In the standard proof showing that hermitian operators have real eigenvalues we exploit the symmetry of the operators to show that they only have real eigenvalues.Does this hold for ALL hermitian operators in general?
In an infinite-dimensional Hilbert space a bounded Hermitian operator can have the empty set of eigenvalues. A typical example is the operator of multiplication by t in the space L2[0,1], i.e. the operator T that maps every function f(t) to function tf(t).
That is why in Functional Analysis one defines the spectrum of an operator T: H --> H not as the set of all eigenvalues, but in a more complicated way, namely as the set of those complex numbers a, that (T - aI) is not invertible, where the letter ``I'' stands for the identity operator. Now the basic facts:
1. The spectrum of a bounded linear operator is not empty and is a closed bounded set.
2. The spectrum of a bounded linear operator contains all the eigenvalues of the operator, but also can contain some other points.
3. All the points of the spectrum of a bounded Hermitian operator are real numbers.
Remark: in Functional Analysis Hermitian operators are usually called ``self-adjoint operators''.
Yes. If you want to know the details these can be found in any textbook on functional analysis. My personal favourite introduction to the subject is Introductory Functional Analysis with Applications
by Erwin Kreyszig.
The short answer is yes. Longer answer is that for an operator to be well-defined, the function space on which it operates must be specified (in other words, an operator on itself, without its domain of operation being specified, is a meaningless concept). This applies also to what we call Hermitian operators. The property of Hermiticity of an operator relies on the definition of the inner product in the vector space in which it operates, so that the relevant vector space is required to be an inner-product space. In the case of dealing with differential operators, in order to determine whether an operator is Hermitian or not, we apply differentiation by parts, and more generally the Green theorems. For Hermitian operators the boundary terms must either identically vanish or add up to an identically vanishing contribution. In general therefore the space of functions in which an operator is Hermitian is a proper subset of all functions that have finite norms (the inner product induces a norm in the function space -- inner-product spaces are normed spaces).
I shall not go into further details, and for these refer you to any good book on functional analysis, such as the one by Erwin Kreyszig (Introductory Functional Analysis with Applications, John-Wiley & Sons, New York, 1978). One point to be aware of is that in general the spectrum of an operator consists of three disjoint sets of which one is the set of its eigenvalues. The other two sets may be empty, which is the case in finite-dimensional inner-product spaces (for instance, in dealing with finite-dimensional matrices, one is working in one such space). The spectrum of an operator is defined in terms its resolvent.
In an infinite-dimensional Hilbert space a bounded Hermitian operator can have the empty set of eigenvalues. A typical example is the operator of multiplication by t in the space L2[0,1], i.e. the operator T that maps every function f(t) to function tf(t).
That is why in Functional Analysis one defines the spectrum of an operator T: H --> H not as the set of all eigenvalues, but in a more complicated way, namely as the set of those complex numbers a, that (T - aI) is not invertible, where the letter ``I'' stands for the identity operator. Now the basic facts:
1. The spectrum of a bounded linear operator is not empty and is a closed bounded set.
2. The spectrum of a bounded linear operator contains all the eigenvalues of the operator, but also can contain some other points.
3. All the points of the spectrum of a bounded Hermitian operator are real numbers.
Remark: in Functional Analysis Hermitian operators are usually called ``self-adjoint operators''.
I like Vladimir's expose--it is essentially right.
Subtleties are present--most operators we know of are not bounded, and are not self-adjoint but "essentially self-adjoint" in the usual QM Hilbert space of quadratically integrable functions, i.e., they are self-adjoint on a subset of the Hilbert space.
One of my interests is in resonances in Quantum mechanics. These are states that decay: historically they were defined with outgoing wave boundary conditions, and thus not in the normal Hilbert space L^2(R^n).- They can be defined as complex poles of the resolvent 1/(H-Iz) in the second Riemann sheet.
So with sensible definitions we can generalise the concept of eigenvalues (or rather of the spectrum of an operator) to the complex plane, even with a self-adjoint operator! A very solid discussion can be found in the series of books by Michael Reed and Barry Simon (Methods of Modern Mathematical Physics).
Thank you gentlemen for your answers.Like Walet,I think subtleties do exist.If for example I could define a Hermitian time operator,using the standard proof would it follow that the time operator has real eigenvalues?And if so,why would standard non-relativistic quantum mechanics use time as an external parameter if a hermitian time operator could be defined within the postulates of the theory?
You are welcome Carringtone. In non-relativistic theories (whether classical or quantum) one has space and time, not space-time. In these theories even space coordinates are merely parameters. The only difference between space and time is that time is uniform, space need not, due to presence of matter that, when not uniformly distributed, breaks the continuous translational symmetry.
You are now getting in deeper water: The role of time in NR QM is apparently very different than that of all other variables, but this is a long-standing discussion, and many different views have been expressed on the topic.
I would strongly argue that time should not have an operator associated with it; If there were one, would you argue hat [H,t]=\hbar? I take the position that the time-energy uncertainty relation has a totally different origin from all other ones: it is statement about the duration of a measurement...
Niels,
As you very well know, there is no problem to define a self-adjoint time operator on the time interval (-∞,+∞). Cf. the momentum operator in on dimension.
There is nothing mysterious in treating space-momentum and time-energy as conjugate variables (operators); carry out the mathematics and then do the physical interpretation depending on the situation at hand.
Best regards
erkki
Dear Erkki,
I didn't say it was impossible--it is matter of interpretation!
The mathematics will work--the operator can be defined, but looks rather strange, which is why I prefer the "time is different" interpretation.
Niels
For the distinction between Hermitian and self-adjoint operators and its consequences see the definition in Reed/Simon vol. 1, page 255
Yes; otherwise they would not be hermitian! It is as simple as that!
Yes, and the proof is exactly the same as for the case of Hermitian matrices. This is true for bounded operators (called Hermitian operators) and for their unbounded analogs, called self-adjoint operators. However, in the latter case, there may not be any eigenvalues (think, for example, of the Dirichlet Laplacian on the full Euclidean space R^N, which has for spectrum the closed half-line [0, \infty) ; this spectrum is purely continuous and therefore does not contain any eigenvalues). Nevertheless, the spectrum of a (possibly) unbounded self-adjoint operator (acting in a complex Hilbert space) is always a subset of the real line.
If I understand you correctly, you are worried about the difference between symmetric and Hermitian operators. Here a few remarks:
1) this difference only exists in function spaces. As long as you are working in finite diemnsional space, there is no difference. Both symmetric and Hermitian matrices then have real eigenvalues.
2) In infinite dimensional spaces, the Hermitian matrices are a genuine subset of the symmetric ones. The statement of the spectral theorem implies that Hermitian operators have a real spectrum, and that they can be expressed as a (generalized) linear combination of projectors. For details, see Reed-Simon. Symmetric operators do not, in general, have real spectrum.
