# PHYS771 Lecture 9: Quantum

*by*Phil Tadros

PHYS771 Lecture 9: Quantum

There are two methods to show quantum mechanics. The primary means — which for many physicists in the present day continues to be the one means — follows the historic order through which the concepts had been found. So, you begin with classical mechanics and electrodynamics, fixing a lot of grueling differential equations at each step. Then you definately be taught in regards to the “blackbody paradox” and numerous unusual experimental outcomes, and the nice disaster this stuff posed for physics. Subsequent you be taught an advanced patchwork of concepts that physicists invented between 1900 and 1926 to attempt to make the disaster go away. Then, for those who’re fortunate, after years of research you lastly get round to the central conceptual level: that nature is described not by *chances* (that are at all times nonnegative), however by numbers known as *amplitudes* that may be optimistic, destructive, and even advanced.

At the moment, within the quantum info age, the truth that all of the physicists needed to be taught quantum this fashion appears more and more humorous. For instance, I’ve had consultants in quantum subject principle — individuals who’ve spent years calculating path integrals of mind-boggling complexity — *ask me to elucidate the Bell inequality to them*. That is like Andrew Wiles asking me to elucidate the Pythagorean Theorem.

As a direct results of this “QWERTY” method to explaining quantum mechanics – which you’ll be able to see mirrored in nearly each fashionable e-book and article, down to the current — the topic acquired an undeserved fame for being laborious. Educated folks memorized the slogans — “mild is each a wave and a particle,” “the cat is neither useless nor alive till you look,” “you’ll be able to ask in regards to the place *or* the momentum, however not each,” “one particle immediately learns the spin of the opposite via spooky action-at-a-distance,” and many others. — and likewise realized that they should not even attempt to perceive such issues with out years of painstaking work.

The second technique to educate quantum mechanics leaves a blow-by-blow account of its discovery to the historians, and as a substitute *begins instantly from the conceptual core* — particularly, a sure generalization of likelihood principle to permit minus indicators. As soon as what the speculation is definitely *about*, you’ll be able to *then* sprinkle in physics to style, and calculate the spectrum of no matter atom you need. This second method is the one I will be following right here.

So, what *is* quantum mechanics? Regardless that it was found by physicists, it is *not* a bodily principle in the identical sense as electromagnetism or basic relativity. Within the standard “hierarchy of sciences” — with biology on the high, then chemistry, then physics, then math — quantum mechanics sits at a degree *between* math and physics that I do not know a very good identify for. Mainly, *quantum mechanics is the working system that different bodily theories run on as software software program* (apart from basic relativity, which hasn’t but been efficiently ported to this explicit OS). There’s even a phrase for taking a bodily principle and porting it to this OS: “to quantize.”

But when quantum mechanics is not physics within the standard sense — if it is not about matter, or power, or waves, or particles — then what *is* it about? From my perspective, it is about info and chances and observables, and the way they relate to one another.

**Ray Laflamme:**That is very a lot a computer-science standpoint.

**Scott:** Sure, it’s.

My rivalry on this lecture is the next: *Quantum mechanics is what you’ll inevitably give you for those who began from likelihood principle, after which stated, let’s attempt to generalize it in order that the numbers we used to name “chances” will be destructive numbers. As such, the speculation may have been invented by mathematicians within the 19 ^{th} century with none enter from experiment. It wasn’t, but it surely may have been.*

**Ray Laflamme:**And but, with all of the buildings mathematicians studied, none of them got here up with quantum mechanics till experiment compelled it on them…

**Scott:** Sure — and to me, that is an ideal illustration of why experiments are related within the first place! Most of the time, the *solely* motive we’d like experiments is that we’re not sensible sufficient. After the experiment has been performed, if we have realized something value figuring out in any respect, then *hopefully* we have realized why the experiment wasn’t needed to start with — why it would not have made sense for the world to be some other means. However we’re too dumb to determine it out ourselves!

Two different excellent examples of “obvious-in-retrospect” theories are evolution and particular relativity. Admittedly, I do not know if the traditional Greeks, sitting round of their togas, may have discovered that these theories had been *true*. However actually — *actually!* — they may’ve discovered that they had been *probably* true: that they are highly effective rules that will’ve no less than been on God’s whiteboard when She was brainstorming the world.

