...and why (near) energy eigenstates are the "most real" among equally allowed states in the Hilbert space
Many physics beginners, physics fans, armchair physicists, young undergraduate students, as well as assorted physics Nobel prize winners who used to be the best quantum mechanical practitioners in the world but who have already forgotten all the basic physics have a problem with some fundamental, rudimentary, and universal features of the laws of Nature, namely with the postulates of quantum mechanics (QM).
They are ready to tell you that they want to construct – or, in the more hopeless cases, they have already constructed – a theory that is able to do everything that quantum mechanics can but it only allows some preferred states to be "truly realized"; the superpositions are less real. Or they tell you that they may reproduce quantum mechanics even though they only deal with real i.e. noncomplex superpositions.
Every single comment of this kind is totally childishly wrong, of course. The superposition principle – which says that every linear combination of two allowed states (e.g. initial conditions) is equally allowed – is a totally rudimentary principle of quantum mechanics. That's also why Paul Dirac dedicated Section I/1 in his Principles of Quantum Mechanics to this insight.
Why real numbers aren't good enough in QM
In the past, I have discussed why complex numbers are fundamental in physics several times. The readers were reminded of the important properties of complex numbers such as the fundamental theorem of algebra i.e. the existence of \(n\) roots of any \(n\)th order polynomial with complex coefficients (it wouldn't work if we demanded real solutions). Complex numbers are important even if one wants to find real solutions of a real polynomial (i.e. a cubic one) so we're getting more than what we insert. Holomorphic (natural) functions of a complex variable have many important mathematical properties that turn complex numbers into useful if not essential tools, e.g. in the case of twodimensional conformal field theories. In many of the applications, the complex numbers may be viewed as nonessential but very useful technical tricks.
But in this article, I want to focus on quantum mechanics where complex numbers are more than just useful tools; they are really crucial for the theory to work. So why must the wave functions (and therefore matrix elements of all observables) be allowed to be complex? Why do we have to allow complex superpositions? Let me begin with enumerating three fundamental laws in which we see the imaginary unit: Schrödinger's equation, Feynman's path integral, and Heisenberg's "uncertainty principle" commutator.
Schrödinger's equation
Fine. The timedependent Schrödinger's equation tells us that\[
i\hbar \ddfrac{\ket\psi}{t} = \hat H \ket \psi.
\] The time derivative of the state vector \(\ket\psi\) is proportional to the action of the Hamiltonian on the same state vector. The coefficient is proportional to the tiny reduced Planck's constant; that has to be the case because the wave function must oscillate very quickly for it to be undetectable in macroscopic situations.
But the coefficient also includes the factor of \(i\): it is pure imaginary. Why does it have to be pure imaginary? Well, it's necessary to preserve the norm of the state vector. Let us calculate the time derivative of the norm, using the Leibniz's rule for the derivative of the product \((uv)'=uv'+u'v\):\[
\ddfrac{\braket\psi\psi}{t} = \frac{1}{i\hbar}
\bra\psi \cdot \hat H \ket \psi
+
\frac{1}{(i\hbar)^*}
\bra\psi \hat H \cdot \ket \psi = \dots
\] The first term, \(uv'\), was obtained by simply multiplying the original Schrödinger's equation by \(\bra\psi\) from the left. The second term, \(u'v\), was obtained by multiplying the Hermitian conjugate of the original Schrödinger's equation by \(\ket\psi\) from the right. A funny thing is that the time derivative of the norm vanishes because the result is \[
\dots = 0.
\] The two terms cancelled because the \(\cdot\) multiplication played no role – it's still an ordinary multiplication of matrices which is associative – and because\[
i^* = i.
\] That's great because the total probability wouldn't be conserved if the imaginary unit were omitted: the wave function would exponentially increase (or exponentially decrease). Pure imaginary numbers are the only ones whose complex conjugates are equal to minus the original numbers and that's exactly the virtue that we needed here. Also, if \(\ket\psi\) is an energy eigenstate whose eigenvalue is \(E\),\[
\hat H \ket \psi = E\ket\psi,
\] the timedependent Schrödinger's equation reduces to the timeindependent Schrödinger's equation or its solution\[
\ket{\psi(t)} = \exp\left(\frac{Et}{i\hbar}\right) \ket{\psi(0)}.