3) What is the difference between sellf adjoint and symmetric? That is not easy. Symmetric is easy to define: it means in essence that, by claculating formally, you can move the operator from one side of the scalar product to the other. In formulae (f, A g) = (Af, g), where the scalar product is, for example, the usual L^2 scalar product defined by the integral. This, however, is not enough to guarantee reality of the spectrum. A neat example is A = p^2 - x^4 = -d^2/dx^2 - x^4. Using semiclassics, you can easily see that the asymptotic behaviour of *any* solution of Af = Ef for x->infinity goes as x^{-1}*exp(+/- i x^(3/2)). This is an L^2 function, so at the naive level you have a valid eigenfunction for every E, including complex ones. How to settle such issues is what the definition of Hermitian operators is all about. It concerns being really careful about the domain of definition of your operatos and not doing things just formally. Note further that the Hermitian/symmetric issue only arises if your operators are unbounded. If you can define your operator for every L^2 function and get another L^2 function, you are OK.
4) The issue of continuous spectrum is yet another problem: when you go to infinite dimensional spaces, apart from the trouble with the difference between Hermitian and symmetric, you have the fact that you may not have eigenvectors. This is the problem with continuous spectrum, which arises for Hermitian operators, whether or not they are bounded.
To speak about sejl adjint operators one needs a scalar product g, that is either a symmetric bilinear map or a sesquilinear map, depending if the space H is real or complex. If the vector space H is infinite dimensional then one requires it to be a Hilbert space (that is complete). An operator f can be continuous only on a vector subspace of H. It is symmetric if g(fu,v)=g(u,fv). It can then be extended to the whole of H, but the extension is not necessarily unique (if it is so it is said essentially symmetric). In all cases the extension is self adjoint and, as any self adjoint operator, its spectrum is real. The spectrum is the equivalent of eigen values for infinite dimensional vecor spaces : the eigen values belong to the spectrum, and they coincide for finite dimensions.
A simple example may illustrate the issue: Consider quantum mechanics on the real line and the quantum mechanical operator O = qqqp+pqqq . This operator is hermtian (or symmetric) but it still has an eigenvalue -i\hbar with a square-integrable eigenfunction. O is symmetric but not self-adjoint.
Stupid question if you do not clarify which definition of "Hermitian" you are using. Typically in QM, it is used as a synonym for self-adjoint.
Sorry Golin, but hermitian operators are well defined in mathematics. The usual QM (in its traditional, ill defined, flavor) is not all science,thanks God !
By the way who has anonymously down voted my answer in this thread ? Could he explain himself ? Or is this just another clanish petty vengence from some academic gang ?
@colin I think we should ignore the trivialities of definitions and focus on the axioms.Here is my problem in a reductio ad absurdum form;
1 all 'self adjoint' operators have real eigenvalues.
2A hermitian time operator will by definition have real eigenvalues
3A Self Adjoint operator can describe time in QM
It follows that the parametrization technique used by standard non- relativistic QM is redundant and possibly harmful to quantum theory.If not,then there is something manifestly false in the above syllogism.My question is:which is it?
Hardly "stupid" ...
To Carrington,
As long as you use mathematics, you have to deal with its definition. Iin the common formalization of QM, you have observables associated to physically measured quantities. These observables are hermitian operators in Hilbert space, and moreover they are compact (so their spectrum is discrete). Time is not an observable, but you have an operator which appears in the Schrödinger equation, which is unitary, and so cannot be hermition (its eigenvalues are imaginary), and the derivative of this operator (that coudl be seen as associated to time) is anti hermition. And actually this operator is not compact (the set of its eigen vectors is continuous)
If you want a clear rigorous definition of observables in QM see my paper QM revisited on this site.
Dear all,
Why making the answers to this question so complicated. As a consumer and practitioner of quantum mechanics one can go along way towards the simple definitions given by Schrödinger in his pioneering papers in 1926. He simply demanded that the solution to his equation should be continuous, single valued and finite throughout the configuration space of the system.
Of course, he was familiar with the classical theory of ordinary differential equations due to Weyl, extending Sturm-Liouville theory for finite intervals to include intervals where the differential operator has singular behaviour (e.g. at origin at infinity). This extension lead to the famous limit-point and limit-circle classifications already before 1910.
The generalization to topological spaces can then be done in several ways – one is e.g. in terms of so-called Gelfand triples and rigged Hilbert spaces. If one takes care of domains and ranges one can for instance make rigorous sense of Dirac’s formalism.
Important generalizations can also be made by defining so-called self-adjoint analytic families of operators through the theorem of Balslev and Combes, 1973 (a very popular method in chemical physics). In particular these techniques allow generalised eigenvectors corresponding the absolutely continuous spectrum as well as allowing rigorous quantitative and qualitative interpretations of the infamous Gamow vectors.
Dear Erkki,
Do you really believe that your answer is simpler ??? In mathematics the whole thing takes 2 or 3 pages, no more.
Yes I believe so, since “the whole thing” presupposes a lot of advanced math, which of course are fundamentally important, nevertheless might become inaccessible for a theoretical natural scientist who needs to apply rigour, quantitity and quality (cf. the story of Einstein and general relativity).
I think your “Quantum Mechanics Revisited” is an excellent example of providing a fertile ground for consolidation and further development.
I stand by my earlier answer that all hermitian operators do indeed have real eigenvalues; otherwise they would not be hermitian! Hermitian operators represent our 'direct' link to Nature, so to speak. For me as a physicist, this is a satisfactory answer. However, I cannot help making the following remark. Clearly, as we can readily deduce from other colleagues' answers and as we know very well, mathematicians and physicists often use different 'languages'. The best results are attained when both join forces. This is a plea to mathematicians to go ahead and continue doing their wonderful work; but to leave us -- we humble physicists -- to get on with our intuitions, methodologies and methods.
Dear Humam,
Of course in any way I (or any other mathematician) would see Physics and Physicists as unable to face mathematics tools. However the fact is that, in theoretical physics, the papers which are written by physicists are for the most part, all about mathematics...QM has open, from my point of view, a very bad way, in which physicists have surrendered their true mission, that is the definition of objects and concepts which can be associated to physical phenomena, for abstract mathematical objects. An example : a system is defined by a set of hermitian operators in a von Neumann algebra. What is a material body, what is a force field ?? It is really difficult to have any grasp of spinors without the knowledge of Clifford algebras, of of gauge theories without understanding fiber bundles. So it is better if physicists do not reverse the link, and define mathematical objects from, ill defined, physical concepts.
Dear Jean,
Many thanks. I agree with you that (at least, leading) physicists should revisit the foundations and fundamentals of QM. Alas! Most practising phsicists have neither the time nor the will to think of these matters.