On this lecture, I’ll attempt to persuade you — with none recourse to experiment — that quantum mechanics would *additionally* have been on God’s whiteboard. I’ll present you why, if you need a universe with sure very generic properties, you appear compelled to one in all three decisions: (1) determinism, (2) classical chances, or (3) quantum mechanics. Even when the “thriller” of quantum mechanics can by no means be banished solely, you may be shocked by simply how far folks may’ve gotten with out leaving their armchairs! That they *did not* get far till atomic spectra and so forth compelled the speculation down their throats is among the strongest arguments I do know for experiments being needed.

**A Much less Than 0% Likelihood**

Alright, so what wouldn’t it imply to have “likelihood principle” with destructive numbers? Effectively, there is a motive you by no means hear the climate forecaster speak about a -20% probability of rain tomorrow — it actually *does* make as little sense because it sounds. However I would such as you to set any qualms apart, and simply assume abstractly about an occasion with N attainable outcomes. We are able to categorical the chances of these occasions by a vector of N actual numbers:

(p

_{1},….,p

_{N}),

Mathematically, what can we are saying about this vector? Effectively, the chances had higher be nonnegative, and so they’d higher sum to 1. We are able to categorical the latter reality by saying that the 1-norm of the likelihood vector needs to be 1. (The 1-norm simply means the sum of absolutely the values of the entries.)

However the 1-norm just isn’t the one norm on the earth — it is not the one means we all know to outline the “dimension” of a vector. There are different methods, and one of many recurring favorites for the reason that days of Pythagoras has been the *2-norm* or *Euclidean norm*. Formally, the Euclidean norm means the sq. root of the sum of the squares of the entries. Informally, it means you are late for sophistication, so as a substitute of going this fashion after which that means, you chop throughout the grass.

Now, what occurs for those who attempt to give you a principle that is *like* likelihood principle, however based mostly on the 2-norm as a substitute of the 1-norm? I’ll attempt to persuade you that quantum mechanics is what inevitably outcomes.

Let’s think about a single bit. In likelihood principle, we will describe a bit as having a likelihood p of being 0, and a likelihood 1-p of being 1. But when we swap from the 1-norm to the 2-norm, now we not need two numbers that sum to 1, we would like two numbers whose *squares* sum to 1. (I am assuming we’re nonetheless speaking about actual numbers.) In different phrases, we now need a vector (α,β) the place

α^{2} + β^{2} = 1. After all, the set of *all* such vectors kinds a circle:

The idea we’re inventing will *one way or the other* have to hook up with remark. So, suppose we’ve got a bit that is described by this vector (α,β). Then we’ll must specify what occurs if we *look* on the bit. Effectively, because it *is* a bit, we must always see both 0 or 1! Moreover, the likelihood of seeing 0 and the likelihood of seeing 1 had higher add as much as 1. Now, ranging from the vector (α,β), how can we get two numbers that add as much as 1? Easy: we will let α^{2} be the likelihood of a 0 end result, and let β^{2} be the likelihood of a 1 end result.

However in that case, why not overlook about α and β, and simply describe the bit *instantly* by way of chances? Ahhhhh. The distinction is available in how the vector modifications once we apply an operation to it. In likelihood principle, if we’ve got a bit that is represented by the vector (p,1-p), then we will characterize any operation on the bit by a *stochastic matrix*: that’s, a matrix of nonnegative actual numbers the place each column provides as much as 1. So for instance, the “bit flip” operation — which modifications the likelihood of a 1 end result from p to 1-p — will be represented as follows:

Certainly, it seems {that a} stochastic matrix is the *most basic* kind of matrix that at all times maps a likelihood vector to a different likelihood vector.

**Train 1 for the Non-Lazy Reader:** Show this.

However now that we have switched from the 1-norm to the 2-norm, we’ve got to ask: *what’s probably the most basic kind of matrix that at all times maps a unit vector within the 2-norm to a different unit vector within the 2-norm?*

Effectively, we name such a matrix a *unitary matrix* — certainly, that is one technique to outline what a unitary matrix is! (Oh, all proper. So long as we’re solely speaking about actual numbers, it is known as an *orthogonal matrix*. However similar distinction.) One other technique to outline a unitary matrix, once more within the case of actual numbers, is as a matrix whose inverse equals its transpose.

