## Friday, April 12, 2013 ... /////

### Why quantum mechanics can't be any different

Classical physics and quantum mechanics are the only two frameworks for physics that are worth mentioning. And it's quantum mechanics that is more true in Nature, that is more fundamental, and that is more legitimate as the starting point. Classical physics may be derived as a limit of quantum mechanics but quantum mechanics can't be obtained by any similarly straightforward, guaranteed-to-succeed procedure from classical physics.

And yet, quantum mechanics remains wildly misunderstood and underestimated. Many people, including professional physicists, can't resist their primitive animal instincts and they keep on trying to rape quantum mechanics, insert their prickly objections and modifications into it, and make it more classical. However, quantum mechanics is well protected and it can't get pregnant with bastards. It's just patiently saying "f*** off" to these deluded non-physicists and equally deluded physicists.

Even those who realize that quantum mechanics – the framework respected by Nature – is fundamentally different than classical physics and that there won't be any counterrevolution that would make physics classical once again often underestimate the rigidity and uniqueness of the universal postulates of quantum mechanics. They think that many things could be altered, mutated, and quantum mechanics has many possible cousins and it's an accident that Nature chose this particular quantum mechanics and not one of the cousins.

They're wrong, too. In this text, I will demonstrate why certain properties of quantum mechanics are inevitable for a consistent theory.

Complex numbers are the only allowed number system for amplitudes

First, let us imagine a cousin of quantum mechanics where wave functions $\ket\psi$ take values in a Hilbert space that isn't complex: let's try to replace $\CC$ by $\RR$, $\HHH$, or something else.

If we're not allowed to multiply the state vector by the imaginary unit $i$, e.g. if we try to work with the real numbers $\RR$, we're immediately in trouble. Schrödinger's equation says that$i\hbar\frac{\dd}{\dd t}\ket\psi = \hat H \ket\psi$ and the coefficient is pure imaginary. This pure imaginary character of the coefficient is what is needed to preserve $\braket\psi\psi$, the norm that is interpreted as the probability. For energy eigenstates, the equation says that only the phase is changing with time. With a real coefficient, the wave function would exponentially increase or decrease with time – and so would the total probability of all mutually excluding properties of the physical system.

I will discuss the need for "unitarity of the evolution" momentarily.

We began with Schrödinger's equation as a place where the imaginary unit $i$ appears but as you know, I don't consider this equation to be excessively "superfundamental" in quantum mechanics. One may show – and Dirac has shown – that this equation is equivalent to the Heisenberg picture in which the state vector is constant but the operators evolve according to the Heisenberg equations of motion$i\hbar\frac{\dd}{\dd t}\hat L = [\hat L, \hat H]$ Needless to say, the coefficient $i\hbar$ in this equation may be shown to be the same $i\hbar$ we had in Schrödinger's equation. In this picture, we may offer many independent explanations why the coefficient has to be pure imaginary. For example, if the operator $\hat L$ is required to be Hermitian at all times – as appropriate for observables, as we will discuss – its time derivative has to be Hermitian, too.

However, the commutator of two Hermitian operators is anti-Hermitian, i.e. it obeys$\eq{ [\hat L, \hat H]^\dagger &= (\hat L \hat H - \hat H\hat L)^\dagger =\\ &= \hat H\hat L - \hat L\hat H = [\hat H,\hat L] = -[\hat L, \hat H] }$ where I have used $\hat H^\dagger = \hat H, \quad \hat L^\dagger = \hat L, \quad (\hat X\hat Y)^\dagger = \hat Y^\dagger \hat X^\dagger.$ If we want to express a Hermitian operator using this anti-Hermitian commutator – a candidate for the time derivative of a Hermitian operator has to be Hermitian – we have to multiply the commutator by an imaginary constant, one we call $i\hbar$, which erases "anti-" from the adjective.

We don't really need to discuss the Hamiltonian and time evolution at all. Think about Heisenberg's "uncertainty principle" commutator$[\hat x,\hat p ] = i\hbar.$ A few paragraphs above, I proved that the commutator of two Hermitian operators is actually anti-Hermitian. So if the commutator of these two particular operators is a $c$-number, i.e. a multiple of the unit operator, then the $c$-number has to be pure imaginary. Again, it's called $i\hbar$ using the usual symbols and unit conventions of quantum mechanics. And once you accept that the commutator is a pure imaginary i.e. non-real operator, it follows that there can't be a basis in which both $\hat x$ and $\hat p$ would be expressed by real matrices; the commutator of any two real matrices is real as well which is no good to satisfy the relationship above!

So the imaginary unit $i$ is clearly needed. You may try to go from $\CC$ to the opposite direction than to $\RR$, i.e. to larger number systems such as $\HHH$ and $\OO$. If you pick the quaternions $\HHH$, it won't be lethal but the non-complex Hamilton numbers will be redundant. There are various ways to see it. For example, Schrödinger's equation or Heisenberg's equations will have one particular pure imaginary unit which we may still call $i$ without a loss of generality. If we pick some "orthogonal" imaginary unit in the quaternions such as $j$, the hypothetically quaternionic wave function will effectively split to two complex ones,$\ket{\psi_\HHH} = \ket{\psi_\CC}_1 + j \ket{\psi_\CC}_2$ and these two state vectors labeled by the subscripts $1,2$ will evolve independently from each other. The only physically meaningful interpretation of the wave function above will be equivalent to a density matrix that is obtained by mixing the two pure density matrices:$\rho_{\HHH,\rm equiv} = \ket{\psi_\CC}_1 \bra{\psi_\CC}_1 + \ket{\psi_\CC}_2 \bra{\psi_\CC}_2.$ You don't get anything fundamentally new. The "quaternionic wave function" will be intrinsically "reducible" and you may always study the elementary building blocks that the wave function may be reduced to – and they're complex. At least with a single time coordinate, you can't get anything really new that could be called "quaternionic quantum mechanics".

Only the complex numbers are tolerable as the "fair number system" for the coordinates of the state vector. Real numbers are complex numbers that are constrained by an extra condition – one that is lethal for a physical interpretation, as we have pointed out; quaternions can't really show their muscles beyond their being a "pair of complex numbers".