Jean,
I treat the assertion that 'time is not an observable' as an assumption since I have not found reasonable justification for it .Could you please justify your assumption?
Thanks Lutz
Another way to see this : when you study the evolution of a system, represented by some map X, measured over some interval of time (0,T) the observable is the map X, not the value X(t). So t is not an observable, and the only operator which could be associated to it would be not compact, so it has a continuous spectrum and no discreet eigen value. This is the same for any map F(x) where x is some coordinates : you cannot associate a compact operator to a coordinate, so no eigen value. This is one of the major misunderstanding in the common presentation of QM.
Dear Luiz,
Yes, asymptotic completeness in the case of long-range interaction potentials is an open problem. The Coulomb case is also technically challenging as rigorous treatments here demands care both at origin and infinity. For physicists this is, however, not as serious as it is for mathematicians. In realistic cases the interactions come with asymptotic screening and quantum chemists are usually able to take care of cusps etc. correctly.
Since you mentioned complex situations, like DNA etc. in biological systems, our focus must change to incorporate quantum mechanical extensions to so-called open dissipative systems. Fortunately quantum aspects can be portrayed effectively by exploiting the recently very popular theorem for dilation analytic operators (Balslev-Combes) or Weyl’s theory for Stark perturbations. Of course the problem you mentioned above comes back in connection with how to employ dilated Coulomb hamiltonians as generators for contractive time evolution (since the theorem cannot be rigorously proven).
I agree with your point regarding Rigged Hilbert Spaces although it may serve some pedagogical purposes for “physical understanding”. Therefore I see no problem to call the energy values associated with the absolutely continuous spectrum as eigenvalues in the sense of Schrödinger, see also my previous comments. Actual numerical calculations of spectral properties of physical and chemical systems do yield an improved feeling of what is going on as well as elucidate understanding as well, see my RG page for details.
I also agree that pioneering quantum mechanics treats time inappropriately as a parameter, which is inconsistent since it is conjugate to the energy operator. This does not usually lead to problems in treatments of stationary states of atoms, molecules and in solid-state theory. However, in an extended framework to dissipative dynamics and relativity, see my comment above, the concept of time must be given appropriate attention in e.g. contractive evolution of the resonance structures associated with “unstable states in the continuum”, see e.g. two recent volumes (#60, # 63) of the Advances in Quantum Chemistry. There is also a similar inconsistency in connection with the Born-Oppenheimer approximation that treats the electron quantum mechanically but the nuclei classically, which becomes inappropriate in dissipative dynamics.
Jean
''Time in non relativistic quantum mechanics is not an observable , since it is an evolution parameter''
Let's examine this statement for a little bit.It runs on the idea that parameters have an external existence in QM at least in the non-relativistic case.
Now if we consider the steady state schrodinger equation,we have a parameter, the position of which the state vector is a function of.BUT the relation of the state vector to position is described by the laplacian which is in fact hermitian.
Why can we not describe time in the same sense?Why keep it external?It all seems unnatural.
Carrington,
I understand that you feel a bit cheated, but this is the sad consequence of the way QM is usually explained. It is not consistent, as shows the fact that there is no observable associated to coordinates. If you want another presentation of QM look at my paper on QM on this site.
Dear Carrington, I think you are on the right track, see my comments to Luiz, about 22 hours ago.
Dear Jean, I notice that we are using a somewhat different “language”, but since you are using modern concepts that are particularly relevant for QFD readings, it is good if we nevertheless could basically agree on the correctness of our inadvertent dissimilarities.
Dear Luiz,
The field on non-self-adjoint problems in chemical physics centers around key papers by Erik Balslev, Jean Michel Combes, Barry Simon, Clasine van Winter, etc. particularly on dilatation anlytic operators and there extended spectra. There are not yet many textbooks for practitioners on these extensions Quantum Mechanics (the Reed-Simon 4 volumes are of course a fundamental platform, but they are not the kind of product I have in mind for general open systems).
In 2011 the pioneering work by Nimrod Moiseyev, “Non-Hermitian Quantum Mechanics” Cambridge University Press was published. As I also mentioned in my previous posting, collections of theoretical approaches to physical and chemical systems appeared in Advances in Quantum Mechanics vol. # 60 “Unstable States in the Continuous Spectra, Part I: Analysis, Concepts, Methods, and Results” in 2010 and vol. # 63 “Unstable States in the Continuous Spectra, Part II: Interpretation, Theory and Applications” in 2012. Both edited by C. A. Nicolaides and myself.
You can also download my presidential address: The Statement of Goals of the International Society for Theoretical Chemical Physics, Int. J. Quant. Chem. (2014).
I am not too excited about problems formulated in terms of quantum systems coupled to (classical?) heat baths. A quantum system is “contagious” as everything “it touches” becomes “quantum” as it de-coheres and everything becomes classical.
My view is that a correct treatment of thermodynamics should here be formulated in terms of (reduced) density matrices in Liouville space and employing the Bloch equation for thermalization. This is of course easy to say but complicated in practise.
For some ideas see my paper with Larry Dunne on “Bardeen-Cooper-Schrieffer (BCS) Theory and Yang's Concept of Off-Diagonal Long-Range Order (ODLRO), Mol. Phys. (2014). Can be downloaded directly from my RG page.
Single quantum particle interactions with an environment is conceptually complicated but not insurmountable as it could in principle be handled by various ensemble representable density matrices, starting from full coherence via Yang’s concept of ODLRO to various thermalized versions of relevance e.g. in biological systems, see also my RG page. Since these “states” evolve dynamically commensurate with Poisson statistics, the damping problem becomes particularly interesting carrying additional biological information. I guess I should stop here.
Luiz,
The Wheeler-De Witt equation, however interesting, is inconsistent in that the conjugate of energy, that is the time "disappears". Even in a zero-energy universe scenario the conjugates imparts certain consistency requirements.
Let me try to discuss whuy the way in which one describes time in quantum mechanics may not be so unnatural. I limit myself to system of finitely many paricles (in particular not fields) and hence non-relativistic.
Start with classical mechanics, since we have a good intuition there. A problem in CM is set up as follows: I am given positions and velocities of a set of particles---what in CM is called the state---and then to work out what the state---again positions and velocities---will be (or have been) at later (or earlier) times. Trying to replace time in this set up by, say, the x axis leads to truly unnatural results. The fact is, time plays a special role with respect to space, because of its peculiarly irreversible nature. Wanting to know what happens at a kilometer distance is (non-relativistically) qualitaively distinct from wanting to know what will happen in an hour. You need not apply equations of motion for the former (all you need is a telephone to communicate with an observer a kilometer away, or else a telescope), for the latter you do.
