Quantum mechanical position and momentum operators are not field operators of a relativistic field theory and therefore no micro-causality condition applies to their commutator. It is important to notice that the field operators of a quantum field theory are labeled by space-time points (or by space-time based test-functions) and that this labeling (a mapping in terms of mathematics) makes up an indispensable part of the physical meaning of a quantum field theory. Here I can't take the time to explain how the commutation (or anti-commutation) of space-like separated field operators is related to the physical concept of causality. The probably best book on the matter is R. Haag: Local Quantum Physics, Fields,Particles, Algebras, Springer, second edition 1996.

As to the second part of the question: Since the Schrödinger picture is related to the Heisenberg picture by an obvious mapping (formal, as all field-theory) there is no need for a independent formulation.

Quantum mechanical position and momentum operators are not field operators of a relativistic field theory and therefore no micro-causality condition applies to their commutator. It is important to notice that the field operators of a quantum field theory are labeled by space-time points (or by space-time based test-functions) and that this labeling (a mapping in terms of mathematics) makes up an indispensable part of the physical meaning of a quantum field theory. Here I can't take the time to explain how the commutation (or anti-commutation) of space-like separated field operators is related to the physical concept of causality. The probably best book on the matter is R. Haag: Local Quantum Physics, Fields,Particles, Algebras, Springer, second edition 1996.

As to the second part of the question: Since the Schrödinger picture is related to the Heisenberg picture by an obvious mapping (formal, as all field-theory) there is no need for a independent formulation.

Charles and Ulrich already gave good answers.

Both in the Heisenberg and Schrödinger pictures the fundamental field operators must be imposed equal time commutation relations such that two local observables, constructed from the fundamental fields, commute at different points (at equal time).

For fundamental bosonic fields this is implemented by imposing commutation relations on these fields and their time derivatives. For fundamental fermionic fields this is implemented by imposing anticommutation relations on these fields, in which case local observables must be quadratic in the fermionic  fields.

There was a time (before the introduction of color in strong interactions) when fundamental fields were assumed to obey parastatistics of some kind; I have not investigated how commutation relations were supposed to be implemented. With QCD, parastatistics may be a closed idea, but the idea of intermediate (effective) statistics prevail, for particles in (effectively) two space dimensions. I am proud of my two Norwegian colleagues and friends, Jon Magne Leinaas and Jan Myrheim, who first pointed out this geometric possibility. In this case, the question of commutation relations can apparently be circumvented by the introduction of one or more (bosonic) Chern-Simons fields.

For a quantum field theory the equal time commutation relations are the equivalent of the initial conditions on classical field equations. It is a requirement on the dynamical equations that the property of commutation between two local observables is extended to all pairs of points with spacelike separation. This is the equivalent of the requirement of causal propagation in a (well-behaved) classical field theory.

The construction of local observables from several fundamental fields is no trivial matter in continuous space-time. I think this can be done in a satisfactory manner  in two space-time dimensions, but I am unaware of extensions of these methods to higher dimensions. 

Microscopic causality refers to the quantum commutator [s(x),s(x')], for a quantum field s, that must be a distributive kernel with support in E^+(x')\bigcup E^-(x'), for any x'\in M, where M is a classic hyperbolic space-time and E^+, (resp. E^-) is the positive (resp. negative) half cone pointed in x'. This interprets the belief that no propagation of interaction can exist between two events x and x' that do not respect the standard causality, namely there exists a time-like (or light-like) curve jointing them. When one quantizes a classical PDE, by means of the canonical quantization, one assures that such microscopic causality is respected. (See for details http://dx.doi.org/10.1016/0034-4877%2891%2990063-S and http://dx.doi.org/10.1016/0034-4877%2896%2988921-X). Note that the introduction of massive neutrinos is compatible with the principle of the microscopic causality, but has not a natural counterpart at the semiclassical level, where massive neutrinos should be identified with takions, hence with no physical particles. (More information on causality and structure of space-time can be found in publication [51] in https://docs.google.com/viewer?a=v&pid=sites&srcid=dW5pcm9tYTEuaXR8YWdvc3Rpbm9wcmFzdGFyby1lbmd8Z3g6M2FiZGVjZWI0NDgyODgyNw. or in the book http://www.worldscientific.com/worldscibooks/10.1142/5477.)