This fundamental character of complex numbers holds even in "deep enough mathematics" that is detached from the physical conditions we have discussed. For example, if we talk about representations of groups – and the Hilbert spaces in any quantum mechanical theory are representations of groups and algebras of operators – the "default" character of a representation is always complex, i.e. $\CC^n$. The real representations $\RR^n$ and the pseudoreal representations, which include the quaternionic ones $\HHH^{n/2}$, may be interpreted as ordinary complex representations $\CC^n$ with an extra "structure map" $j$ acting on the representation that is antilinear (differing from a linear map by an extra complex conjugation in a defining "scalar linearity" condition) and that commutes with the action of the group.

Real representations are those whose structure map obeys $j^2=+1$ while the pseudoreal (including quaternionic) representations are those that obey $j^2=-1$. At any rate, the representation may always be viewed as a "complex representation with some extra structure". For $j^2=+1$, the structure map allows us to prove that there is a basis in which all the matrices are real; for $j^2=-1$, we may prove that all the matrices representing the group elements may be organized into $2\times 2$ blocks $a+ b\sigma_y$ where $a,b\in\CC$ and these blocks effectively represent $1\times 1$ quaternionic entries $a+jb$.

Real numbers and quaternions are just "cherries added on a fundamental pie" and the fundamental pie is always complex. It's not smaller and it's not larger. At the end, this fundamental position of complex numbers boils down to the fundamental theorem of algebra: every algebraic equation of $n$-th degree has $n$ roots. But this theorem only holds for $\CC$.

While the quaternions as components of a state vector were just "redundant" but non-lethal, octonions $\OO$ would be lethal as matrix entries of operators because octonions are not associative (they break the rule $(ab)c=a(bc)$) while the matrices – something identified with observables and evolution operators etc. – have to be associative e.g. because the evolution is associative.

You could try to modify $\CC$ in a different way – for example, you could try to pick all the "rational complex numbers". This would also be bad, at least in theories with a continuous time coordinate. In some not-quite-physical toy models, the amplitudes could happen to be rational for "rational questions" but it's an extra coincidence, or an "extra structure", and it doesn't hurt if you simply use wave functions in $\CC^n$.

Paradoxically enough, the most tolerable "number system" in which you could try to pick your state vector are deeply esoteric systems such as the so-called $p$-adic numbers. Quantum mechanics based on such numbers could obey some consistency rules but it would certainly be very different from the theories we use to describe Nature around us.

Linearity of evolution operators

Schrödinger's equation is linear in the wave function. This also implies that the finite-time evolution operators are linear:$\ket{\psi(t_1)} = U(t_1,t_0) \ket{\psi(t_0)}$ Could we make the future wave function depend on the initial wave function in a nonlinear way? We could try but we would quickly run into some serious trouble. What kind of trouble?

Quantum mechanics and any other "at least remotely similar" hypothetical cousin of it describes the state "A or B", with some probabilities, as a superposition$\ket{\psi(t_0)} = c_A \ket A + c_B \ket B$ Assume that someone may "perceive" whether the state of the physical system at time $t_0$ is A or B; the "A or B" information is a legitimate information that may split consistent histories. Without a loss of generality, imagine that she learns that the state is A. Such a state will evolve into $c_A \cdot U(t_1,t_0)\ket A$ at time $t_1$. Similarly for B.

Now, it's important that her consciousness or the absence thereof remains undetectable. After all, no one has ever experimentally demonstrated whether women have consciousness much like men. ;-) And it's true for men, too. It's important that someone's "conscious" learning about the result of a measurement doesn't modify the system in any further way. The procedure needed to measure may impact the measured physical system of interest; however, the mental processes that this measurement causes remain subjective and inconsequential for the rest of the world. We don't want a qualitative "wall" separating conscious and unconscious objects or subjects. Observers are dull physical systems, too.

We're really discussing "Wigner's friend" scenario here. It's important that Wigner is allowed to ignore the "A or B" realization and continue to work with the whole initial state $\ket{\psi(t_0)}$ above. Because the evolution operator is linear, this state evolves to$\ket{\psi(t_1)} = c_A U(t_1,t_0) \ket A + c_B U(t_1,t_0) \ket B.$ That's great because these two terms (and it could work for many terms, too) are sharply separated from one another. Wigner may calculate the probability of a property at time $t_1$ and there's a chance that the "A or B perception" at time $t_0$ only has a tolerable impact on Wigner's predictions: it suppresses the history with A and B at $t_0$ by their probabilities $p(A),p(B)$, respectively.

If the evolution operator were nonlinear, Wigner would get various terms that depend both on $c_A$ and $c_B$, e.g. that would be proportional to $c_A^m c_B^n$ with some positive powers. These terms would be there and nonzero if he used the full wave function with both possibilities; but if he accepted that his female friend made a measurement at time $t_0$, they would disappear because $c_A^m c_B^n=0$ if either $c_A=0$ or $c_B=0$! So he would get different predictions depending on the question whether his female friend "perceived" something or not.

In other words, souls and ghosts would become physical and they would start to fly everywhere. This is lethal for a candidate theory of mutated quantum mechanics not only because we dislike souls and ghosts. It's lethal because the measurement – that would tangibly affect Wigner's predicted probabilities – could occur at huge distances, at a spacelike separation, and the influence proved above would be a genuine, detectable, faster-than-light signal that would demonstrably violate Einstein's special theory of relativity. We would enable not only souls and ghosts; we would enable superluminal voodoos. You should understand that this would lead to real trouble in the predicted phenomena which is a genuine, objective problem with a candidate theory; your unfamiliarity with a mathematical framework to describe Nature (quantum mechanics) is not a genuine problem, it is just your subjective, psychological problem.

So we have to keep the alternatives that may decohere from each other separated, even after some extra evolution in time; linearity is needed for that. Because the evolution operator $U(t_1,t_0)$ is a linear operator on the Hilbert space, so is its $t_1$ derivative near $t_1\to t_0$ – and it's the Hamiltonian that enters Schrödinger's or Heisenberg's equations (up to a factor of $i\hbar$). So the Hamiltonian has to be a linear operator, too.

Similarly, we may see that all other observables representing Yes/No questions have to be linear operators. These linear Hermitian projection operators $P$ are operators of the type that Wigner's female friend actually applied at time $t_0$ to simplify her further thinking about the system (the "collapse" of the wave function). If the operator were not linear, one would get a similar interference between the possibilities that should be mutually exclusive.