\] That's nice because the only thing that is oscillating is the phase. Note that the complex exponential \(\exp(i\omega t)\) allows us to distinguish positive frequencies (energies) and negative frequencies (energies), something that \(\cos(\omega t)\) or \(\sin(\omega t)\) wouldn't be able to do. The same comment applies to many other dependencies of the wave function. For example, the plane wave \(\exp(+ipx/\hbar)\) differs from \(\exp(ipx/\hbar)\) which is why it is able to distinguish a particle moving to the left with \(p\lt 0\) from a particle moving to the right with \(p\gt 0\).
So the imaginary unit is totally essential for Schrödinger's equation to work. If you omitted it, you would get a totally different equation with a totally different behavior – it would be utterly ludicrous to claim that you are successfully imitating Schrödinger's equation – and you would easily find out that none of such equations could accurately describe processes in Nature that depend on quantum mechanics.
Needless to say, the usual proof of the equivalence of the Schrödinger picture and the Heisenberg picture may be employed to show that for the same reason, the imaginary unit also has to be present in the Heisenberg equation of motion\[
i\hbar \ddfrac{\hat L}{t} = [\hat L,\hat H].
\] I could also optimize arguments directly for the Heisenberg picture that wouldn't depend on the Schödinger picture.
The path integral
Analogous comments apply to Feynman's path integral,\[
{\mathcal A}_{i\to f} = \int{\mathcal D}\phi\,\exp(iS[\phi]/\hbar).
\] Again, you see an imaginary unit in the complex exponent. This imaginary unit is totally essential because the absolute value of the weight of a history is always the same which is a good thing. In fact, this \(i\) is totally equivalent to the \(i\) in Schrödinger's equation. When you are proving the pathintegral formula for the evolution amplitude from the Hamiltonian evolution, you will ultimately use the relationship between the Hamiltonian and the Lagrangian\[
H + L = \sum_j \dot q_j p_j
\] Because there is an \(i\) in front of the Hamiltonian in the Schrödinger's equation, there has to be an \(i\) in front of the action which is \(S=\int\dd t\, L\). Needless to say, the complex exponential in Schrödinger's equation or Feynman's path integral is also needed for the interference to exist; think about the double slit experiment. There are other "detailed situations" in which the key role of the imaginary unit may be seen. Also, one may easily argue that once the wave function for a subsystem is allowed to be complex, the wave function for the whole Universe has to be allowed to be complex, too – because in the "clustered" situations, the wave function of the whole system factorizes into the product of wave functions of the subsystems and the product of a complex number and a number from any other "field" is a complex number.
The commutators
Finally, let me discuss the commutator\[
\hat x \hat p  \hat p \hat x = i\hbar.
\] There is a simple reason why the \(c\)number on the right hand side has to be pure imaginary. The reason is that the left hand side is antiHermitian i.e. it obeys\[
(\hat x \hat p  \hat p \hat x)^\dagger = \hat p^\dagger \hat x^\dagger  \hat x^\dagger \hat p^\dagger = \hat p \hat x  \hat x \hat p = (\hat x \hat p  \hat p \hat x).
\] The antiHermiticity means that the Hermitian conjugate of the left hand side is equal to minus the left hand side. We used the Hermiticity of \(\hat x,\hat p\) in the proof (it's needed because they have real eigenvalues, the measured positions and momenta), aside from the identity \((AB)^\dagger = B^\dagger A^\dagger\). So the imaginary unit inevitably has to appear in this commutator and most other commutators.
As I mentioned, the commutator involving \(i\) is also needed for the \(\hat p\) eigenstates to have the form of the plane waves\[
\psi_p(x) \sim \exp(ipx / \hbar )
\] which don't pick a specific place in space and which aren't exponentially increasing or decreasing. Again, the sines and cosines wouldn't be enough because they wouldn't remember the sign of the momentum. I could add comments about dozens of other catastrophic failures that would follow from the omission of the imaginary unit.
Because the commutator \([\hat x,\hat p]\) is an operator \(i\hbar\) whose matrix entries are obviously complex, I mean nonreal, it follows that it can't be the case that all the matrix entries by \(\hat x,\hat p\) are real. At least one of them has to have complex entries. In the position representation and the momentum representation, one of the operators is given by a real matrix and the other one is given by a pure imaginary matrix. But in more general bases, the operators are given by matrices that are complex, none of them is real, and none of them is pure imaginary.