Things are similar in QM: the state is given by the wave function psi(x). By the Born rule, |psi(x)|^2 gives the probability of finding the particle (or more generally the system) in the position x. Note in paticular that |psi(x,t)|^2 refers to the probability of finding the system at x, if you observe the system at time t. It most emphatically does not mean: the probability that the system is found to be both at time t and at position x. A quantum mechanical problem, just like a classical one, involves an initial state, from which you compute future states at later times. Therefore, just like in CM, what you want is a time-dependent state, not a state which describes ``both the time and the position'' of the system. At the formal level, you see this in the fact that for all t, the integral over all x of |psi(x, t)|^2 dx, is equal to one. It would be very different to state that the integral over all x and t of |psi(x, t)|^2 dx dt, would be equal to one. Such a probabilistic view would mean that measuring a particle at time 0 excludes finding it at some other time. Something like this is true for position measurements, but not for time. Particles persist through times and states evolve in time. It is asymmetric, but if you consider everyday life, it is in fact true that making one step to the left is quite distinct from waiting five minutes.
Now you might ask, how can this possibly go over into a relativistic framework, where x and t must play similar roles? My understanding is that in that case *both* space and time become external, and the basic role is played by a field, which takes values as functions of both x and t. But this is not easy, and I have already written too much.
We can take a simple example : a system comprised of a single material body. The purpose of the experiment is to follow its trajectory over a time interval (0,T),, represented by a map (0,T)->X(t). We can assume safely that the map is square integrable (its length is finite), so the map X belongs to a Hilbert space. This map fully represent the state of the system and is a vector of the Hilbert space. This is a function in an infinite dimensional vector space but, as we have only a finite number of measures, we choose a simpler specification Y fo X : it can be straight line, a circle, a parabola, and thie data are adjusted to Y. The map f:X->Y is linear, and this is the observable, this is an operator on the Hilbert space, and it can be proven that it is hermitian. It can be proven also that Y (the result of the experiment) is always an eigenvalue of f.
In this pircture it is clear that the time (or the coordinates) are not observables : what we measure is X, not t. It would not make any sense to estimate t !
Of course one can like more complicated pictures !
all operators are called Hermitian have the following special properties:
Sorry Mohsen, it is not true
The set of eigen values of compact normal operators in Hilbert spaces is either finite or make a sequence converging to zero, the eigen spaces are orthogonal. The fact is that the so called operators associated to the coordinates are not compact, whence troubles in usual QM.
Leyvraz ,
You provide reasonable arguments for the uniqueness of time.I however to disagree on a few crucial points
''Trying to replace time in this set up by, say, the x axis leads to truly unnatural results.''
I disagree.The description of steady state systems is not unnatural, yet they are not functions of time.We essentially remove t as the independent variable and replace it with x.This then allows us to describe the distribution of particles with changes in x in the system a ''position evolution system''.This is hardly unnatural,in fact it is inevitable.
Now, to Quantum Mechanics,
You argue that-: '' It would be very different to state that the integral over all x and t of |psi(x, t)|^2 dx dt, would be equal to one. Such a probabilistic view would mean that measuring a particle at time 0 excludes finding it at some other time. ''
No it would not mean that.In fact the Born square condition does not imply that your measuring of the particle excludes the possibility of it existing anywhere else.(This interpretation is inconsistent with nonlocality.Strangely,even single particles can be nonlocal ).The born rule only says that the particle exists somewhere in space.
So it is with a hypothetical Born rule for time.It only implies that the particle should have existed at some point in time for you to have measured it.It categorically does NOT rule out the existence of the particle at another point in time.IF a particle is non local w.r.t to space it can be non local w.r.t to time,yes?
@Jean, Nevertheless x can be interpreted as a Dirac q-number symbolized as |x>, which was later made mathematical sense in terms of so-called Gel’fand triples. To extend definitions of t similarly in a relativity framework is no grand tour.
The fact that they do not correspond to a compact operator is not forbidding since these representations permit very efficient spectral expansions of any self-adjoint operator in Hilberts space of exceedingly practical use in fundamental problems in chemistry and physics.
@ Mohsen. I know what you want to express, but Jean is correct. In mathematics, as a tool for us as consumers, one has the duty to prepare the tool or “language” as succinct and rigorous as possible and to avoid contradictions and surprises. The consensus is that this is the best approach forward (irrespective of Gödel’s theorems) for everyone involved from creators of math to consumers.
To Erkki
Actually as far as we look to the evolution of a system, t appears in the Schrodinger equation, and we have the spectral decomposition of a unitary (and so not hermitian) operator, sometimes seen as associated to t. And this result is by far more useful thant to try to associate an operator to the coordinates..
Dear Jean,
As the self-adjoint Hamiltonian is the generator of time evolution of the system, the usefulness of spectral decompositions using the "eigenvalues" and their Dirac q-number representations remains.
This is also commensurate with an anologous treatment of the scattering problem, with efficient representations of the scattering matrices, yielding explicit solutions to the Lippman-Schwinger equation etc. Furthermore this formalism acts as a consistent and general extension to rigorous analytic continuations procedures onto the so-called "unphysical sheet"of the energy plane, with interesting consequences for the time evolution picture.
Drar Jean,
Since you do not consider operators with absolutely continuous spectra to have any eigenvalues (despite they are given a mathematically rigorositet definition), I wonder how you will approach typically scattering situations, like electron-atom scattering or orbiting resonances in hydrides? Please be specific.
The spectrum of any linear map on a Banach vector space is well defined. An eigen value belongs to the spectrum but the converse is not true if the space is infinite dimensional.
As for your other question....I am not a chemist (at least not a QM chemist) and I know very little about scattering. I believe that this is the chemist's job to deal with that. My ppurpose is only to help others with dealing with mathematical issues in theoretical physics. Excuse me, but for me mathematics is there to help physicists, but nowadays it seems that physicists try to invent a new mathematics through ill defined physical concepts (say hamiltonian, energy,...). I know that this is the way careers are built, but I am not longing for a careeer or a tenure, and I am a bit old to battle on, from my point of view, are void issues.
Dear Carringtone: let me try to clarify what I meant. You say: ``The description of steady state systems is not unnatural, yet they are not functions of time.We essentially remove t as the independent variable and replace it with x.'' From a physical viewpoint, I disagree: steady state systems are functions of both time and position. It is just that, as a function of time, they happen to be constant. They are neither timeless nor instantaneous. A marble at the bottom of a cup exists through time, it just happens to have no time-dependent characteristics. The same holds in quantum mechanocs
What I had in mind was rather the following: consider a harmonically bound particle making oscillations of frequency one and amplitude one. If you view this as a function of x, but looking at all t at once, you start at x=0 with infinitely many equispaced particles, since the particle is at x=0 for infinitely many equispaced times. Now if you proceed to the right (positive x) infinitely many pairs of particles start moving towards each other. Eventually, at x=1, all particles annihilate. You can do this, of course, but to me at least, it does not seem natural. Whereas saying that one single particle changes its value of x and of xdotAdd your answer as t proceeds, seems to me to describe rather simply what happens.