The Yes/No operators have to be projection operators, $P^2=P$ – yes, I started to drop the silly hats at some moment, I hope that you survived that (everything in Nature has hats and we should, on the contrary, invent bizarre accents for things that aren't operators, to emphasize that they're not fundamental physical quantities!) – because we want their eigenvalues to be $0$ and $1$. Also, we need $P^\dagger=P$ because we want all the eigenvectors with the $0$ eigenvalue to be orthogonal to (i.e. mutually exclusive with) those with the $1$ eigenvalue.

Yes/No operators must be represented by linear Hermitian projection operators. Similarly, operators such as $X$ are linear Hermitian operators because they may be constructed out of the Yes/No operators by the following sums:$X = \sum_i X_i P_{X=X_i}^{0/1:\rm No/Yes}.$ Note that this formula doesn't really depend on any conventions in quantum mechanics. It just says that the value of $X$ is the value of $X_i$ of the only allowed (eigen)value of the coordinate for which the projector $P_{X=X_i}=1$; the other projection operators are effectively equal to zero.

Fine. We see that all observables with real measurable values are represented by linear Hermitian operators acting on a complex Hilbert space.

Probabilities as squared amplitudes

Born's rule tells you that the probabilities – the only kind of numbers that quantum mechanics may predict in the most general situations – are calculated from the complex numbers, the amplitudes, by squaring their absolute values. We have$p_i = |c_i|^2, \quad c_i\in \CC.$ That's obviously another favorite target of the rapists I mentioned at the beginning. Why wouldn't we use $|c_i|$ or, more naturally, $|c_i|^4$ or any other function of the amplitudes (perhaps not necessary a phase-independent function)? If you pick the fourth power, for example, you may surely get an equally good cousin of quantum mechanics – or mutated quantum mechanics – and our Nature has just picked the second power due to some random subjective choices, hasn't it?

Not really.

When you decompose a wave function into some components that are eigenvectors of $L$$\ket\psi = \sum_i c_i \ket{\ell_i},\quad L\ket{\ell_i} = L_i \ket{\ell_i},$ we want to say that the probability that $L=L_i$ is equal to $p_i=|c_i|^2$, assuming that the basis of vectors $\ket{\ell_i}$ is orthonormal. We need it for the total probability of all possibilities, $\sum_i p_i$, to be conserved. So if it is 100 percent at the beginning, it is 100 percent at the end.

This conservation law follows from $H$ that is a Hermitian operator as we have already demonstrated; the evolution operators are unitary, $UU^\dagger=U^\dagger U = {\bf 1}$, as a result. And what is conserved is $\braket\psi\psi$ which may be proved to be equal to $\sum_i |c_i|^2$ by pure algebra i.e. without any assumptions about physics. There can't be an equally general sum that is conserved in the general situation so the two sums must be functions of one another and $p_i=|c_i|^2$ follows from that (up to the freedom to insert an illogical universal multiplicative coefficient into this relation).

This argument holds for any Hermitian operator $L$ and the corresponding decomposition of the state vectors into its eigenvectors. The probabilities have to be given by the squared amplitudes, otherwise the "total probability of all mutually excluding alternatives" can't be conserved.

You could try to keep on struggling and proposing various creative loopholes. For example, you could say that this whole quantum mechanics is based on "unitary evolution operators" and the unitary groups just happen to have a bilinear (well, sesquilinear) invariant given by the complexified Pythagorean theorem. But there may be other groups that have higher-order invariants, right?

Well, there exist groups with higher-order invariants but these invariants aren't guaranteed to be positive so they can't play the role of probabilities. This is enough to kill these possibilities but there are actually many other ways to kill it. We simply want simple enough state vectors – energy eigenstates – to evolve simply. The change of the phase with time is what this change has to look like.

There are various other ways to attack this loophole but I don't want to spend too much with it. You should just realize that in proper quantum mechanics – whatever the Hamiltonian is: non-relativistic quantum mechanics, quantum field theory, string theory, whatever you like – pretty much any "physical transformation" of the physical system (evolution in time, translation in space, rotation, parity, and so on) is expressed by a unitary operator on the Hilbert space. If you want to change something about this rule, you are really building an entirely new theory from scratch.

Fixing the norm of the state vector along the way

Another group of "anything goes" rapists could propose a universal cure for all the non-unitary, nonlinear, and other theories. They could say that the only "constraint" we faced was the condition that the sum of probabilities had to remain 100 percent. Can't we just rescale the wave function – that may evolve according to any non-unitary, non-linear equation of motion – at each moment to manually guarantee that the sum of probabilities remains equal to 100 percent?

We may do it but we will run into conflicts with other basic physical or logical requirements that these rapists might be willing to overlook but that are paramount, anyway. What do I mean?

Imagine that you start with $\ket{\psi(t_0)}$ and evolve it to$k(t_1,t_0)\cdot U(t_1,t_0) [ \ket{\psi(t_0)} ]$ where I wrote the ket vector as an argument in the square brackets to indicate that the operator $U$ may be nonlinear. Also, the added coefficient $k$ is there to keep the total probability equal to 100 percent according to your own formula for the total probability, one that may differ from Born's rule.

That may look fine to you but we resuscitate ghosts and voodoo again. The required "renormalization constant" $k(t_1,t_0)$ actually has to depend on the initial state as well if it's able to preserve the total probability in the general case – it was fraudulent to suppress this dependence. And if the initial wave function describes the "A or B" state, this $k$ will inevitably depend on $c_A$ and $c_B$ again. The possibilities "A or B" will refuse to split in the final expression for $\ket{\psi(t_1)}$. Again, it will be important whether Wigner's female friend at $t_0$ "eliminated" the other possible outcomes or not. The eliminated outcomes will still affect the outcomes for the moment $t_1$ that remain viable; equivalently, consciousness will become physically measurable and it will violate the laws of special relativity again.

So it's important not to attempt to "renormalize" the formulae for probabilities by additional ad hoc fudge factors. One may argue that such fudge factors would damage the very logical structure of the theory but even if you were OK with it, you will ultimately see that your alternative theory allows the female observers to send superluminal signals by the "power of her will" (a superluminal form of telekinesis combined with telepathy) and violate the rules of relativity which seem to hold, according to observations and a robust symmetry principle extracted from all these observations.

So the probabilities have to be what they are according to the unadjusted formulae and because their sum has to remain equal to 100 percent and because the bilinear invariants are the only universally non-negative (for all states) invariants one may find for general classes of transformations, it follows that all "physical transformations" are encoded by unitary linear transformations on the Hilbert space and the squared complex amplitudes have to be interpreted as probabilities.