It is worthy to emphasize that in my model of quantum-gravity does not exist the concept of time and hence does not exist the concept of causality. By the way, when one introduces the observed quantum-gravity we refer to an hyperbolic space-time endowed with a frame, namely with a proper-time. In such a situation, one can talk again of particles that stay in the future of some other ones, and hence of microscopic causality.  However, quantum entanglement phenomena, encoded by observed nonlinear quantum propagators, allow to state that qualities of particles that are in the future of other ones, with respect to the observer, can instantaneously be changed with the qualities of the corresponding entangled ones, that instead are in their past. This of course does not contradict microscopic causality, whether observed nonlinear quantum propagators are time-like.

A virtual propagator in a quantum field theory, which mediates an interaction,  may travel faster than the velocity of light for a short time. This is a purely quantum effect, which does not have a classical analog. Therefore as a quantum effect microcausality may be violated for a short time in a quantum field theory even though in the zero'th order causality is respected. 

The concept of a propagator, which mediates an interaction exists in quantum mechanics and quantum field theory. The propagator connects cause to the effect. At the same time, being inherently a quantum object, the propagator obeys uncertainty relations. Therefore one is led to a dilemma. On one hand, one has to invoke quantum theory to define propagator, and on the other hand, virtual propagators may violate causality as a consequence of the uncertainty relations. 

An example of virtual propagator: Consider the scattering e+e- --> q qbar, with CM energy much below mz. A part of the amplitude is due to the exchange of virtual Z bosons.

The Feynman propagators do not vanish outside the light cone, but they are not the (anti-)commutator objects used to check microcausality, as Charles points out. The Feynman propagators smartly takes account of (virtual) creation and annhilation processes in a  manifestly covariant way. And without violating microcausality.

A relativistic quantum particle may suddenly seem to travel backward in time, but this can be reinterpreted as the annihilation with its antiparticle, which was created some time before. The particle of that pair creation process can move forward in time, outside the light cone of the first particle. In a Feynman diagram this is represented by a single (causality violating) world line.

Of course, this is all poetry, which we can think of while we shut up and calculate...

 is not Lorentz invariant in general. The prerequisite for Lorentz  invariance is precisely the condition that field operators commute with each other when the points x and y are separated by a space-like interval. This condition is quite sacred. The Lagrangian density, from which the theory is derived is Lorentz invariant by construction, but apparently, solely that does not  guarantee Lorentz invariance of the propagator.

Intuitively it is not very difficult to understand. Lagrangian is a classical quantity, which respects Lorentz invariance, whereas the propagator is a quantum mechanical object, which comes to exist only after the classical theory is quantized. In the process of quantization classical fields are elevated to the level of operator (which acts on a set of orthonormal basis states including vacuum state |0>; The vacuum state has zero occupation number whereas other basis states have non zero occupation number), and commutation relations are brought in.  Therefore to guarantee Lorentz invariance of the propagator, additional restrictions  are needed on commutation relations.  

In general, causality in physics means that two events outside each other's light-cone cannot be causally connected.  In QFT, observables in Schrodinger picture are hermitian operators that depend on space and time coordinates. The microcausality in QFT states that the commutator of any two operators at (r1,t1) and (r2,t2), respectively, must vanish if (r1,t1) and (r2,t2) are outside each other's light-cone. This is required in order that  measurement of one observable at (r1,t1) does not affect the other at (r2,t2) as demanded by relativistic causality.