In quantum mechanics, it is the same, and has little to do with complex mathematical issues. The main point I was making is an issue of normalization: psi(x,t) is normalized so that |psi(x,t)|^2 adds up to one when summed over all positions, not when summed over all instants. This is precisely because, if you measure the position of a particle, you will only get one result. You never obtain the same particles in two positions when you measure it. Nonlocality is soemthing different and does not affect this claim. For this reason, we say that position measurements at different positions are exclusive, so their probabilities must add up to one. This is in fact what actually happens. On the other hand, you can very well measure the same particle at different times. And the integral over all times and all positions of |psi(x, t)|^2 is infinite, even for a stationary state. In other words, there is no such thing as space-time symmetry in classical non-relativistic physics.
To understand what happens in relativity, you need field theory. Again, classical field theory is easier than its quantum analog. If you think about it, you will see that the field configuration can now be viewed as a state propagating through time. Again, some amount of asymmetry is needed and provided by the theory: the wave equation is such that you should specify initial conditions in the time variable for the solution to be well defined.
Dear Jean,
Mathematicians and natural scientists need to communicate. At the same time they also need each other: the mathematician will tell the consumer what is to be expected from theoretical models and simulations, while sometimes a physicist may guide math into new unexpected domains.
I mentioned Dirac as a case where something very useful was formulated, even if at the time this was not made mathematically rigorous. Why deny the extremely useful and practical Dirac formalism, which is not as you imply bad math, and assert that modern scattering theory is an illdefined physical concept?
In physics as in any scientific theory we need a balance between
i) the concepts, which strive to describe physical phenomena, and laws, represented in formal mathematical models
ii) experimentation, organised around the concepts and using mathematics to compte and check the laws.
Physics does not produce mathematics, it uses them. Mathematics have greatly progressed in 50 years and offers now new tools which make the work of physicists much easier (I think in particular to fiber bundles). The problem is that many physicists do not have, or do not want to acquire the necessary knowledge to use these tools. And besides QM has introduced a bizarre mixture of formal procedures, unexplained concepts and traditional concepts, so that we are submerged by a litterature that is neither maths nor physics. And goes nowhere. It is a fact that the standard model goes back to 1973, that we know nothing about the weak and strong interactions, or gravitation. Theroretical physics has completly stalled. It suffices to have a look at the equations of the standard model (see Wikipedia), or to know why the Higgs boson has been introduced, to realise that it is not going better.
What we need from physicists is that they think in depth about their concepts, mass, matter, force field, space and time, and not that they rush brandishing some badly understood mathematical formula. Just anoher example : it is easy to prove that the universe (sapce time) is a manifold. And the fact that for any observer past and future are clearly distinct makes that for each observer the structure of a fiber bundle, and a specific real valued function (the time at any point). This would be worth a look before introducing propagator or anything else.
Dear Jean,
I think you are way too pessimistic! True that present communication technologies allow many non-serious actors to fill our already crowded showground, but at the same time it provides immense possibilities.
I agree with you regarding the balancing argument if taken to include fertile communication between the parties involved. My own experiences in communication with leading mathematicians related to my work and interests, have been exceedingly positive. I have benefited greatly from their generous advice, adapted my approach and developed my act accordingly. At the same time I did note their positive, surprising and encouraging reactions in that their theorems could lead to concrete applications in chemistry and physics as well as revealing results that points at further conceptual understanding and development.
Your negative evaluation of the progression of physics during the last 50 years may depend on your predisposition to look in the wrong direction. Mathematical applications to physics, chemistry and biology do have ups and down and certain areas have indeed stalled, while others are mushrooming.
Take e.g. the developments of the new sciences of photonics (quantum optics, quantum electronics), quantum computation and quantum information (including all devise technologies using photons), dispersive Fourier transform spectroscopies, or highly efficient mathematical simulations in biology and medicine, the simulations of tsunami waves – the list will never end! Here physics in addition provide novel ways to calculate!
In Uppsala we have created (I did belong to the pioneers that created this centre) a very successful Centre for Interdisciplinary Mathematics, with members from e.g. Material Chemistry, Dept. of Ecology and Genetics, Earth Sciences, Cell and Molecular Biology, Physics and Astronomy etc. etc. Do check internet and you will see that these type of Centres pop up all over the globe!
As our material culture gets in contact with science and consolidates our body of knowledge both science as a whole benefits as will do our fundamental knowledge.
I think the way to proceed is to leave the ivory tower and, as we say in our community, get our fingers dirty. If you feel that fiber bundles have been under-represented in modern theoretical physics – go out and promote the cause. It sounds as an excellent idea to me, but in the end the “proof is in the pudding”. A mathematical or theoretical tool will overhaul “ballparks as it increases the batting average and the number of homeruns and lowers the earned run average”.
Dear Erkki,
You are kind and professional. I have no doubt that in many fields (notably biology and chemistry) there has been tremendous progress the last decennies. But we are still reaping the fruits of electromagnetic theory. It is quite clear that one can manage with theusual QM (after all, if people are used to it as it is I am not the one to discourage anybody) with electhe tromagneticfield. But we have fundamental forces,and there is so much to do ! Anyway I do not want to perturb people, debause i am free to tell what Iwant, I just feel sometimes as a teacher, telling people : do not rush towards hastely mixed symbols and mathemtical tools, think ! thin as a physicist : what is that ? why does it behave this way? What, on the mathematical shelves, is the right object to represent that ?
I just hope to be useful to others in this way.
Good luck !
Leyvraz ,
Taking into account the principle of relativity I have ;to disagree with your representation of classical steady state systems.Let us imagine a closed steady state system such that external parametrization is conceptually infeasible.There would be no way to describe time within the system since no such thing exists unless we invoke Newton's absolute time arguments,which I think is not a very good idea.
In your argument in quantum mechanics you say that your interpretation of the born square rule is not inconsistent with nonlocality.How then does it explain single particle non locality where a particle is measured to be in two places at the same instant?
Now,for you to sum up the probability of existence over time for a particle and find it to be greater than one,you have to presume its existence before and after measurement.For you to be sure of its existence a priori and a posteriori,you have to measure it!We are thus in an infinite regress.
I find this alternative much simpler.The hypothetical born rule for time can only be defined over an instant of measurement where the probability of existence of the particle is one or it to be measured over the duration of the experiment.
Hello Luiz,
Of course I am fully aware that many scientists know of fiber bundles ! And that there are still a lot of work to do in this field to understand how to get a sensible representation of the phenomena. But this is a fact that the standard model still lies in Special Relativity, that it pretends to explain mass, the gravitational charge, without gravitational field, and that too many papers are still focused on playing with a patchwork of physical concepts and mathematical objects taken as granted.