I feel that I have forgotten some other "popular" ways to rape quantum mechanics. But it's been enough so far and if I recall what I have forgotten, I will update this blog entry.

#### snail feedback (60) :

Hi Lubos, If we think of the wave function as a 2 component spinor for the real and imaginary parts then I can write the Schrödinger eq. as follows:

\hbar\frac{d}{dt}\begin{pmatrix}\psi_{1}\\
\psi_{2}
\end{pmatrix}=\begin{pmatrix}0 & -\mathcal{H}\\
\mathcal{H} & 0
\end{pmatrix}\begin{pmatrix}\psi_{1}\\
\psi_{2}
\end{pmatrix}

which looks real but messy. Presumably the physics should be the same.

"the equation says that only the phase is changing with time"

Dear Lubos, A naive question, but does this have anything to do with so-called gauge invariance? I've read (or mis-read?) that changes in phase don't effect anything measurable the same way that rotations or translations in space don't, and that this is what is meant by local symmetry. I've also read that calling it a symmetry is something of a misnomer, that redundancy would be a better word, which (I am guessing) means that information about the phase of a quantum state (elementary particle?) is superfluous when it comes to predicting the probabilities of a measurement of its state.

Dear Luke,

it's hard to deal with such questions. Does the changing phase of the wave function have "anything" to do with gauge invariance?

Well, Yes, No, it depends on what you count as "anything". Most importantly, the phase of the wave function is *not* gauge invariance. There are lots of reasons why they're not the same thing in general. First, gauge invariance exists even in classical, non-quantum physics; wave functions and their phases only exist in quantum mechanics.

With this being said, the phase of the wave function may emulate the phase of a classical field if the classical field becomes quantum and is used to create a particle, via creation operators. So then these two phases are related.

However, this relationship still doesn't mean that the change of the phase of the wave function is a gauge invariance. For example, as you notice, we must physically identify states related by gauge invariance; we must declare that their difference is physically a zero vector of the Hilbert space.

But if we identified wave functions differing by a phase in this mathematical way, it would be like identifying each of them with zero. There would be no states left. So while state vectors with different phases are equally good for physics, we must always treat this as a global symmetry, not a gauge symmetry, otherwise we eradicate the whole Hilbert space.

Also, the overall phase of the wave function is just one phase. Gauge invariance typically allows us to change infinitely many phases, one phase per each point of the space or spacetime. So they're not the same thing.

Please don't take it too personally but the degree of confusion behind your question "is A related to B" is so deep that you should start with learning what the damn A and B mean. If you have no clue what they mean, and all the data suggests that you have no clue, then it's very likely that convoluted questions such as "do A and B have anything to do with each other" end up being completely unconstructive and not leading to any useful answer that you may understand.

It's like asking "does this chromosome you talked about have anything to do with stomach?" Yes, no, what of it. There are surely some relationships between chromosomes and stomachs but the question makes it pretty clear that the author of the question probably has no clue what either of these two concepts mean, otherwise he wouldn't ask this strange question. If it's so, why doesn't he start to learn the basic things about the chromosomes and stomachs first, before trying to construct would-be advanced and would-be creative questions involving both concepts - that are in reality completely silly?

Cheers
LM

Hi JR, 1/2 is written as \frac 12 in TeX! Your very way how you use pmatrix to write this simple thing, a fraction, suggests that you're using TeX to make your comment look wiser than it is. Why not write it in words? Well, DISQUS doesn't do any TeX, anyway. ;-)

Yes, the imaginary unit may be "replaced" by the 2x2 matrix ((0,-1),(+1,0)). This matrix squares to -1 (times the unit matrix), too. It doesn't really look messy. It's the same thing. You just used a different notation for the imaginary unit. What's important is that the components of the wave function will always come in pairs - we call the members of the pair "real and imaginary part of the amplitude" - and the equations will satisfy the complex structure behind these amplitudes which means that all the operators in your equations will commute with the matrix ((0,-1),(+1,0)) because this matrix represents "i" and C is a commuting field.

Your way of writing it obeys it which is why you haven't gotten rid of the complex numbers in any sense. You just obscured the structure of the equation but it didn't stop being an equation for a complex wave function. What's critical here is that you can't "relax" the special properties that follow from the complexity - you can't replace the equation above by one that *couldn't be* rewritten in terms of complex components.

Dear Lubos,

Is it possible to reformulate QM with real numbers only, and Born's rule substituted by p_i=c_i, by introducing unobservable ghost-states with negative amplitudes? Is it true that Dirac tried to introduce such a notion, or maybe this is nothing more than a fairy-tale?

My apologies, Lubos, for asking a dumb question. In my amateurish way and with a rickety old brain I have been trying to better appreciate QM. And I've learned a lot (well, a little bit) by reading you, from Leonard Susskind's series on quantum entanglement, most recently by watching Ramamurti Shankar's video lectures on OM which are on YouTube. Maybe one of your readers -- Dilaton perhaps? -- could point me to a place where Leonard Susskind discusses these issues. I know I'm a fool but at least I'm a humble fool wanting to learn, and I hope you will suffer me in small doses.

(Besides -- now, please, don't hit me! -- when it comes to economics I think I may know a thing or two you don't -- which is not to say I don't learn things from you too.)

Lubos, what about the fact that Bohmian Mechanics exactly reproduces the predictions of quantum mechanics?

Hi Lubos,

Could you explain why this statement is true?

> these two state vectors labeled by the subscripts 1,2 will evolve independently from each other.

Because $i$, $j$, and $k$ don't commute and the Schroedinger equation involves an $i$, I thought a state of type 2 would mix with a state of type 1 after evolution. I don't think this affects your conclusions, though.

What the hell makes a gauge theory? Why is Maxwell's electromagnetism a gauge theory but Newton's mechanics is not?

Why the hells? Why don't you try to find at least one sentence of a definition e.g. at

http://en.wikipedia.org/wiki/Gauge_theory

instead of spreading hells? Your question is analogous to the question: What the hell makes a mammal? Why is squirrel a mammal while an eagle is not?

Sorry, I think I'm responsible for that hell. Was just trying to be colloquial.

Dear Anon,

this is kind of the same question, fundamentally, as JR's question below.