Dear Biswajoy,

from your answer I understood that perhaps we are talking about different mathematical objects. Really my observed nonlinear quantum propagator (say NQP) cannot be identified with the usual quantum propagator of QFT, (say QP). NQP is a compact solution V of the observed quantum super Yang-Mills PDEs, (say (YM)[i]), such that \partial V=N_0\bigcup P\bigcup N_1, where N_0 and N_1 are respectively the initial and final Cauchy data representing initial and final particles entering in the reaction. These are 3-dimensional, space-like, non-point-like objects, such that their times, with respect to the observer, are t_0

Sub causality of an event.

@Ayman

Perhaps causality of events related to microscopic or quantum scales. For simplicity let us take the QFT of one free real scalar field.  Let us take two space time points x and y. If the two point correlation of the field is zero, then these two points are not causally connected.

@Patrick

Perhaps you mean to say that operators in Heisenberg picture depend on space and time points. Kindly clarify.

Biswajoy> If the two point correlation of the field is zero, then these two points are not causally connected.

I am sure you didn't mean to say that (because of what you have written before).

1) The correlator can be nonzero even if the points are not causally connected.

2) The correlator could be zero, while the commutator [phi(x), phi(y)] were nonzero (and hence the points were causally connected). 

Kare: You are absolutely right on your point (2). However,  I was considering only one field. How can an operator not commute with itself? Please explain. Many thanks in advance.

Perhaps the commutator the field with Hamiltonian is something non-trivial ?

Dear Biswajoy,

 OK the old correlation function, can help you to enter in my subject ...

Biswajoy> How can an operator not commute with itself

For different points x and y, phi(x) and phi(y) is not the same. 

Kare> For different space points the equal time  commutator between phi(x) and phi(y) is zero in the Heisenberg picture. But then perhaps they may not commute at different points in time. See for example: L. Ryder's book, Eqn. 4.9 page 128.                                                                                     

Microcausality can be formulated this way: the averages of field observables (e.g. average of electric or magnetic field operators) commute if they are separated by light cone where the border of the cone is smeared with fuzzy region of size of Compton wave length where the mass in Compton wave length is the characteristic mass of the system. Good point to start is Bohr&Rosenfeld, Bjerken&Drell -"physical  point of view", or Wightman-Bogoliubov - "mathematical point of view"

The (particle) coordinate can be defined in relativistic quantum theory (say, as Newton and Wigner did), but can not be defined in quantum field theory.

All these without accounting for gravity.

G.S.> the averages of field observables (e.g. average of electric or magnetic field operators) commute

This is not a correct formulation. Averages (expectation values in some state) are complex numbers, which always commutes. 

Kare Olaussen - that's why I used word "observables". The averages of observable electric and magnetic fields are real numbers and the reason why observables are presented by real numbers is exactly what you said: complex numbers may not represent observables since their measurement requires two simultaneous observations.

Real numbers, as a subset of a complex numbers, always commute also...  

Dear Prof. Brahmachari,

Your question has been the bread and butter of my research over last 25 years [see Ann.Phys. 338, 207 (2013) and references therein]. The brief answer to your question is, "Strict microcausality is inbuilt in quantum field theory, and may be revealed by a suitable one-to-one transformation of Schwinger-Perel-Keldysh's closed-time-loop formalism". There are subtleties associated with renormalisation rooted in so-called Schwinger terms (see the quoted paper), but, by and large, it is as simple as that.

Best

LP

@ Plimak, We are fortunate to welcome someone who is an active researcher in this field of discussion.

Could you give us an example where a two point correlation function vanishes and the operator valued functions at these two space time points (x,t1), (y,t2) do not commute. We are considering a single real and free scalar field only. And we are working in the Heisenberg picture where equal time commutation relation is [phi(x,t),phi(y,t)]=0.

Limit it to two-pole (two-point) functions. Generally, you need two of them: Kubo's response, and the normally-ordered operator product. The former vanishes for all space-like intervals. The latter is identically zero if the field is in a vacuum state. For the algebra, see Ann.Phys. 323, 1963 (2008).