**Train 2 for the Non-Lazy Reader:** Show that these two definitions are equal.

This “2-norm bit” that we have outlined has a reputation, which as is *qubit*. Physicists prefer to characterize qubits utilizing what they name “Dirac ket notation,” through which the vector (α,β) turns into . Right here α is the *amplitude* of end result |0〉, and β is the amplitude of end result |1〉.

This notation often drives pc scientists up a wall once they first see it — particularly due to the uneven brackets! However for those who keep it up, you see that it is actually not so unhealthy. For example, as a substitute of writing out a vector like (0,0,3/5,0,0,0,4/5,0,0), you’ll be able to merely write , omitting all the 0 entries.

So given a qubit, we will rework it by making use of any 2-by-2 unitary matrix — and that leads already to the well-known impact of *quantum interference*. For instance, think about the unitary matrix

which takes a vector within the airplane and rotates it by 45 levels counterclockwise. Now think about the state |0〉. If we apply U as soon as to this state, we’ll get — it is like taking a coin and flipping it. However then, if we apply the identical operation U a second time, we’ll get |1〉:

So in different phrases, making use of a “randomizing” operation to a “random” state produces a deterministic end result! Intuitively, despite the fact that there are two “paths” that result in the end result |0〉, a kind of paths has optimistic amplitude and the opposite has destructive amplitude. In consequence, the 2 paths *intervene destructively* and cancel one another out. In contrast, the 2 paths resulting in the end result |1〉 each have optimistic amplitude, and subsequently intervene *constructively*.

The explanation you by no means see this kind of interference within the classical world is that chances cannot be destructive. So, cancellation between optimistic and destructive amplitudes will be seen because the supply of *all* “quantum weirdness” — the one factor that makes quantum mechanics completely different from classical likelihood principle. How I want somebody had instructed me that after I first heard the phrase “quantum”!

**Combined States**

As soon as we’ve got these quantum states, one factor we will at all times do is to take classical likelihood principle and “layer it on high.” In different phrases, we will at all times ask, what if we do not *know* which quantum state we’ve got? For instance, what if we’ve got a 1/2 likelihood of and a 1/2 likelihood of ? This provides us what’s known as a *blended state*, which is probably the most basic type of state in quantum mechanics.

Mathematically, we characterize a blended state by an object known as a *density matrix*. Here is the way it works: say you’ve gotten this vector of N amplitudes, (α_{1},…,α_{N}). Then you definately compute the *outer product* of the vector with itself — that’s, an N-by-N matrix whose (i,j) entry is α_{i}α_{j} (once more within the case of actual numbers). Then, you probably have a likelihood distribution over a number of such vectors, you simply take a linear mixture of the ensuing matrices. So for instance, you probably have likelihood p of some vector and likelihood 1-p of a distinct vector, then it is p occasions the one matrix plus 1-p occasions the opposite.

The density matrix encodes all the data that would ever be obtained from some likelihood distribution over quantum states, by first making use of a unitary operation after which measuring.

**Train 3 for the Non-Lazy Reader:** Show this.

This means that if two distributions give rise to the identical density matrix, then these distributions are empirically indistinguishable, or in different phrases are *the identical blended state*. For example, as an example you’ve gotten the state with 1/2 likelihood, and with 1/2 likelihood. Then the density matrix that describes your information is

It follows, then, that no measurement you’ll be able to ever carry out will distinguish this combination from a 1/2 likelihood of |0〉 and a 1/2 likelihood of |1〉.

**The Squaring Rule**

Now let’s discuss in regards to the query Gus raised, which is, why will we sq. the amplitudes as a substitute of cubing them or elevating them to the fourth energy or no matter?

**Devin Smith:**As a result of it offers you the precise reply?

**Scott:** Yeah, you do need a solution that agrees with experiment. So let me put the query otherwise: why did God select to do it that means and never another means?

**Ray Laflamme:** Effectively, provided that the numbers will be destructive, squaring them simply looks as if the best factor to do!

**Scott:** Why not simply take absolutely the worth?

Alright, I may give you a few arguments for why God determined to sq. the amplitudes.

The primary argument is a well-known outcome known as Gleason’s Theorem from the 1950’s. Gleason’s Theorem lets us assume *half* of quantum mechanics after which get out the remainder of it! Extra concretely, suppose we’ve got some process that takes as enter a unit vector of actual numbers, and that spits out the likelihood of an occasion. Formally, we’ve got a perform f that maps a unit vector to the unit interval [0,1]. And let’s suppose N=3 — the concept truly works in any variety of dimensions three or better (however apparently, *not* in two dimensions). Then the important thing requirement we impose is that, each time three vectors v_{1},v_{2},v_{3} are all orthogonal to one another,

_{1}) + f(v

_{2}) + f(v

_{3}) = 1.