When we say that quantum mechanics has to be complex, we don't mean that one can't obscure this fact. One may always obscure this fact or any other fact.

We may say that [a particular woman] is a woman but one may obscure this by a dildo. But she's still a woman. In the same sense, quantum mechanics' being complex is some well-defined intrinsic property that doesn't depend on makeups, dildos, and reformulations. You may reformulate quantum mechanics in any way but if it's still a physically equivalent theory, one may still show that there is a "complex structure" on the space of possible states. There is a way to multiply the state vectors by complex numbers.

You may write their coordinates as 3 horses on one side of a coin and 4 cows on the other instead of 3+4i, but I still know how to use the complex number "reformulated" in this way. It is still complex despite the "makeup" and attempts to obscure it.

Now, concerning your particular "new rule", the probability has an invariant physical meaning. One can't redefine it; it is defined by something that may be operationally measured - by repetitions of the same experiment. So "p_i" means what it means. If you say that p_i=c_i, it means that you want to parameterize the states by what we call |c_i|^2 and use a new (redundant, deliberately confusing) symbol for it. Great but then you also need to remember the phase that doesn't affect the absolute value. You may surely write all complex amplitudes in the polar form, r * exp(i.phi). But that doesn't affect the fact that all the numbers will still be complex and quantum mechanics will still be linear although the people using nonlinear parameterizations such as polar coordinates will have a harder time to understand this fact. But it's their problem. The fact that it's harder for them to see that the operators etc. are linear doesn't mean that the linearity is invalid.

Concerning negative probabilities, a reformulation that survived because it makes sense is Wigner's quasi-probabilistic distribution

http://en.wikipedia.org/wiki/Wigner_quasi-probability_distribution

which is a way to rewrite/encode density matrices as functions of classical commuting phase space coordinates. This distribution is the closest quantum mechanics' definition of the "probability distribution on the phase space". However, it may be negative in small regions and this allowed negativity effectively enforces the uncertainty principle and all the wonderful new features of quantum mechanics.

Dirac and others may have emitted lots of other speculations but nothing else meaningful linked to "reformulations using negative probabilities" came out of it. So we just ignore it. Physics isn't like religion where a Moses can say an arbitrarily bizarre sentence that makes no sense and his followers spend thousands of years by attempts to decode what he meant. He just didn't mean anything that made sense at that moment, otherwise it could have been formulated meaningfully.

There are also lots of negative probabilities in physical theories that aren't quite consistent etc. So for example, Dirac and others tried to formulate theories of extended objects such as membranes and they found that they would allow the probabilities to be negative - "ghosts" - which is bad. That's why they abandoned it. But they overlooked string theory along the way - a theory of strings may be completely meaningful because the ghosts may be tamed by new gauge symmetries. Needless to say, they therefore overlooked the whole structure and many extra things they were just vaguely envisioning but something similar turned out to be possible. But many particular claims they did were still wrong.

Cheers
LM

You might be interested in Kapustin's recent paper, which addresses some of the same questions in a mathematically precise way: http://arxiv.org/abs/arXiv:1303.6917

Fun paper!

No prob! Still, sometimes devils need to be regulated. Here it looked like a leftist rally complaining about an essential thing (gauge theory in this case) and it's always right to shoot into such rallies. ;-)

Dear jy,

(e+fi) (a+bi + cj+dk) = ((ea-bf)+i(eb+fa)) + (j(ec-fd)+k(ed+fc))

Note that a,b only appear in the first part of the final result while c,d only appear in the second half. They never mix up and in fact, the rules of multiplication of a quaternion by a complex number (e+fi) obeys the fact that the quaternion is composed of two complex components, one for the 1,i units and the other for the j,k units.

It doesn't matter that i,j and i,k don't commute. In this picture, j and k is always on the right side from the "i", so we never see what the product in the other order actually is. That's really the point: the operators are always acting on the (quaternionic) ket vectors from the left side.

Cheers
LM

This "fact" is completely false. What is true is that one may write down a classical theory with hidden variables in which the probability distribution for a single particular quantity, the position X, will be the same as predicted by quantum mechanics if it was the same at the beginning.

But this isomorphism breaks down if we switch from this toy model of spinless nonrelativistic particle to any other quantum mechanical system - particles with spins; quantum fields, strings; anything else - or if we measure different observables than X and we surely measure different ones most of the time, usually various terms in the energy operator. The need of the Bohmian theory to separate the observables to those that really exist and those that must be faked is really a proof of the inconsistency of the theory, see

http://motls.blogspot.com/2009/01/bohmists-segregation-of-primitive-and.html?m=1

More importantly, we can't ever make a functioning theory out of "Bohmian Mechanics" because 1) the pilot wave in this picture is a real classical wave and needs to be "swept under the rug" after the measurement. 2) But there is no "broom" to do so that could be compatible with basic principles of physics such as relativity - any "broom" would mean a superluminal action at a distance.

de Broglie-Bohm pilot wave theory may have been an admirable attempt to rewrite a toy model of quantum mechanics in a classical framework but it's incompatible with everything that was discovered after the mid 1920s and it's really incompatible with principles that have been known since 1905, too.

Thanks for your answer. You are absolutely right. That was a stupid question.

This blog explains why current quantum mechanics has to be formulated on complex unitary Hilbert spaces, given the current axioms of quantum mechanics and it is right. But all the arguments given here do not rule out the possibility, that some other axioms and a different theory could be found that would equally describe the world. QM as currently formulated might not be the ONLY POSSIBLE framework to describe the world. There might be others. Maybe we find some new axioms and derive a different theory with different mathematical structure from them. The only constraint is that it has to be consistent with all experimental results. The wave function itself is just an axiom. There might be some other theory that doesnt have the wave function but some other object. You cannot prove that QM is the only possible framework, because you would need to prove that the axioms of QM are the onnly possible axioms. And axioms are by definition unprovable, but rather confirmed by experience and observation

I'm not sure how your understanding of QM is (and if someone were to ask me about "guage invariance", btw, it'd be a short conversation), but as a student/layman who has had a 'popular' interest in such things for many years, I would suggest Albert's "Quantum Mechanics and Experience." It's no text book, and it doesn't give you the REAL maths involved, but it explains QM from the ground up, and presents simplified versions of all the key tools/formalism/notation IN MATHEMATICAL FORM, which to me makes it significantly different from your standard '1st-timers' popularization, where everything is explained through ANALOGY, instead. So in other words there are equations, and he takes you through how QM systems are analyzed and manipulated through the methods and conventions of the field. I'm going to assume you know a lot of that stuff if you've listened through stuff like the Susskind lectures, but Albert's book gave me the tools to 'play around with' the ideas mathematically as he (and others since) discuss the more philosophically interesting implications of QM's revelations about the universe around us.