Biswajoy, it does not matter whether the field operators are viewed in Heisenberg picture or the Schrodinger, as the commutators do not depend on the pictures. Main point that I wished to emphasize is that simultaneous accurate measurements of two observables at two distinct space-time points require that they commute (only then one can have common set of eigenvectors for the two observables).

Since causality implies that if the two distinct space-time points are space-like separated, measurements of the two observable cannot influence each other, and therefore one should be able to measure both accurately, hence it follows that the two observables at these space-like separated events  must commute. That is the argument for microcausality. 

I would like to highlight an interesting point:  When compatibility of space-like separated observables is being considered, this seems to imply macroscopic size of  involved devices and would probably allow for violations at a scale set by the Compton wavelength of an electron. Nevertheless nature seems to be very distinctive here: If we construct free quantum fields with the wrong connection between spin and statistics (e.g. Bose statistics for electrons) the incompatibility of spacelike separated observables actually appears only on the scale of Compton-wavelength of the particles belonging to the fields. In short: Only strict micro-causality enforces the well-established connection between spin and statistics.

A final remark: For commuting observables A and B we know that the statistics of measurement results of A remains unaltered, whether simultaneously B is active or not. It does not say that the results obtained from A and B are statistically independent. To the contrary, they may be strictly correlated if it is the case if B is a function of A in the sense of the functional calculus of operators.

Ulrich highlights a good point, which also exhibits the difference between microcausality (strict commutation outside the lightcone) and macrocausality (which is the concept also described in an earlier post by G.S.Japardize --- and wrongly referred to as microcausality). The observed fact that nature obeys the spin-statistics theorem is a good indication that microcausality is the physically correct concept.

Dear All,

I gave the initial question and my own remarks more thought, and now believe that a few reservations are in order. The inbuilt (kinematical) property of QFT is actually NONRELATIVISTIC causality. This is not what prof. Brahmachari asked. Relativistic causality, IMHO, is a property of models. I am wondering if prof. Brahmachari would agree with the following reformulation of his question: "Can we construct an example of a relativistically covariant quantized field not obeying microcausality". For all fields we know, their commutators may be expressed applying local transformations (differentiations, etc.) of the Pauli-Jordan function. Such local transformations do not change its fundamental property, that it is zero outside the light cone.

L.I.Plimak> "Can we construct an example of a relativistically covariant quantized field not obeying microcausality"

I think Ulrich already gave the answer. A massive scalar field theory quantized with anti-commutation relations (demanding all observables to be quadratic in the fields) would not obey microcausality, only macrocausality.  Such objects are in common use in QFTs, as ghost fields.

Isn't this just a basis problem?  Once constructs nonlocal constructions in physics all the time (eg using Fourier expansions) but the true microscopic reality is local.  Doesn't manifestly covariant QFT give microcausality?  To answer this you need a real space formulation of the theory and a subset of allowable initial data.  I made some progress in this direction for QED very recently.  

It is interesting someone brought up Haag.  Haag proved any interacting QFT is inconsistent.  This paper is motivated by a need to reformulate it so this does not happen.  In the mass of perturbation terms one needs to find if localized initial data remains in the light cone of its future.  Since the usual QFT is asymptotic this is problematic because you can't utilize an infinite number of terms.  

Article Beyond Quantum Fields: An Operator-Free Covering Theory for QED

Dear Clifford,

microcausality is automatically satisfied in observed nonlinear quantum propagators in my quantum-gravity theory. Therefore this problem has been just solved !...

All the best,

Agostino

Condition for microcausality, requires existence of antiparticles. Therefore single particle relativistic wave equations do not satisfy microcausality condition. Even relativistic theories with a fixed number of particles will also not satisfy microcausality. Quantum Field theories have the advantage of having a mechanism for creation and destruction of particles and antiparticles. This feature is needed in a theory for it to satisfy microcausality.

This was pointed out in this thread by Charles. This is indeed the truth.