My critic is fundamentally that physicists should devote more energy to ain depth assesment of the concepts, meaning physical concepts, before wrapping them quickly into mathematical objects. Mathematics offers tools, perhaps still no as powerfull than what is needed, but by themselves they will never provide physics.
As an example, among many others, I have been lead to think about the concept of motion, meaning the combination of translation and rotatiion. It is seen usually through the displacement of frames (notably in special relativity), but actually this is not as physicists (say in mecanics) see it. And this is, from mypoint of view, a mistake to rush to a Lie algrbra (of Poincarré group) before really understanding the true nature of motion ofmaterial body. Or similalry very little is said about the most fundamental symmetry breakdown : that felt by any observer between past, future and present.
Dear Carringtone: I like to have concrete examples. A marble at the bottom of a cup seems to me to be an easily visualized concrete instance of a classical system in a steady state. As a matter of curiosity: what is a similarly easy instance of ``a closed steady state system such that external parametrization is conceptually infeasible''? I am not trying to make fun, but I would like to understand what you have in mind. And at this level of abstraction, I do not.
Similarly for your quantum mechanical issue: you say ``How then does it explain single particle non locality where a particle is measured to be in two places at the same instant?'' I am not aware of a single experimental instance where the same particle is measured to be in two places at the same time, or even rather distant at times close by, or where a particle is measured and only half a patricle is found. If such things were possible, then the purely wave-like intepretations of the wave function might not be the nonsense which they in fact are. The crucial point is that, whenever you measure the position of a particle as described by a wave function, you get exactly one specific answer every time, except that it is not the same answer every time, but you rather get different results with a frequency proportional to the corresponding squared ampliteude of the psi function.
@jean claude “….Or similarly very little is said about the most fundamental symmetry breakdown: that felt by any observer between past, future and present……”
I agree with your critique of the standard model and the special relativity. In an evolution perspective “past” and “future” are illusions only “present” is meaningful – this would be one natural way to get rid of McTaggart’s paradox (aiming to prove the unreality of time), see Stanford Encyclopedia of Philosophy on Time. How would fiber bundles deal with this conundrum?
Leyvraz,
It was just a thought experiment.I don't think thought experiments are illegal in physics.In fact they help us gain insights about real physical systems.We MUST imagine the possibilities.
Please refer to http://phys.org/news113824784.html for an actionable proposal to demonstrate single particle non locality.
To Erkki,
This a bit out of the scope of this thread, but let us see how one could see GR :
i) the fact that the universe is a 4 dimensional maniffold M is easy (one needs 4 coordinates, compatible charts define e structure of manifold)
ii) any observer can distinguish a present, which is a 3 dimensional submanifold. So we have a collection of manifolds with boundary. This collection can be characterized by a single function f whose derivative is never null, and of course it defines a fiber bundle whith base R.
iii) This structure is specific to each observer. The issue is How the existence of such a collection is cmpatible with the structure of the universe.? The usual cosmology is based upon a single fiber bundle (the warped universe) : it sums to assume that there is special observer (God ?). But we can do better. First we have to characterize observers. They have a trajectory meaning a map R->M. Second because M is defined through charts on R4 its topology is such that it has a positive definite (!) scalar product, third we never see material bodies going towards the past, which implies that the scalar product of the velocities of two material body is always positive, and fourth this implies that there is also a lorentzian scala product on M, and we have the usual structure, but without any assumption about the speed of light (wwhichis difficult to define).
iv) So we have for each observer the structure of an associated fiber bundle, with the lorentz group. And this is the basis for the gauge representation of the gravitational field.
The simple answer is , "No!" To have only real eigenvalues an operator must be self adjoint (see my book–– Nonrelativistic Quantum Mechanics for the difference) .
An example of a hermitian, but NOT self adjoint operator is
p = -i d/dx defined on the interval (0, infinity). Here the function
f(x) = exp(-x) is an eigenfunction with the imaginary eigenvalue i.
Carringtone:
I have no problem with thought experiments. But the thought experiments you gave me a link for are, in fact, based on standard quantum theory. They try to imagine what outcomes would arise if certain setups were made with a given system. And since they follow standard quantum mechanics, they never claim that a given particle is *measured* at two places at the same time. What happens is that, using some locality assumptions, you might get into difficulties.
But this is not what I was saying or talking about: it was all much simpler. I said a wave function in quantum mechanics is normalized with respect to space, that is, the integral of |psi(x, t)|^2 over space is one. That follows from the fact that |psi(x, t)|^2 denotes the probability of measuring the particle at x, and to measure the same particle at x and at x' are two exclusive events, so th etotal probability of measuring a particle somewhere must add up to one. If you actually measure a particle's position, you will according to quantum mechanics always only get one answer. Non-locality is different, and significantly more complex. It deals essentially with the positions where the unmeasured particle might have been, in view of the results of later measurements. I was only speaking of probabilities of actual measurements
On the other hand, the same particle can readily be followed through time. That was the very simple point I was making: we see the same particle changing through time. A particle persists through time, which translates into the fact that |psi(x, t)|^2 integrates to one over all x for all times, and that this integral does not change. That is what the formalism of quantum mechanics says. Of course, in such a framework, there is no place for the concept of ``probability to find the particle at time t''. Similarly, thisis the logic underlying the claims that ``time is a parameter, not an operator''.
If you want to change the formalism of quantum mechanics, that is fine. But then you should have some kind of experimental reason to change it, that is, your improved approach should yield results that explain things which standard quantum mechanics cannot. There are not many such, however.
ALL hermitian operators have real eigenvalues. This is a mathematical theorem without exception
Jean Claude,
Yes – so far so good! How do you include light in the fiber bundle? Would you need to define a particular finite speed for light or can you do it in some general way?
Anton,
To define a differential operator over an interval, cf. the Sturm-Liouville problem, you need to assign proper boundary conditions. The operator p on [0, b] with solutions equal to zero at the end points is certainly a hermitian operator with real eigenvalues.
The problem here is what happens when b—> ∞. It turns out that p on [0, b] cannot be extended to a self adjoint operator on (0, ∞) but p on [-b, b] can, when b—> ∞!
Anton: Some questions are issues of mathematics, some others are nomenclature. In mathematics, as far as I know, one distinguishes between symmetric operators and self-adjoint ones. The latter are a special case of the former. And indeed, as you say, only self-adjoint operators have real spectrum. Symmetric operators that are not self-adjoint always have at least one complex eigenvalue. Your example, as well as my earlier one, p^2 - q^4, show it clearly.