Intuitively, if these three vectors characterize “orthogonal methods” of measuring a quantum state, then they need to correspond to mutually-exclusive occasions. Crucially, we do not want *any* assumption apart from that — no continuity, no differentiability, no nuthin’.

So, that is the setup. The wonderful conclusion of the concept is that, for *any* such f, there exists a blended state such that f arises by measuring that state in accordance with the usual measurement rule of quantum mechanics. I will not have the option show this theorem right here, because it’s fairly laborious. Nevertheless it’s a method which you could “derive” the squaring rule with out *precisely* having to place it in on the outset.

**Train 4 for the Non-Lazy Reader:** Why does Gleason’s Theorem *not* work in two dimensions?

In the event you like, I may give you a way more elementary argument. That is one thing I put it in one of my papers, although I am certain many others knew it earlier than.

For instance we need to invent a principle that is not based mostly on the 1-norm like classical likelihood principle, *or* on the 2-norm like quantum mechanics, however as a substitute on the p-norm for some . Name (v_{1},…,v_{N}) a *unit vector within the p-norm* if

_{1}|

^{p}+…+|v

_{N}|

^{p}= 1.

Then we’ll want some “good” set of linear transformations that map any unit vector within the p-norm to a different unit vector within the p-norm.

It is clear that for any p we select, there will likely be *some* linear transformations that protect the p-norm. Which of them? Effectively, we will permute the idea components, shuffle them round. That’ll protect the p-norm. And we will stick in minus indicators if we would like. That’ll protect the p-norm too.

However this is the little remark I made: *if there are any linear transformations apart from these trivial ones that protect the p-norm, then both p=1 or p=2.* If p=1 we get classical likelihood principle, whereas if p=2 we get quantum mechanics.

**Train 5 for the Non-Lazy Reader**: Show my little remark.

Alright, to get you began, let me give some instinct about why my remark *would possibly* be true. Let’s assume, for simplicity, that the whole lot is actual and that p is a optimistic even integer (although the remark additionally works with advanced numbers and with any actual p≥0). Then for a linear transformation A=(a_{ij}) to *protect the p-norm* implies that

each time

Now we will ask: what number of constraints are imposed on the matrix A by the requirement that this be true for each v_{1},…,v_{N}? If we work it out, within the case p=2 we’ll discover that there are constraints. However since we’re making an attempt to select an N-by-N matrix, that also leaves us N(N-1)/2 levels of freedom to play with.

However, if (say) p=4, then the variety of constraints grows like , which is *better* than N^{2} (the variety of variables within the matrix). That means that it will likely be laborious to discover a nontrivial linear transformation that preserves 4-norm. After all it does not *show* that no such transformation exists — that is left as a puzzle for you.

By the way, this is not the one case the place we discover that the 1-norm and 2-norm are “extra particular” than different p-norms. So for instance, have you ever ever seen the next equation?

^{n}+ y

^{n}= z

^{n}

There is a cute little reality — sadly I will not have time to show it at school — that the above equation has nontrivial integer options when n=1 or n=2, however not for any bigger integers n. Clearly, then, if we use the 1-norm and the 2-norm greater than different vector norms, it is not some arbitrary whim — these *actually are* God’s favourite norms! (And we did not even want an experiment to inform us that.)

**Actual vs. Complicated Numbers**

Even after we have determined to base our principle on the 2-norm, we nonetheless have no less than two decisions: we may let our amplitudes be actual numbers, *or* we may allow them to be advanced numbers. We all know the answer God selected: amplitudes in quantum mechanics are advanced numbers. This implies which you could’t simply sq. an amplitude to get a likelihood; first you must take absolutely the worth, and you then sq. *that*. In different phrases, if the amplitude for some measurement end result is α = β + γi, the place β and γ are actual, then the likelihood of seeing the end result is |α|^{2} = β^{2} + γ^{2}.

*Why* did God go along with the advanced numbers and never the actual numbers?

Years in the past, at Berkeley, I used to be hanging out with some math grad college students — I fell in with the incorrect crowd — and I requested them that precise query. The mathematicians simply snickered. “Give us a break — the advanced numbers are algebraically closed!” To them it wasn’t a thriller in any respect.