Spin1/2 particles and complex spinors
The electron is the most famous particle whose spin is \(j=1/2\). The three components of the intrinsic angular momentum \(\hat S_x,\hat S_y,\hat S_z\) have eigenvalues \(\pm \hbar/2\) – just like any other observables, these observables are normalized to have real eigenvalues – and their commutators are\[
[\hat S_a,\hat S_b] = \sum_c i\hbar \epsilon_{abc} \hat S_c.
\] The explanation of the imaginary unit \(i\) in the equation above is totally analogous to the explanation of the imaginary unit in the commutator \([\hat x,\hat p]\). Some previous blog entries tried to explain why are there spinors and what are their basic properties.
A funny thing about all twocomponent spinors – the normalized (to unity) column wave functions \(\ket\psi\) with two complex components – is that we may calculate a direction in the threedimensional space\[
\vec V = \bra\psi \vec\sigma \ket \psi
\] where the triplet of matrices sandwiched between the bra vectors and ket vectors is made out of the Pauli matrices. The electron whose spin wave function is \(\ket\psi\) is guaranteed to be spinning "up" with respect to the axis \(\vec V\). Note that \(\vec V\cdot \vec V = 1\) is guaranteed if \(\braket\psi\psi=1\). Up to the overall phase, the (normalized) state vector \(\ket\psi\) is uniquely determined for every (normalized) vector \(\vec V\).
Once again, the important properties above couldn't be obeyed if you required the wave function to be real. For example, if you rotate \(\vec V\) by the angle \(\gamma\) around the \(z\)axis, the components of the twocomponent wave function \(\ket\psi\) are multiplied by \(\exp(+i\gamma/2)\) and \(\exp(i\gamma/2)\), respectively. If you banned complex components, you would ban rotations around the \(z\)axis – and in fact, rotations around any other axis – as well. That would be too bad.
As we have already seen in several examples, complex numbers and phases are not optional luxuries in QM. Very basic and key transformations of any physical system – evolution in time, translation in space, rotation around an axis – are inevitably expressed by the change of the phase of the probability amplitudes (and, in more general bases, by unitary transformations). Moreover, the superposition principle holds; we may always mix two allowed wave functions into their sum. So physically vital operations force us to add and multiply the probability amplitudes according to the rules for complex numbers; up to a change of terminology or notation, the probability amplitudes are and have to be complex numbers! It's the complex numbers \(z\in\CC\) that are the natural values for the wave functions. Requiring them to be real i.e. any condition of the sort \(z=z^*\) would be unnatural (i.e. nonholomorphic) and would prevent the system from doing elementary operations such as rotations, translations, and evolution in time (systems wouldn't be allowed to wait!).
We may design procedures to prepare the electron in any spin state, combine two components of its wave function into general complex superpositions, and verify that all the statements above hold. So all the complex combinations of any two allowed states must be allowed for the laws of physics to be rotationally invariant. So "something" that describes the electron's spin and behaves as a twocomponent complex spinor has to exist; a set of simple direct experiments is enough to establish that this "something" has a probabilistic interpretation (the probabilities are given by the squared absolute values of the complex amplitudes). One may also see that this framework – the general framework of QM – is actually the only framework among a priori similar candidates that makes any sense.
If someone is telling you that he or she may reproduce all of quantum mechanics while demanding that all the wave functions are always real, you may be sure that he or she has always been or has become a confused amateur regardless of the number of the Nobel prizes he or she has received in the past.
Why (near) energy eigenstates are "somewhat more real" than other bases
Such people often tell you that the physical system is really allowed to be found in some specific basis vectors but the general complex superpositions are not allowed or less real. We have discussed two major examples showing that such an assumption is completely nonsensical.
If you allowed a positionrelated "preferred basis", it's clear that particles could never move to the right (or to the left) if you banned the complex superpositions of your "preferred basis". In particular, the plane wave describing a particle that moves to the right has the form \(\exp(i p x / \hbar)\). As emphasized above, this function is complex (nonreal) and has to be complex (nonreal) for it to remember the direction of the motion.