Now to the question that is nomenclature. In quantum mechanics, we always say that the Hamiltonian is ``Hermitian''. This terminology runs somehow parallel to the mathematical one. But, in quantum mechanics, Hamiltonians are always viewed as generators of one parameter unitary groups. So to maintain consistency and in deference to Stone's theorem, we should take the charitable view towards physicists and accept that, when they use the nonstandard term ``hermitian'', what they mean is ``self-adjoint''. Non self-adjoint symmetric operators do not, to my knowledge, play any role in physics.
To Erkki
Because we do not see matterial bodies coming from the future, the velocities must belong to a cone, and one gets the lorentz scalar product. It implies the existence of a universal constant relating lengths and time, which has the dimension of a speed. That this is the speed of light is an experimental fact, and also from the fact that one synchronizes clocks using light signals, as Einstein showed in 1905. But I would be careful about the speed of light, because this is the speed of propagation f light. It is well defined in special relativity, less so in GR (the propagation of a field is not an easy subject).
To Leyraz,
I believe that the best way to deal with mathematical objects is to let mathematicians do the definitions....Anyway in studies of systems defined through von Neumann algebras, as it is common in QM, there are synnetric operators defined over a subset of the Hilbert space. This is the main motivation to introduce symmetric operators (see the precise definitions somewhere above).
Leyvraz
I have got no experimental justification for introducing the time operator.My motivations are purely mathematical and conceptual
First is the issue of consistency;
If you are going to address a parameter within the context of the theory by using its standard formalisms,it follows that one ought to also include all related parameters within the context of the theory.The failure of standard QM to do this for time may not have hurt QM,but it has made it very difficult for the theory to understand time yet it is fundamentally a theory of mechanics.
Second is the issue of completeness
Ideally you would like a theory to generate its own parameters and variables and describe the interaction of these variables in (and with) the parameters to produce laws which describe the behaviour of the system.
Now, if a theory did not generate some of the variables or parameters would it be complete?If it depended on the implicit existence of an externally generated parameter to work,would it be accepted as a final theory?I flatly wouldn't.
PS;
The article says " the nonlocality must stem from the original single-particle state."
I don't agree with your stataement that " A particle persists through time, which translates into the fact that |psi(x, t)|^2 integrates to one over all x for all times, and that this integral does not change. "
Paricles decay,interact and get absorbed(for photons) surely,it cannot be for ALL times.We can only define a particle during its so called lifetime and trivially, you can only measure it in this period,It follows that the probability that you'll measure it in this period is one.If I remember correctly,you were saying that the probability will sum up to greater than one.Change of heart? :)
Hi
a simple remark. A hermitian matrix is by defintion such that the element in the i-th row and j-th column is equal to the complex conjugate of the element in the j-th row and i-th column, for all indices i and j:
Hermitian matrices are named after Charles Hermite, who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of having eigenvalues always real. It is a theorem.
So the problem is :Is the operator hermitian or not?
If hermtian is the operator, then eigenvalues are real.
Carringtone:
Yes the article says that. But you should make an effort to understand what non-locality means in this context. It emphatically has nothing to do with a paticle being measured in two places at the same time. Rather, it is linked to deductions as to where the particle could have been before it was measured.
I guess I was careless there when I spoke of ``a particle''. I should have said ``a state''. A one photon state together with one atom can change into a zero-photon state with the atom excited. For simplicity's sake I was limiting myself to non-relativistic QM. But the main point is not affected: detection of a particle at two different points are mutually exclusive events, the probabilities of which add up to one, detection of the same particle at two different times are by no means exclusive, and thus the probabilities do not add up to one. That is what leads to the concpet of a state evolving through time, which is important both in classical and quantum mechanocs, and which, additionally, corresponds rather well to the way in which we perceive things.
jean claude,
In what way would a “black hole”, if the possibility emerges, become discernible in this portrait?
I stick to the observable world (this is alreay complicated enough). As for black hole I do not deny their possibility, but obviously they require some discontinuity. So the normal physics does not apply anymore, and I have nothing new to tell on this topic.
Leyvraz
I agree,I don't understand nonolocality(who does?) but let's find out.In the same article the conclusion goes thus:
"According to the principle, a pair of entangled quantum particles must be indiscernible from a single particle, since both objects have in common all the same properties—this is the only stipulation of the principle, number being irrelevant."
Now,if,as the authors of the paper say number is irrelevant,how comes your interpretation of single particle non locality is radically different from the interpretation of multipartite non locality?Why is the number relevant?
Carringtone:
Again, the whole issue of nonlocality is besides the point. If you want to understand it in detail, look, if anything, at the original paper. But note well that, in the summary you quote, there is *nothing* speaking of measuring two particles at different positions at the same time. That is not what nonlocality is about. What the authors of that paper want in detail is not altogether clear to me, but it is emphatically not this. In fact, the whole logic of nonlocality always assumes that the calculational procedures of quantum mechanics are exact. So what is done is always to perform a long-winded *traditional* quantum computation and then to verify experimentally that it is indeed observed. So far, this program has always met with success, and quantum mechanics has never been found at fault. The issue is usually that you get a contradiction if, to the formal quantum calculation, you add some seemingly plausible assumptions of locality, no action at distance or something of the kind. What this means, of course, is that those seemingly plausible assumptions cannot be right, since experiment yields a result in contradiction with these assumptions. But it is a constant feature of these calculations that they always use strictly traditional quantum mechanics in their arguments. It is thus clear enough that they could not use an assumption completely at variance with ordinary quantum mechanics, such as the observation of the same particle at two different positions at the same time. .
But once again, I am not dealing with the issue of nonlocality. I am simply saying how a state is ordinarily described in quantum mechanics. The description is that a state describes mutually exclusive outcomes for space measurements, whereas the probabilities for these outcomes vary with time. Measuring the same particle at two different times, however, are not exclusive events and adding probabilities in that case is not correct.
If you wish to develop a formalism in which things are otherwise, you should first try and be very clear in your own mind as to what you wish to attempt, and then try it---with formulae, not words---in the simplest case you can possibly think of. Good luck!
Dear Jean Claude,
I understand your point of course. My “insistence” is not to be critical, on the contrary I am very much interested in grasping new ideas and do not request that they should give answers to all kinds of difficulties. My hunch, nevertheless, is that some discontinuity is needed to get the limiting velocity of light and ensuing consequences from there. My hunch is also that such a discontinuity should not “disturb” the “normal physics”.
I have outlined a possible way in some recent articles that utilizes a simple matrix of operators like energy and momentum (and their conjugates) with its determinant yielding a Klein-Gordon equation (or Dirac eq.). This is a simple trick to be commensurate with the Minkowski (also the Schwarzschild gauge) metric. I am not sure how this property is modelled with fiber bundles.
When an operator is Hermitean, then in particular its diagonal elements have to be real. Then also its eigenvalues (as diagonal elements) are real.
Does that answer your question?
Dear Erkki,
Physics is a work in progress !!