[Actually, on a quite ironic note, I'm now remembering that one of the concessions Albert makes for the elementary reader is abandoning use of complex numbers all together when first presenting the state space. Perhaps Lubos or someone else familiar with the book could critique this approach to keeping things accessible and uncluttered.]

But on the original subject of bringing oneself up to speed in the sciences, I keep discovering that biting the bullet and engaging some real academic material (textbooks) has again and again ended with me asking myself, "Why the hell did I wait so long?!" and telling myself, "Next time I'll surely be smarter.." ...And then I end up reading blogs/research papers/message boards/etc. for 6 months intensely on complexity theory, or neuroscience on brain metabolism before finally hazarding a crash course on the subject ;-), and suddenly noticing the incredible amount of deserving words/terms/concepts that I was somehow "reading through"--often without even realizing that I was missing anything! [And just a reminder: google is your friend when it comes to textbooks...]

http://thisquantumworld.com/wp/a-critique-of-quantum-mechanics/the-standard-axioms/

This blog explains why current quantum mechanics has to be formulated on complex unitary Hilbert spaces, given the current axioms of quantum mechanics and it is right. But all the arguments given here do not rule out the possibility, ...

Right, I introduced the purpose of this particular blog entry in the same way...

that some other axioms and a different theory could be found that would equally describe the world. QM as currently formulated might not be the ONLY POSSIBLE framework to describe the world.

It "might not be" except that it is.

My no-math brain seems to draw from analogies systematically. QM vs classical physics looks like accountancy. The whole account must strictly balance with the existing. With QM the Universe Company is safe from any bankruptcy.

Imaginary is ONLY possible in math and fantasy!

Qt does NOT exist OR ELSE Light of the hubble deepspacefields WOULD HAVE proven any existence of quantummechanics and heisenberg. 13 BILLION YEARS of flight through millions of gravityfields and time WOULD have had AND MUST had an accumulating einsteinian and accumulating quantumdistorting influence on the light from those Galaxies. NONE is seen! Perfect Imagery!

So also deepspace Proves einstein is History and hoax!

Besides quantummechamics calculates for example orbitdistorsions at satellites BUT WITH Newton ANY of them CAN ALSO BE Explained AND Calculated.

STOP whoreshipping old fossiles AND USE YOUR SELF For a Change!

Btw NONE of ANY qt claim IS found in Nature AND ALL qt including higgsbosons ARE fantasised AND BASED ON that qt. And higgsbosons are NO particles.

Since I had No opposition at http://motls.blogspot.nl/2013/04/ams-02-dark-matter-is-composed-of.html#comment-854510110 Does This Mean You Agree with My Conclusions?

The history of physics (or mathematics) shows that axioms are not some absolute truths, although they might be held as such over many centuries. It took 2000 years to realize that the Euclid axioms are not some self-evident truths but that one might modify them to arrive to non-euclidean geometries that can serve as a basis for physical theories.
The axioms of Newtonian physics are only 3 (Newton laws) and are pretty straight-forward (intuitive). They were held as sacred truths for over 300 centuries.

The axioms of QM are maybe 80 years old and they are not as intuive and self-evident. I mean there is nothing intuitive about complex-valued coefficients that have to be squared to get the probabilities of measurements. There is nothing inuitivie in the Schrödinger equation (Feynman himself wrote in his Lectures, that SE was "born in the head of Schrödinger" and that there is no justification for it)

I still consider QM to be the highest achievement of science. It is just brilliant that physicist were able to find these axioms and formulate a theory that describes all the experiments so wonderfully.

Mephisto, in your exposition you are ignoring a very big fact: Euclid's axioms still hold for euclidiean spaces, Newton's axioms still hold for classical mechanics. It is the field of definition that changes, and new fields need new axioms. New theories for different realms should blend with the old ones at the boundaries, and this is true with quantum mechanics and classical mechanics.

What is the new realm you envisage for quantum mechanics where there will be data that will require new axioms? At the moment the deeper we go into the particle world as far as sizes go and the higher as energies quantum mechanics reigns. Even with the air showers of cosmic rays which go http://en.wikipedia.org/wiki/Cosmic_ray where the energies reach 3*10^20eV nothing unusual is seen.

If in the future, some ingenious setup can deliver us in the lab such energies, and if discrepancies are found with QM, a big if, still any new theory and its axioms should merge smoothly with QM and its axioms in the realm where it has been validated by an enormous amount of data..

I think that the linearity in Quantum Mechanics and linearity in Special Relativity (Poincaré transformations) have the same origin, that is the invariance of the non-correlation of systems.

More precisely, take two independent systems S1 and S2. I can consider the whole system S = (S1, S2)
Thus we consider an additive quantity A (like information, energy/impulsion, angular momentum, etc...)
So we have A(S) = A(S1) + A(S2)

So we can make time evolution in Quantum Mechanics or Poincaré transformations in Special Relativity, but all these transformations do not change the fact that S1 and S2 are independent, it is a physical fact, independent of the point of view of a particular observer or repository, (including translations in time).

So, For instance, after a Poincaré Transformation, you must have A'(S) = A'(S1) + A'(S2)
So the only possiblity is that A'(X). is a linear function of A(X), for all sytems X

This is a nice reminder about why QM exactly has to be what it is :-)

And it contains some cool new to me issues that I have not yet seen before, and that give me something to think about, for example the thing bout real and quaternionic representations being interpretable as "ordinary" complex representations with additional sturcture maps etc ...

I'd like to make the case that while necessary, complex numbers are not as fundamental to quantum mechanics as one might think and I argue that looking beyond quantum mechanics does not necessarily make you a crackpot (though that possibility of course remains):

First, the fact that the commutator of Hermitian operators is anti-Hermitian is somewhat misleading: The algebra of observables is a _real_ Lie-algebra, and obviously not a complex one (if H is Hermitian, iH cannot be). It's just that unitary representations highlight the wrong Lie-bracket, ie the commutator instead of Dirac's quantum Poisson brackets.