The case of the spin is even more clear. You could pick a "preferred basis" for a spin1/2 particle, e.g. the \(\ket{{\rm up}}\) and \(\ket{{\rm down}}\) states. However, those states pick a preferred axis in the space, the \(z\)axis. The observations show that the laws of Nature are rotationally symmetric so you must be able to prepare corresponding states with respect to any other axis in space. As discussed above, the relevant wave functions (spinors) that are the eigenstates of the projection of spin with respect to a general axis are general complex linear superpositions of the \(\ket{{\rm up}}\) and \(\ket{{\rm down}}\) states. Real combinations only – or the ban on any combinations, if you want to make things really bad – would produce a sick theory that would brutally disagree with the rotational invariance of the laws of Nature.
As I have discussed in the text about the diversity of observables in quantum mechanics, there's nothing special about the position eigenstates, of course. The eigenstates of other but equally good observables such as the velocity are complex linear superpositions of the position eigenstates. The wave function in the momentum representation is the Fourier transform of the wave function in the position representation and vice versa. It's clearly wrong to say that one of them is "more real" than the other. All of the complex linear combinations of such state vectors must be allowed. They must be and they are as allowed as the state vectors you started with. That's what the superposition principle of quantum mechanics means.
If you were looking for a basis that is "a little bit preferred", after all, you could find one. But it wouldn't be composed of position eigenstates. Not even momentum eigenstates. It would be made out of energy eigenstates, i.e. the eigenstates of the Hamiltonian operator (or states that are close to them),\[
\hat H \ket \psi = E\ket\psi.
\] Why? It's because Nature tries to minimize the energy; and it's also because these states are stationary (or close to be stationary) so they tend to conserve the "identity of the physical system". We may also say that the basis (or bases) of energy eigenstates are the only ones whose elements evolve to multiples of the same bases vectors. For all other bases you could think of, a general initial "basis vector" always evolves (via Schrödinger's equation) into a linear superposition of several vectors. So even if you decided to ban all states except for basis vectors at \(t=0\), the general complex linear combinations would inevitably occur for almost any later value of \(t\); it is not really possible to "ban" general complex superpositions because the time is working against you. The energy eigenstates are the only counterexamples.
Consider a very slow proton and a very slow electron in a box. After some time, they approach each other and form a bound state, the Hydrogen atom. Chances are substantial that they immediately form the Hydrogen atom in the ground state and emit a \(13.6\eV\) photon. If they produce an excited state of the Hydrogen atom, it will eventually emit a photon and fall to the ground state, anyway.
So the most relevant wave function of the relative position between the proton and the electron is the wave function for the ground state of the Hydrogen atom. I want to emphasize that this wave function is rather complicated if you express it in the position representation. For example, it exponentially decreases as you increase the distance between the proton and the electron (the \(e\)folding distance is the Bohr radius). In other words, it is not one of the wave functions that someone could include into a "preferred basis" at the very beginning. It is very far from any element of the position eigenstate basis. The ground state depends on the Hamiltonian – on the dynamics – and requires you to make a nontrivial calculation to find out its "geometric shape".
But it's the energy eigenstates, especially those with a low energy eigenvalue, that are most relevant in realistic situations. Especially when we talk about "very fast" degrees of freedom that have the potential to dramatically increase the energy (and there are infinitely many such degrees of freedom in a quantum field theory and even a higher number of them in string theory – e.g. particles i.e. excitations of a quantum field with a high momentum; or excitations of an individual string that turn it into a much heavier particle), Nature always tries to avoid such excessive energy increases. So most of such potentially highenergyadding degrees of freedom are always described by the groundstate wave function, by an energy eigenstate. Such an eigenstate is totally different from any positionlike "preferred basis" that someone may want to prescribe bureaucratically.
The total wave function at least morally follows the Ansatz of the BornOppenheimer approximation,\[
\ket{\psi_{\rm total}} \sim \ket{\psi_\text{slow, general}} \otimes \ket{\psi^0_\text{fast, ground state}}
\] and when we're discussing the evolution of such a system, we simply ignore the last factor (the highenergy, shortdistance inner structure of particles, for example) and deal with the first factor only. In chemistry, we ignore the possible internal excitation of the nuclei. Even if we work on nuclear physics or the quarkgluon plasma at RHIC, we ignore the possible additional internal excitations on the strings "inside" each quark and other degrees of freedom.