Ref: Quantum field theory by Peskin and Schroeder,

Dear Biswajoy,

I agree with your last post and remark about Charles,s post in this thread. However, it should be important to underline also that with the classic quantum field theory it is impossible justify quantum entanglement. For that my nonlinear quantum propagator is necessary.

My best regards,

Agostino

@Biswajoy, that is not really true and has certainly not been proved.  I agree this the approach that is used in P&S but that is far from saying that such an approach is *required*.  QFT is a formal theory.  Since it lacks any rigorous foundation you can't make statements about what is really proves.  It derives numerical results.  That is really the extent of it.  You can get microcausality and entanglement in a classical field theory if it is 3N dimensional.  The trouble is in dealing with the necessary many time variables that arise in the process and interpreting "causality" in a meaningful way in such a theory.  One can produce a causal theory without antiparticles.  It is problematic for operator and path integral based approaches.  That is true.  

 Charles,

thank you for this interesting publication.  It seems you have accomplished your results by a careful order of limits.  Am I right?  I tried a radically different foundation using a purely initial data formalism above and found that the photon norm arose as a conservation law on its own without the need to impose any commutation relations or operators at all.  I will review your paper more closely later.  

There is a subtle difference between vanishing of the commutator outside light cone and the vanishing of propagator. If we start from the commutator then denoting the scalar field by p

= -

the right hand side is the difference between two propagators. What Peskin and Schroeder's book has done is that they have computed the propagators in the right hand side which shows that individual terms in the RHS are nonzero for a space-like separation between x and y (there is an exponential factor signifying that they vanish when x and y are infinitely separated). Apparently this may look like a violation of causality. However, because the terms are equal to each other (only for space-like separation) they do cancel, therefore RHS vanishes. This shows that the VEV of the commutator is zero for such a space like separation. Thus causality is restored. 

Physically it means that if two points x and y in MInkowski space are separated by a space-like interval then any measurement at the point x cannot influence another measurement at the point y.

Almost all books that attempt to make peace between causality and QFT rely on this somewhat strained argument about commutators.  I worry that the enthusiasm to embrace them is driven by the highly credulous inclinations of people who can stomach the current state of the theory.  What is different about Schro and Heis approaches to QM and QFT is that the QFT approach generally is built on operators in either case.  We only really need operators in QM for measurements and some of us believe the measurement actions arise from the details of the device and so the operators are not really even necessary here.  A theory built on a tower of actual many body wavefunctions that interact along the two particle diagonals should leave no such formal constructions.  I have found that causality in the actual wavefunctions do hold in this case but the theory requires some slight extension to make such a construction possible.  

   Please read my discussion of this tenuous point in my paper.   Causality Is Inconsistent With Quantum Field Theory.   

 @Fred,

I took a quick look at your paper.  I have always hated the way negative energy solutions are finagled into positive energy positrons with commutation relations.  I dealt with this in my attempt by introducing a different lagrangian for positrons altogether.  Localization of an electron packet beyond the classical electron radius results in some necessity for negative energy solutions in the classical Dirac field approach.  Higher particle number spaces seem to help the situation.  QFT is known to be inconsistent in the case of any interaction so having causality problems is not surprising but I think it is just a matter of the right tweaks to justify the spectacularly successful calculations of QED.  Anyhow, that is what I have worked to accomplish.  

Finding some problems in a strong limit for an asymptotic field theory seems inevitable. Where would you like to go from here is what I'd like to know.

I agree.  Feynman's approach was simple:  I call it NONEFIT (no negative energy forward in time).  Following that to its logical conclusion provides a sensible meaning to QFT namely antimatter moving forward in time with positive energy is the same as matter moving backward in time with negative energy.  As I see it, the direction of time is arbitrary along with the sign of charges and energies.

The inconsistency arises when you allow for propagation in the elsewhere. Such propagation can be viewed as either propagation backward in time with negative energy or forward in time with positive energy. Hence a Feynman zig-zag arises naturally with no need to "invent" antimatter. This gives rise to the Feynman propagator.  For more please read Causality Is Inconsistent With Quantum Field Theory.   where I go over these considerations in great detail.