Second, quantum mechanics without complex numbers makes a lot of sense geometrically. Remember, the quantum-mechanical phase space is not the complex Hilbert space, but rather its projective version, which is a (real) Kähler manifold. It comes with three compatible structures - a Riemannian metric, a symplectic product and an almost-complex structure. However, any two of these are enough to define the third one. The symplectic product gives the dynamics via Hamilton's formalism, the metric gives probabilities. The almost-complex structure on the other hand has (as far as I know) no fundamental role, even if it is necessarily present if we require symplectic product and metric to be compatible.

Third, using an axiomatic approach like Kapustin's paper is worthwhile to figure out why exactly the quantum and classical world are incompatible, but ultimately the fact remains that classical physics is a perfectly fine theory even if it violates a set of axioms tailored for the quantum world; personally I do not see what makes quantum mechanics the more natural choice except for the fact that reality works this way at a more fundamental level.

Fourth, the Pawlowian LOLWHAT whenever someone questions quantum mechanics is a bit premature: Once quantum mechanics has been around as long as classical mechanics has, we can talk. I do not see why quantum mechanics can't have underpinnings that look decidedly non-quantum. Yes, there are some no-go theorems, but some of them might not turn out as severe as one might think (after all, the second law of thermodynamics and the associated arrow of time didn't stop us from coming up with time-symmectric foundations), and there's still the possibility that reality is only approximately quantum, same as the real world only obeys the laws of thermodynamics in the thermodynamic limit. There's a lot you can do if the sub-quantum theory operates at or even below Planck-scale levels. Sadly, I don't expect that we'll find an underlying theory during my lifetime and perhaps even not ever, which of course doesn't imply that such a theory does not exist.

Dear Christoph,

the algebra of observables may be interpreted as an algebra over reals or over complex numbers. The latter is *necessary* if the operation is the commutator because the commutator of two anti-Hermitian operators *is* demonstrably anti-Hermitian, and the blog entry contains the elementary proof. Your questioning of this elementary fact is exactly as dumb as if you questioned 1+1=2.

The algebra of observables must also be considered a complex algebra - and not a real algebra - if we mean the algebra with the operation "product" and not a "commutator" because general products of observables, even Hermitian ones (e.g. XP), are neither Hermitian nor anti-Hermitian in general.

It's also nonsensical to call the Hilbert space or the "projective space" constructive out of it as the "phase space" of quantum mechanics. Quantum mechanics isn't a classical theory so it doesn't have a phase space - and the quantum counterpart of the phase space is actually neither the Hilbert space nor its quotient but a basis of the density matrices (the density matrices themselves generalize the probability distributions on the phase space in classical physics).

There isn't any natural universal Kahler metric on the Hilbert space (or its projection version) except one that boils down to K = .

Fourth, the Pawlowian LOLWHAT whenever someone questions quantum mechanics is a bit premature: Once quantum mechanics has been around as long as classical mechanics has, we can talk.

This will never happen. Quantum mechanics was developed 2+ centuries after classical physics, so it will always be historically 2+ centuries younger. Does that mean that we will never be allowed to point out that stupid comments denying basic insights of modern physics such as yours are stupid? It was possible to point out this fact already in the mid 1920s, shortly after QM was discovered.

Cheers
LM

I find the opening sections of this paper on gauge theories to be within reach -- now that I've viewed Shankar's Yale lecture series on electromagnetism and QM:

http://www.scholarpedia.org/article/Gauge_theories

Hope I'm not being misled by either of these sources. Thanks for your trouble.

It's tiresome, let me just say that pretty much every sentence in your comment is either demonstrably wrong or morally wrong

Let's stick to the demonstrably wrong things. I claim the following:

* Dirac's quantum Poisson bracket makes the space of observables into a real Lie-algebra

* the space of observables cannot be made into a complex vector space without introducing non-observables

* the actual phase space of QM (in the Schrödinger picture) is the projective Hilbert space

* the projective Hilbert space is a principal bundle

* both Hilbert and projective Hilbert space are Kähler manifolds and Schrödinger dynamics on both of them can be realized via Hamiltonian vector fields

Lubos: You seem to be dominated by magical thinking viewing complex numbers as carrying deep physics, while a complex number is just a pair of real numbers, allowing quantum mechanics to be expressed equally well using real numbers, and the linearity of the Schrödinger equation as reflecting a fundamental aspect of reality, while there is no reason to expect physics to be linear. From where did you get your conviction that the basics of quantum mechanics is given once and for all?

Dear Claes, I noticed that this concept is extremely difficult for eternal laymen but it's the other way around.

Complex numbers are fundamental - they enter the fundamental theorem of algebra; fundamental theorems of everything; representations of groups, as explained in my newest blog entry just posted. Real numbers are just complex numbers constrained by an extra reality constraint which makes the structure "more adjusted" and less universally applicable.

From where did you get your conviction that the basics of quantum mechanics is given once and for all?

From indisputable logical arguments applied to unquestionable empirical evidence.

The big trouble with QM is the high space dimensionality of the wave function, which defies physical interpretation because physics is 3d. The way out of this paradox, which is the paradox that made Schrödinger abandon QM, is to give the wave function a statistical interpretation. But statistics is not physics, because statistics is what accountants do at an insurance company as business, and business is not physics.

This is a sequence of prejudices and animal instincts.

First of all, physics isn't 3D but 10D or 11D, you forgot to count not only the nice tiny 6 or 7 curled-up dimensions but even time. But this is still just the geometry that decides "where" events take place. One must also describe which events and this inevitably brings many more "dimensions" of new types.

The laws of physics makes probabilistic predictions only. It wasn't obvious before the mid 1920s but it's been clear since the mid 1920s. Whether Nature's inner workings resemble insurance companies or bakers or anything else or - most realistically, nothing humans know well - is up to her.

http://www.nature.com/nphys/journal/v8/n6/full/nphys2309.html

Anton Kapustin's father is an absolutely great composer who straddled the jazz-classical boundaries, Lubos....listen to some of his stuff--http://www.youtube.com/watch?NR=1&list=RD02vDWeGp4UE6M&feature=endscreen&v=Yn9fTO7zp5Q
or

Anton's paper looks interesting.

BTW yet another great didactic blog post that I will have to read more carefully. "They" should get you to teach a Coursera course.

P.S. I am constantly amazed at the rate of production and quality of your blog posts.