(The ground state wave function therefore depends on the Hamiltonian. The Hamiltonian is essential for many other "classical properties" of a physical system, too. For example, the Hamiltonian governs the process of decoherence and because decoherence may be viewed as a process picking a "somewhat preferred basis" for macroscopic objects, this "somewhat preferred basis" depends on the Hamiltonian, too. None of these "somewhat preferred state vectors" can be determined bureaucratically or kinematically, i.e. before we study the Hamiltonian and the evolution! The decohered states usually have sharp positions of macroscopic objects but that's not because position is preferred from scratch; instead, it's because the Hamiltonian is approximately or exactly local, i.e. an integral of a Hamiltonian density.)
Only when it comes to the degrees of freedom whose "most intrinsic" Hamiltonian term adds a low enough energy to the total Hamiltonian, much more general wave functions that differ from the energy eigenstates become "typical" states of the given physical system. And of course, some deviation of the state vector from any energy eigenstate is needed, otherwise Schrödinger's equation would imply that the world is stationary and nothing ever changes about it. That would be bad, too.
But physical systems in the real world don't want to be in position eigenstates (strict position eigenstates are not even normalizable and by the uncertainty principle, they carry a hugely undetermined momentum and therefore a divergent average kinetic energy). Instead, they want to minimize their energy so they tend to organize themselves into energy eigenstates corresponding to low eigenvalues of the energy. This point is really rudimentary and if someone thinks that it's OK to assume that physical systems are "actually" (in the classical or "realist" sense) found in a positionlike predetermined bureaucratic "preferred basis vectors", it means that they don't understand that every physical system in Nature is actually doing something completely different and prefers to sit in one of the "complicated" energy eigenstates, probably the ground state, a state that no one could guess from the beginning (before she calculates anything that depends on the Hamiltonian).
The people who tell you that they may do or emulate quantum mechanics by banning complex linear combinations or even all linear combinations and people who tell you that the physical systems may be assumed to "objectively be" in one of the basis vectors of an easytosee, "kinematical" basis have completely lost it. They don't understand basic physics; either they have never understood it or they don't understand it anymore. We may feel compassion with these people but because the ignorance of basic modern physics doesn't really "hurt", we shouldn't exaggerate the compassion. So instead of excessive compassion, we should better protect our journals and websites against excessive flooding by these delusional folks.
And that's the memo.
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snail feedback (40) :
as the Spin example shows  we should not ignore quaternions or complex numbers  since Pauli matrices marry them, and all of a sudden, electrons make sense.
For some reason there is an itch to think that Reals are more 'fundamental', but actually Reals are just a subalgebra, and it is the whole structure that reveals what is going on  to the extent one can make sense of it  as Nature evidently sees the algebraic structure.
If you do not see the "i" explicitly, it does not mean that it is not there. The asertion " that a polynomial equation of degree n has n solutions" does not mention complex numbers. By your argument, such assertion is a fake, an emulation of the fundamental theorem of the algebra that will never work. But what happens is that you discover the complex structure once you start to use the assertion.
Other example, closer to the path integral equation, is the regularisation of the delta distribution, or even better of the delta' distribution. The definicion < d'  f > = f'(0) does not contain any imaginary number; the regularisation contains an exp(i ...)
Off Topic: Can someone explain how this is possible?
http://www.gizmag.com/radioactivesolarflarewarningsystem/23702/
"Scientists may have hit upon a new means of predicting solar flares more
than a day in advance, which hinges on a hypothesis dating back to 2006
that solar activity affects the rate of decay of radioactive materials
on Earth."
Lumo
Do any theorists take Hilbert Space to be real instead of just a mathematical formalism for QM?
Hi, first I thought you meant real as in "real numbers". Now I think you meant a different meaning of "real" but I don't know what it exactly is. Concepts in physics aren't divided to "real" and "unreal". The Hilbert space is surely "real enough" so that we need it to describe physics properly; but it is surely not "too real" so that it would imitate objective quantities we knew in classical physics.
I mean physically real, like quarks are considered real even though we cannot observe them alone, but only as composites.
And then you have also Real spaces, meaning topological spaces with an involution.
Sounds like nonsense to me . In order to significantly affect decay of radiactive elements flux of "bursts" of solar radiation have to be so high that it would kill everything on face of Earth.
Although I am not quite 100% sure, I agree with Vlad.
Decay rates on Earth can't be detectably affected by some fields inside the Sun which is 150 million km away. It's disputable even when we may create fields strong enough so that the change of the decay rate would become measurable.
This stuff looks like astrology, a method to make prophesies out of noise and incorrectly measured quantities.