However to me it *is* kind of unusual. I imply, advanced numbers had been seen for hundreds of years as fictitious entities that human beings made up, so that each quadratic equation ought to have a root. (That is why we speak about their “imaginary” elements.) So why ought to Nature, at its most basic degree, run on one thing that *we* invented for our comfort?

**Reply:**Effectively, if you need each unitary operation to have a sq. root, you then

*have*to go to the advanced numbers…

**Scott:** Dammit, you are getting forward of me!

Alright, yeah: suppose we require that, for each linear transformation U that we will apply to a state, there have to be one other transformation V such that V^{2} = U. That is principally a *continuity* assumption: we’re saying that, if it is smart to use an operation for one second, then it must make sense to use that very same operation for less than half a second.

Can we get that with solely actual amplitudes? Effectively, think about the next linear transformation:

This transformation is only a *mirror reversal* of the airplane. That’s, it takes a two-dimensional Flatland creature and flips it over like a pancake, sending its coronary heart to the opposite aspect of its two-dimensional physique. However how do you apply *half* of a mirror reversal with out leaving the airplane? You’ll be able to’t! If you wish to flip a pancake by a steady movement, then you might want to go into … *dum dum dum* … THE THIRD DIMENSION.

Extra typically, if you wish to flip over an N-dimensional object by a steady movement, then you might want to go into the (N+1)^{st} dimension.

**Train 6 for the Non-Lazy:** Show that *any* norm-preserving linear transformation in N dimensions will be carried out by a steady movement in N+1 dimensions.

However what if you need each linear transformation to have a sq. root within the *similar* variety of dimensions? Effectively, in that case, you must permit advanced numbers. In order that’s one motive God might need made the selection She did.

Alright, I may give you two different the explanation why amplitudes must be advanced numbers.

The primary comes from asking, what number of impartial actual parameters are there in an N-dimensional blended state? Because it seems, the reply is precisely N^{2} — offered we assume, for comfort, that the state does not should be normalized (i.e., that the chances can add as much as lower than 1). Why? Effectively, an N-dimensional blended state is represented mathematically by a N-by-N Hermitian matrix with optimistic eigenvalues. Since we’re not normalizing, we have N impartial actual numbers alongside the principle diagonal. *Under* the principle diagonal, we have N(N-1)/2 impartial advanced numbers, which suggests N(N-1) actual numbers. For the reason that matrix is Hermitian, the advanced numbers beneath the principle diagonal *decide* those above the principle diagonal. So the overall variety of impartial actual parameters is N + N(N-1) = N^{2}.

Now we herald a facet of quantum mechanics that I did not point out earlier than. If we all know the states of *two* quantum programs individually, then how will we write their *mixed* state? Effectively, we simply type what’s known as the *tensor product*. So for instance, the tensor product of two qubits, α|0〉+β|1〉 and γ|0〉+δ|1〉, is given by

Once more one can ask: did God *have* to make use of the tensor product? May She have chosen some *different* means of mixing quantum states into greater ones? Effectively, perhaps another person can say one thing helpful about this query — I’ve hassle even wrapping my head round it! For me, saying we take the tensor product is sort of what we *imply* once we say we’re placing collectively two programs that exist independently of one another.

As you all know, there are two-qubit states that *cannot* be written because the tensor product of one-qubit states. Essentially the most well-known of those is the EPR (Einstein-Podolsky-Rosen) pair:

Given a blended state ρ on two subsystems A and B, if ρ will be written as a likelihood distribution over tensor product states , then we are saying ρ is *separable*. In any other case we are saying ρ is *entangled*.

Now let’s come again to the query of what number of actual parameters are wanted to explain a blended state. Suppose we’ve got a (possibly-entangled) composite system AB. Then intuitively, it looks as if the variety of parameters wanted to explain AB — which I will name d_{AB} — ought to equal the *product* of the variety of parameters wanted to explain A and the variety of parameters wanted to explain B:

_{AB}= d

_{A}d

_{B}.

If amplitudes are advanced numbers, then fortunately that is true! Letting N_{A} and N_{B} be the variety of dimensions of A and B respectively, we’ve got

_{AB}= (N

_{A}N

_{B})

^{2}= N

_{A}

^{2}N

_{B}

^{2}= d

_{A}d

_{B}.

