be measured simultaneously, because operators of Lx and L^2 commute.

The question asked to list out pairs of physical quantities that are relevant. Can we exhaustively list them?

Then those variables are not in that set are interesting for discussion.

the simplest variable is a spin 1/2 component. It gives just two results UP and DOWN.

Each pair of spin components (x,y,z) for am CONJUGATE pair , they do not commute.

All examples that you list have a vectorial nature.

Components of angular momentum do not commute, etc. Every variable that you list HAS a conjugate one.

If one defines as a conjugate pair operators with [A,B] = i, then it is easy to find operators which are not member of any conjugate pair. The reason is that the spectrum of operators with [A,B] = i has to be continuous and reach from -oo to oo. This can be easily proven, because a (generalized) eigenstate of A with eigenvalue a can be transformed by the operator exp i b B/hbar into another eigenstate with eigenvalue a+b (modulo signs, too lazy to check).

So, if the spectr of an operator does not fit into this scheme, there can be no conjugate.

The simplest example is energy, with a spectrum, whatever it is, which has a lower bound. And there is, correspondingly, no operator T of time measurement in qantum theory. Operators with a discrete spectrum cannot have conjugates too.

Of course, if you define conjugate pairs simply as [A,B] =/= 0, there will be no operators without conjugates. But there will be many "conjugates" for every operator, which sounds quite strange, language intuition suggests that a conjugate has to be something unique.

Conjugate pairs signify the reality of duality in physical variables.