Imaginary numbers don't really exist. Quantum mechanics is incomplete, and will be replaced by a deeper set of ideas involving only things that actually exist. Science is naive, only philosophy and religion is truth.
Others who say thinks like "there's something deeper than QM" or "it's heretical to work with complex numbers" may think that the commenter above is nothing more than a hopeless moron. However, it may be a good idea for them to try to look in the mirror and impartially answer to the question whether they're too difference from "Knowledge is power".
presumably, changes in neutrino flux could lead to changes in the rate of certain feynman diagrams that involve a neutrino coming along, and the neutrinoes would reach us in advance because much of the sun is opaque. however, sunspots and flares happen because of the magnetic field which neutrinoes don't interact with
in the rational case there can be extension fields of any dimension formed by adding roots to irreducible polynomials, but once you admit the real numbers (through geometric arguments) the fundamental theorem of algebra becomes inevitable (and can be proven by several methods all of which need at least an epsilon of analysis)
It's a very interesting idea that someone is detecting some modified neutrino flux but if this plays a role, then the process that is being detected is an interaction of 2 bodies, a neutrino and a nucleus, and it shouldn't be called "decay". which is a process with 1 particle in the initial state that decays "spontaneously", not because of a collision with another particle such as neutrino.
Dear Sage, you seem to be confused about a basic thing. The fundamental theorem of algebra doesn't hold for the rational numbers and doesn't hold for the real numbers, either. Also, It doesn't hold for extension fields with extra real numbers added to the reals.
The reason is simple. x^2+1 has no rational or real roots. You may create an extension field in which you allow the root of this equation, i.e. sqrt(1). Indeed, this is the same thing that is normally called "i" these days and the extension field with this "i" is dense in the complex numbers.
But you won't get it right without some obviously complex numbers, named in one way or another, simply because x^2+1=0 has no real solutions.
sure. we can restate the fundamental theorem of algebra as claiming that any proper field extension of the real numbers with the roots of a polynomial is the complex numbers, and the only analysis required is already implicit in the construction by Dedekind cuts that the infimum of the set where 0<f(x) for an odddegree polynomial f is a real number. so that's the sense in which the fundamental theorem of algebra is inevitable in the real numbers
whats the difference between finding a neutrino and emitting an antineutrino, other than losing <1eV?
Thanks for this very nice step by step review of some complex QM issues Lumo, I like this :)
Well, except for all the differences between the two processes, there are no differences. What's your point?
The difference between the absorption of a neutrino and the emission of an antineutrino is that there is a neutrinolike particle only in the initial state in the first case, and it is only in the final state in the latter example.
This difference influences all the kinematic factors that enter the probability of the process. The absorption of neutrinos depends on their flux and one doesn't integrate over their phase space. The emission of an antineutrino doesn't depend on their flux because it doesn't depend on such a flux. Instead, the rate involves the integration over the possible momenta of the emitted antineutrino.
These differences are huge enough so that a good experimental physicist should never confuse the two. Whether something is due to incoming particles/quanta or whether it is because of a spontaneous process whose rate is mysteriously affected by some changes in the Sun is a damn important "detail". If a paper claimed that it was the latter but what the experimenters would be measuring would actually be a detection of incoming neutrinos, I would surely reject the paper if I were a referee.
Let me take a crack at this moron who makes a choice to be ignorant by closing his mind. Science is provable, religion is not, and most of "proven" philosophy are only as good as the axioms, most of which can be false in another philosophical system. So tell me you brain stem, how is philosophy and religion true if it can't be proven?
It looks to me like quaternions would also work in the above equations. Is the something that would rule out quaternions or is a quaternion valued amplitude a possiblity?
thanks for the excellent answer. I apologize for how churlish my question sounded.
religion has been proven to be true in the hearts of all. science has only been proven wrong. microscopic and telescopic studies have detected god throughout the universe. while all discoveries in physics and science has disproven quantum mechanics and science. in fact einstein proved, using his his theory of relativity (special and general), that god exists, as he stated, while disproving quantum theory and physics.
Hi Lubos,
you don't need imaginary i in the evolution equation,you just need an antihermitian matrix
ie dpsi/dt = H.psi where H = 2x2 matrix [0 1] [1 0] will do. In 1 dimension the only antihermitian matrix is +/i so , yes, in 1d you need i.