However what if the amplitudes are actual numbers? In that case, in an N-by-N density matrix, we would solely have N(N+1)/2 impartial actual parameters. And it is *not* the case that if N = N_{A} N_{B} then

**Query:**Can this similar argument be used to rule out quaternions?

**Scott:** Wonderful query. Sure! With actual numbers the left-hand aspect is just too massive, whereas with quaternions it is too small. Solely with advanced numbers is it juuuuust proper!

There’s truly one other phenomenon with the identical “Goldilocks” taste, which was noticed by Invoice Wootters — and this results in my third motive why amplitudes must be advanced numbers. For instance we select a quantum state

uniformly at random (for those who’re a mathematician, underneath the Haar measure). After which we measure it, acquiring end result |i〉 with likelihood |α_{i}|^{2}. The query is, will the ensuing likelihood vector *additionally* be distributed uniformly at random within the likelihood simplex? It seems that if the amplitudes are advanced numbers, then the reply is sure. But when the amplitudes are actual numbers or quaternions, then the reply isn’t any! (I used to assume this reality was only a curiosity, however now I am truly utilizing it in a paper I am engaged on…)

**Linearity**

We have talked about why the amplitudes must be advanced numbers, and why the rule for changing amplitudes to chances must be a squaring rule. However all this time, the elephant of *linearity* has been sitting there undisturbed. Why would God have determined, within the first place, that quantum states ought to evolve to different quantum states by the use of linear transformations?

**Reply:**As a result of if the transformations weren’t linear, you possibly can crunch vectors to be greater or smaller…

**Scott:** Shut! Steven Weinberg and others proposed nonlinear variants of quantum mechanics through which the state vectors do keep the identical dimension. The difficulty with these variants is that they’d allow you to take far-apart vectors and squash them collectively, *or* take extraordinarily shut vectors and pry them aside! Certainly, that is basically what it *means* for such theories to be nonlinear. So our configuration area not has this intuitive which means of measuring the distinguishability of vectors. Two states which can be exponentially shut would possibly actually be completely distinguishable. And certainly, in 1998 Abrams and Lloyd used precisely this remark to show that, *if* quantum mechanics had been nonlinear, then one may construct a pc to resolve **NP**-complete issues in polynomial time.

**Query:** What’s the issue with that?

**Scott:** *What’s the issue with having the ability to remedy NP-complete issues in polynomial time?* Oy, if by the tip of this class you continue to do not assume that is an issue, I’ll have failed you… [laughter]

Significantly, *in fact* we do not know whether or not **NP**-complete issues are effectively solvable within the bodily world. However in a survey I wrote a pair years in the past, I defined why the flexibility to resolve **NP**-complete issues would give us “godlike” powers — arguably, much more so than the flexibility to transmit superluminal indicators or reverse the Second Legislation of Thermodynamics. The essential level is that, once we speak about **NP**-complete issues, we’re not simply speaking about scheduling airline flights (or for that matter, breaking the RSA cryptosystem). We’re speaking about *automating perception*: proving the Riemann Speculation, modeling the inventory market, seeing no matter patterns or chains of logical deduction are there on the earth to be seen.

So, suppose I preserve the working speculation that **NP**-complete issues are *not* effectively solvable by bodily means, and that if a principle suggests in any other case, extra probably than not that signifies an issue with the speculation. Then there are solely two prospects: both I am proper, or else I am a god! And both one sounds fairly good to me…

**Train 7 for the Non-Lazy Reader:** Show that if quantum mechanics had been nonlinear, then not solely may you remedy **NP**-complete issues in polynomial time, you possibly can additionally use EPR pairs to transmit info sooner than the velocity of sunshine.

**Query:**But when I had been crafting a universe in my storage, I may select to make the velocity of sunshine equal to infinity.

**Scott:** Yeah, you’ve got touched on one other one in all my favourite questions: *why ought to the velocity of sunshine be finite?* Effectively, one motive I would prefer it to be finite is that, if aliens from the Andromeda galaxy are going to harm me, then I no less than need them to should *come* right here first!

**Additional Studying**

See this paper by Lucien Hardy for a “derivation” of quantum mechanics that is carefully associated to the arguments I gave, however a lot, rather more severe and cautious. Additionally see just about something Chris Fuchs has written (and particularly this paper by Caves, Fuchs, and Schack, which discusses why amplitudes must be advanced numbers quite than reals or quaternions).

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