I am thinking more and more about one of the tenets of GR : the existence of the Lorentz metric. Actually the only lead for this assumption comes from Einstein and Poincarré works on the equivariance of the Maxwell's equations : we need the transformation laws to keep the equations in a change of observers. Later Minkowski invented his space. This assumption is strongly supported by facts. However one concern is that it relies only on the electromagnetic field, and more precisely on the concept of propagation of light. But, out of the special relativity framework, this concept is not obvious (what do you call propagation in a curved space time ?). And we know that the weak and strong interactions propagate in a more complex way. And we know almost nothing about the propagation of the gravitational field. Moreover the unification of fields would require to use a larger group (SO(5) and now SO(10)) to account for every thing.
This is probably out of your research, but just to tell you that the fiber bundle formalism is well suited on these issues. Actually the raditional GR formalism is both cumbersom and too narrow. The problem is that there are no good books on fiber bundles (sorry to tell that, but I think that this is a strong point of my mathematics book...).
Leyvraz
Non locality is NOT besides the point.It IS the point.With all due respect,your interpretation of born's square rule is inconsistent with non locality.And since non locality is central to QM,I reject your interpretation.
Several papers on non relativistic time operators have been suggested.I also tried to come up with one using basic algebra but I think it is greatly limited in scope.In any case it has got a structure that is analogous to already published material.I'm not speaking out of hot air.
My concern is this: hermitian time operators are dismissed on rather spurious arguments.Let's consider your argument as an example-;
"The description is that a state describes mutually exclusive outcomes for space measurements, whereas the probabilities for these outcomes vary with time. Measuring the same particle at two different times, however, are not exclusive events and adding probabilities in that case is not correct."
1 I argue against the point of mutually exclusive measurements in space because QM assures no such thing.It only assures that the particle exists somewhere in space.As such I will INSIST that the measurement of a a particle in a point in space will not immediately reduce the probability of measuring the particle at other points to zero.That is the whole point of single particle non locality.
2 Just as we are careful when defining the limits of a particle in space (due to heisenberg uncertainty),so should we when defining limits of a particle in time.An example:
In the usual radioactive lump analogy you can never tell when a particle is going to decay.All you know is that it has increasing probability of decay.Why?Simple because a quantum physicist isn't a prophet(or a historian)!
We van never be sure the particle existed at a specific point or will exist at another point before we measure it,And trivially for us to measure it.It has to exist with probability ONE at that time.its past or future is a game of probabilities.Unless we invoke laplacian determinism or naive realism,your picture of adding probabilities is rejected on these grounds.
PS:I am quite clear(IN MY HEAD) what I want.I want balance,symmetry and consistency,Inconsistency may be tolerable elsewhere,but not in physics.No.
Dear Carringtone,
I have, midly, followed your posts. My feeling is that you are a beliver. QM is right. Nobody knwos why, but it is so. eThe measurement of a a particle in a point in space will not immediately reduce the probability of measuring the particle at other points to zero". No trouble. When something seems weird, it is because the real world is bizarre. And if the maths do not cope, this is because the right definitions are inscrived in QM. So why bother with maths ?
All is a matter of faith. I am a scientist. I do not pretend ti know the truth. I do not pretend to tell what the real orld is. I just want to understand what a theory means.
Carringtone:
As to what ordinary QM says concerning simultaneous measurement of particles at different positions: this is not really a matter of private opinion, or of interpretation. The impossibility of this is contained in the basic postulates of QM. As to the paper you referenced: it speaks in somewhat vague language, as marerial intended for a broader audience often is, but they only speak of nonlocality in general terms, never of simulatneous *actual* measurement. So whether you accept it or not, it is part of QM that a measurement of a particle at one position reduces the probability of finding that same particle elsewhere to zero. If you want to know what one-particle nonlocality is, find out, but use the actual paper, rather than a popular summary.
As for radioactive decay, again keep in mind that what we study in QM are states. It can happen that a state which initially looks like one particle, will suddenly change to a state with two particles. It does not mean that a particle ``is born'' but that one many-particle state has changed into another. The one-nucleus states then evolves in time to a state that is a superposition of a one nucleus state, and another state that contains one nucleus and a decay product. It is not appropriate to say that the decay product ``starts to exist'' with a certain probability. Rather, a state that existed throughout has evolved from containing one set of tightly bound protons and neutrons to having two or more particles, one being a modification of the original nucleus and the other, say, an alpha particle.
As for what you want: I believe we should all look for consistency. But it may not be superfluous to try to understand why a fair number of people consider the actual picture of QM---at the microscopic level, that is, appart from measurement---to be consistent. In fact, there are difficulties with QM, but these are not, as far as we know, associated with time, but precisely issues of measurement.
@Jean
Glad you agree with me on the point of measurement.YES we should not pretend to know the truth.For us to understand QM we should first accept it as it is,start from zero.
@Leyvraz
Sir,I have read the paper.It ends with the same statement "Since entangled quantum particles are indistin
guishable, this principle suggests that a single-particle
system should not behave differently from a multiparticle
one. "
This implies that a single particle has the same nonlocality as an entangled multipartite system.Which trivially implies a particle can be measured to be in two places at the same time. Physical Review Letters 99, 180404 (2007).
My point in the radioactive decay scenario is that your interpretation of born's square rule for time requires more knowledge of the system than can be found at that point in time.Since such is impossible,your interpretation of rule is impossible .One of the reasons there is a problem with measurement because the behaviour of the state vector becomes TIME assymetric.Don't you think we ought to know why,by first understanding time in the context of QM?
With all due respect to the physics community,It really is irrelevant what " a fair number of people think'' .What most people think changes all the time.Science can't rely on that.
Dear Carringtone !
I have a mathematical answer for your question:
The space of all bounded operators on a Banach space (or a Hilbert space) X is a Banch Algebra. Now, Theorem 10.12 from the excelent book of Walter Rudin on Functional Analysis states that the spectrum of any bounded operator on a Bancch space is always compact nonempty set.
Theorem 12.12 in the above mentioned book states that for a normal (TT^{*}=T^{*}T) bounded operator T on a Hilbert space H:
T is self-adjoint if and only if the spectrum of T (which is compact nonempty) lies in the real axes.
I hope that I helped you in any way.
All The Best And Success !
Bounded hermitian operators of a Hilbert space have always a real spectrum. For unbounded operators this is in general no more true, as simple examples show. For these operators one has to distinguish between hermitian and self-adjoint operators. The latter have always a spectral resolution with a real (unbounded) spectrum. All this goes back to John Von Neumann. For physicists an excellent reference is Vol. I (Functional Analysis) of M. Reed and B. Simon on "Methods of Modern Mathematical Physics", Academic Press.
refer to page 40 of "Gerard J. Murphy, C-Algebras and Operator Theory"
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