Where are 't Hooft's postings here? Another blog suggested he posted here about his deterministic theory, I wanted to tell him that theories of everything involving permutations are so OLD :)
(but thank you Gerad for enabling everyone to think about stuff more fundamentally :) )
er, my comment disappeared??
You are cherry picking the data and not taking in the whole story. I have no problem with philosophy and religion, just uninformed, under informed, or just plain wrong arguments that contradict well proven science fact. In terms of Einstein, he was wrong about Quantum Mechanics. It has never failed a test. In fact Einstein's Special Relativity has been combined with Quantum Mechanics to make Quantum Field Theory, which, so far has passed all tests. Although, many physicist's believe it will break down to string theory at much higher temperatures.
yes, quantum field theory will breakdown. that is the one point you made which is right. but god and philosophy will never breakdown. thank you for helping to prove my point.
Please name some names! Which Nobel prize winning physicist has forgotten the basics?
Dear Benjamin, you may follow the "recent Disqus comments" in the right sidebar. If you did so, you would know that the answer to your question, and the driver for whom this article was designed, is called Gerard 't Hooft (Physics 1999, for things in the 1970s). You didn't expect me to be reticent while answering your question, did you? ;)
It would be nontrivial to make the list much larger but a name that has forgotten much more than that, Brian Josephson (Physics 1973), would be easy to add. He heads a mindmatter unification (paranormal) group.
I am not against God or philosophy, as I stated. I am against incorrect thinking and arguments on what has been proven. That leaves a multiverse of room for things based in reason and faith beyond what science has proved.
yes, it is the socalled scientists that today speak of the "multiverse". It is physics that it is unscientific to do so. You are sounding very unscientific to use the multiverse in your argument.
Knowledge is power, according to you which religion and which philosophy "is truth" ?
I don't want to limit the power of the Almighty to just one universe if there are many more. Contradict that!
Damn! t'Hooft is (or was) very, very, very smart! Of course, I don't know at all. But you have succeeded in scaring me!
Regarding the "pure imaginary" i in Schroedinger's equation, I don't see why "preserving the norm of the state vector" is a necessity. The universe has a finite life and so we have no evidence that state vectors are preserved indefinitely. Similarly, we have no evidence that the universe is unchanged by translations in position as we do not know the boundary conditions. So we cannot take as an assumption that the only possible correct theory of quantum mechanics will imply conservation of momentum or energy. These are human restrictions, not necessarily part of nature. Similar arguments apply to commutators of x and p and path integrals.
Meanwhile, I'm having a blast working on gravitational wave detection. This semester I'm supposed to be writing a paper on the subject of calculational techniques.
Dear Carl, the norm is neither energy nor momentum.
The (squared or not, depending on your definition of norm) norm that is preserved is the total probability of all mutually excluding possible conditions that the physical system finds itself in, and it must be equal to 100% at all times. It's pure logic; a violation of this condition means a logical inconsistency.
If you wanted to redefine the relationship between the probabilities and norms (squared amplitudes), which is a postulate of quantum mechanics, you would find out that you won't be able to get a consistent theory, either, at least not a local one.
Those things don't depend on finite or infinite lifetime at all. Even if there would be a dependence, you could at most violate any of these conservation laws by 0.1% per 14 million years, and that's an undetectable rate. It's surely preposterous to suggest that such tiny effects that could be hidden behind cosmology  and I think that they can't be hidden even behind cosmology  would be able to deform "i" to a real number. That's complete nonsense. They could at most modify "i" to "i (1+epsilon)" or something like that which is still complex.
Your comments show that you still don't have the slightest clue how quantum mechanics works but they suggest more than that. You don't understand that small effects can't possibly produce arbitrarily large ones with a large probability. And these basic concepts of "continuity" of physics were understood well before quantum mechanics.
Again you are very unscientific to argue for the multiverse. Science has proven it to be false, a false idol actually. Only god is true, and science is naive and unprovable. This is uncontradictable.
Shannon, your question is wrong. The truth is the answer. There is only one true god and one true religion, there always has and always will be. Listen to your heart and you will see. Science has proven itself to be a false demigod and unprovable. Please open your mind.
Dear Knowledge is Power, could you please reduce your commercials for God at least by an order of magnitude? I am sure that God can finds His or Her own communication channels outside TRF comment sections.
This information is really wonderful and reliable that many students and other people who are still confused regarding on this matter will be able to know on how to solve such problems in physics.
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