Wow, this Kapustin jazz is amazing. I am no true fan of complicated abstract jazz - i.e. jazz except for several well-known melodic themes - and this sounds somewhat close to what I listened to at Klaus' Jazz-on-the-Prague-Castle concerts ;-), but with the same complex music played by a single pianist, it looks even more impressive and controllable, kind of.

Quite generally, I think it's a pity that music composers aren't really famous these days. We rare know who is the actual composer of a song, for example - the interpreter is far more well-known. And it's true even for folks like me who would be interested: I don't know current composers, either. Some people who are not just showmen and showbabes, like Lady Gaga or Habera, enjoy my respect but for a vast majority of the music, and some of it is very good, I don't know who composed it.

I am just applying principles of rational thinking or "instincts" if you prefer that terminology, and 11 space dimensions (with 6-7 curled up) is beyond rationality. It does not help to scream your message into my ear; it only sounds even more distorted. To believe that Nature works like an insurance company computing mean values, is an illusion without other purpose than confusion and rip off of honorable people. If you believe in Einstein, then you cannot adore statistics, and if you don't believe in Einstein, then you have a problem as a physicist.

No, Claes, there is absolutely nothing rational about your thinking. It's purely about random guesses, rationally unsubstantiated prejudices, and childish demonization and humiliation of all the properties of Nature that happen to disagree with your medieval image of the world - which is pretty much all Nature's properties.

Very interesting!

Hearhear WHO IS banning from Logical Dispute???

What is needed mathematically depends upon the way mathematics itself has been formulated. Complex numbers are abstract concepts that must be given concrete interpetations when they are applied.

good article for learning for this ... thanks

"If the evolution operator were nonlinear, Wigner would get various terms that depend both on cA and cB, e.g. that would be proportional to cmAcnB with some positive powers."

What kind of terms are you referring to? You still have two eigenstates. The female friend is the only one that interacts directly with the quantum system, which is a two-state system. When Wigner interacts with his friend, he is interacting with a two-state system as well. Am I missing something?

My text is absolutely explicit about "what terms I am referring to". I am referring to terms in psi(t1), i.e. U(t1,t0)psi(t0). If the evolution operators U are nonlinear, then the coefficients defining psi(t1) won't be linear in the coefficients defining psi(t0).

Ok the term eigenstate might not have been entirely appropiate, but even if evolution is nonlinear you still can pick a basis. In this case what I meant is that the basis is still given by two possible states A and B. I'm just trying to write the expression where you get those c^m_A * c^n_B terms, but I'm not sure how to do that

You may pick a basis in a linear space but if the "operator" is nonlinear, the basis is completely worthless and cannot be used for anything - it really shouldn't be called a basis.

The c*c terms are there simply because one may Taylor-expand a general function. If it is linear, there are only linear terms in "c". If it is not linear, there will be c*c and all other terms, too.

Judging by your bizarre, question, you apparently don't understand what the word "linear" means (it *means* that there are also terms that are not just "c"), a fact that makes me wondered why you are unsuccessfully trying to participate in this discussion at all.

Well, you usually say things that are true, but you just said something completely false, and I need to call you on this. A basis is always possible and is completely unrelated to linearity, just in the same way that a solution to a nonlinear equation can be expanded in a fourier basis just in the same way that a solution to a linear problem. The equation over the expansion might not reduce to a linear matrix just like in a linear equation, but it is far from being 'worthless' as you said.

"the c*c terms are there simply because you might Taylor expand a general function of coefficients c"

A Taylor expansion requires some variables over where the sucessive derivatives are being performed. But you didn't precise what variables is this Taylor expansion happening. I would dare a guess that you are referring to time t1 as independent variable, and evaluating the Taylor series at t0. Now, if that is what you meant (you didn't specify it) I don't see why would the nonlinear evolution be special in the sense of having nonzero terms of high order, since any linear Hamiltonian (even a free Hamiltonian) when exponentiated, will make an infinite Taylor series on derivatives of the quantum state function (which is *always* expressed as coefficients on a chosen basis, regardless if the evolution is nonlinear or linear)

Return to the undergrad courses of linear algebra, aggressive hack, "Basis is completely unrelated to linearity"? Holy crap.

A basis is a set of vector in a *linear* space such that every element of the *linear* space may be written as a *linear* combination of the basis vectors.

The action of an operator on the basis vectors only determines the behavior of the operator on the whole space if it is a *linear* operator.

This was your last comment here.

Banned, eh?

I guess that would be the eigenstate resulting from application of the Motl annihilator operator?

But Loboš, is it positive-definite? I think we need to know.

OK, I can see it's definitely positive for you but is there any risk of quantum-tunnelling back out and stealing your cornflakes? :)

These technical questions are important!

By the way, are they Kellogg's, or the supermarket's own brand?

I'm thinking of changing to Sugar Puffs as precaution against cornflake burglary just in case.

:)

For some weird reason you enjoy distorting other people's words as a way to disqualify them, but that doesn't make you any less foolish. One thing is the differential equation which describes the evolution of solutions, which might be linear or nonlinear, and another thing entirely different is the *space* of functions where the solutions exist. The space of functions that solve a nonlinear differential equation is the same space of functions that solve a linear differential equation, which is the Hillbert space. When I'm talking about basis I'm referring to the space of functions where there nonlinear solution is expressed. Saying that a 'basis' do not exist in a space of functions where a solution of a nonlinear diff equation might be expressed makes you sound like you don't believe in Fourier expansion. But of course that is not what you meant, your only intent seems to be keep distorting what I'm saying, and now you moved the subject to a discussion about functional analysis instead of answering the original question about what Taylor-expansion are you talking about.

So can we skip this silly debate and go back to your explanation about the Taylor expansion? you were about to explain how the "c*c" terms come to be

Lubos , apart from any experimental input , Is quantum mechanics inevitable ? Classical mechanics can be derived from quantum mechanics but quantum mechanics can't be derived from anything else .

Charles, according to all the current evidence, quantum mechanics is indeed the deepest framework which is not an approximation of an even deeper one, so the hierarchy from classical physics to QM ends there.

I think that the words "apart from any experimental input" is pretty much a contradiction. *Every* proof of correctness or inability of a theory in physics has to use some experimental as well as some theoretical arguments. If your word "apart" means that one may ignore everything about the experiments, including the existence of observables, then it's probably impossible to support QM. But the basic framework of QM really depends on a very small number of facts about the observable world.

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