I know some people have played with the Wheeler propagator to give evolution that gives no contribution outside the light cone but have not reviewed it.  I think Fred is using the usual ones though.  I he referring to the propagators themselves into the absolute elsewhere or some gauge independent reality?  

Charles, I believe you mistook my paper.  It is precisely that the Feynman propagator does not vanish that gives us propagation in the elsewhere and therefore a proof that antimatter must exist.  On the other hand if you choose the retarded propagator you get no propagation in the elsewhere--the consistent condition for causality..  My point is that there is no causality argument consistent with the Feynman propagator.  Please read the paper.  I am sorry that the abstract is not clear.  As far as the relationship between propagators and commutators.  In brief:

Causality in quantum field theory means the vanishing of commutators for spacelike separated fields (VCSSF). For VCSSF to be tenable, and therefore, to have both retarded and advanced propagators separately vanish in the elsewhere, a superposition of negative energy antiparticle and positive energy particle propagators, traveling forward in time, and a superposition of negative energy particle and positive energy antiparticle propagators, traveling backward in time, are required. Hence VCSSF predicts non-vanishing probabilities for both negative energy particles in the forward-through-time direction and positive energy antiparticles in the backwards-through-time direction. Therefore, since VCSSF is unrealizable in a stable universe, tachyonic propagation must occur in denial of causality.. 

To have the field commutators vanish in the elsewhere you must use the retarded Gret or advanced Gadv propagator. In either case you are superposing Feynman propagators for both negative energy and positive energy propagators.  That is my point.  Please read my paper and you will hopefully understand my point of view.

Since in calculating Gret, or Gadv, we use superpositions of Feynman and anti-Feynman propagators we need to regard these “causal” propagators with suspicion. The usual reason people use this argument  as proof of causality is that both Gret and Gadv vanish,  when (x – y)2 

Dear All,

the opinion that quantum energy cannot have a spectrum with negative values is completely unfounded and related to the (semi-)classical point of view.

But in the quantum world the situation is completely different ! On the other hand in order to conserve the quantum number of the vacuum, 'negative energy' is a necessity. For more information see the following paper. http://arxiv.org/abs/1206.4856. The microcausality is automatically conserved in the observed nonlinear quantum propagators as introduced in my quantum-gravity theory.

Let us consider a scalar field p

Then = -

In the RHS both terms are Lorentz invariant. One can find a Lorentz transformation for space like separation between x and y, such that x-->y, y-->x. Therefore the two terms cancel and the commutator vanishes. Here is a relation between spin and statistics. Had we used anti-commutators in place of commutators then the RHS would not have vanished for space-like separation. Therefore we are forced to use commutators for the quantization of a scalar field.

I have found a short discussion on this interesting point in Itzykson and Zuber's book. See page 150.

In my view the question of causality in QM takes precedence over the formalism whether it's Schrödinger's wave function, Heisenberg-Born's matrix mechanics or field operators. Since the Bell's inequality experiments and in contradiction with relativity (EPR etc.) we know that entanglement is here to stay. Prior to his wave theory, (in his inaugural lecture at Zurich in 1922), Schrödinger argued that the evidence did not justify our assumptions that physical laws were deterministic and strictly causal. Boy was he right when he wrote:

"If nature is more complicated than a game of chess, a belief to which one tends to incline, then a physical system cannot be determined by a finite number of observations. But in practice a finite number of observations is all that we can make. All that is left to determinism is to believe that an infinite accumulation of observations would in principle enable it completely to determine the system. Such was the standpoint and view of classical physics, which latter certainly had a right to see what it could make of it. But the opposite standpoint has an equal justification: we are not compelled to assume that an infinite number of observations, which cannot in any case be carried out in practice, would suffice to give us a complete determination."

Yet remember that as entangled the Quantum world is you can not tell us about it in a message that will exceed the speed of light. Yes! you can encrypt it but that's about how far your causality goes ;-)