Demystifying Quantum Physics

[zaybu]

What is wrong about it?*

*edit: And could you please specify whether you believe it is wrong according to standard physics as commonly understood by physicists, or based on your own personal take on things?

There is no influence. Suppose I send two entangled particles: one to the left with spin up, and the other to the right with spin down, when observer at both ends make their measurement, they are confirming only what was already prepared. There's no mystery here.

 

[Mr Spinkles]

This shows you absolutely got it wrong. Aspect's experiment showed that Bell was right, and what Bell had claimed was that there are no hidden local parameters. IOW, what Einstein had claimed - spooky action at a distance - is wrong. Get it. What all this demonstrates is that quantum logic is different than classical logic.

It sounds like you might be confused. Einstein's argument was against the spooky action at a distance implied by quantum mechanics, and in favor of hidden variables because he thought QM was incomplete. You realize that, right?

 

[Mr Spinkles]

There is no influence. Suppose I send two entangled particles: one to the left with spin up, and the other to the right with spin down, when observer at both ends make their measurement, they are confirming only what was already prepared. There's no mystery here.

But the key experiment is where you prepare the particles in an entangled state, so that neither particle is spin up or spin down until someone measures one of them.*

*edit: Oh and again, are you talking about standard physics here or are you giving your own personal view?

 

[idav]

But the key experiment is where you prepare the particles in an entangled state, so that neither particle is spin up or spin down until someone measures one of them.

Would you mind explaining more on this? It is already in a certain quantum state (prepared that way) by the time it is measured. It doesn't just choose a state based on whether anyone is looking.

 

[LegionOnomaMoi]

This shows you absolutely got it wrong. Aspect's experiment showed that Bell was right, and what Bell had claimed was that there are no hidden local parameters. IOW, what Einstein had claimed - spooky action at a distance - is wrong. Get it. What all this demonstrates is that quantum logic is different than classical logic.

Since we have established I have no clue what I'm talking about, perhaps you'll be kind enough to explain a few things. We both agree, the Einstein got it wrong. What did he get wrong? The famous paper (EPR) is the clearest example of what Einstein claimed. Why is it titled ). "Can Quantum-Mechanical Description of Physical Reality be Considered Complete?"
Why, if Einstein supported (or tried to claim) that this "spooky action existed", did EPR conclude: "Previously we proved that either (1) the quantum-mechanical description of reality given by the wave function is not complete or (2) when the operators corresponding to two physical quantities do not commute the two quantities cannot have simultaneous reality. Starting then with the assumption that the wave function does give a complete description of the physical reality, we arrived at the conclusion that two physical quantities, with noncommuting operators, can have simultaneous reality. Thus the negation of (1) leads to the negation of the only other alternative (2). We are thus forced to conclude that the quantum-mechanical description of the physical reality given by wave functions is not complete."

Now, clearly I can't understand anything about the above quote. After all, it seems to me that EPR tried to show that if quantum mechanics (in which the wave function completely describes/characterizes the quantum system) is complete, then it must be that "two physical quantities...can have simultaneous reality".

Having "proved" this, the authors then claim not that we should accept this (i.e., nonlocality), but rather that the problem is with quantum mechanics?

And if Aspect et al. proved there were no "spooky actions" and Einstein was wrong, why does their 1982 study in Physical Review Letters "Experimental realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment: a new violation of Bell's inequalities", conclude as follows:
"In conclusion, our experiment yields the strongest violation of Bell's inequalities ever achieved, and excellent agreement with quantum mechanics. Since it is a straightforword transposition of the ideal Einstein-Rosen-Bohm scheme, the experimental procedure is very simple, and needs no auxiliary measurements as in previous experiments with single-channel polarizers. We are thus led to the rejection of realistic local theories"?

Why does "Violation of Bell inequalities by photons more than 10 km apart" (another Gisin study), state: "Quantum theory is nonlocal. Indeed, quantum theory predicts correlations among distant measurement outcomes that cannot be explained by any theory which involves only local variables. This was anticipated by Einstein, Podolsky, and Rosen [1] and by Schrödinger [2], among others, and first demonstrated by Bell in 1964 with his now famous inequality [3]."?

NOTE: please refrain from posting links which you have no clue what these articles asre talking about.

So no links to articles. Got it. Are monographs, volumes, graduate textbooks, and similar physics literature I have (and clearly either not read, or read without ever understanding a single word) off the table?

 

[Mr Spinkles]

Would you mind explaining more on this? It is already in a certain quantum state (prepared that way) by the time it is measured. It doesn't just choose a state based on whether anyone is looking.

The particles are prepared in a definite quantum state, but it's a special state in which neither particle has a particular spin. The act of measurement (or any physical interaction like a measurement) does indeed change the state, for example by causing one of the particles to adopt a definite spin-up state. The probability of the particle making this "choice" is determined by the original state.

 

[zaybu]

It sounds like you might be confused. Einstein's argument was against the spooky action at a distance implied by quantum mechanics, and in favor of hidden variables because he thought QM was incomplete. You realize that, right?

Yes, Einstein was against the notion of spooky action at a distance, or at least abhored it, and believed wrongly that's what QM was suggesting. But the reality is that Einstein misundertood QM, and unfortunately died prematurely in 1954, so to speak, as he died before people made sense of QM, which came about in the 1960-70. Unfortunately, there are those who still carry his torch and perpetuate that misunderstanding.

 

[zaybu]

But the key experiment is where you prepare the particles in an entangled state, so that neither particle is spin up or spin down until someone measures one of them.*

*edit: Oh and again, are you talking about standard physics here or are you giving your own personal view?

An entangled system is one in which you preparte one particle in a state such that you know the state of the second particle. There's nothing personal here. It's basic stuff.

 

[PolyHedral]

"A good way of thinking about such apparent instantaneous “spooky action at a distance” (as Einstein put it) is to realize that the entangled state must have been prepared locally by having let the two subsystems interact at some point in the past. When these two subsystems are then separated from each other, the quantum state is simply “spread out” (i.e., delocalized) over a larger spatial region." p. 31

Schlosshauer, M. A. (2007). Decoherence: and the quantum-to-classical transition. Springer Verlag.

How am I supposed to interpret that when quantum states don't live in 4D space? (they live in Hilbert space)

 

[zaybu]

Since we have established I have no clue what I'm talking about, perhaps you'll be kind enough to explain a few things. We both agree, the Einstein got it wrong. What did he get wrong? The famous paper (EPR) is the clearest example of what Einstein claimed. Why is it titled ). "Can Quantum-Mechanical Description of Physical Reality be Considered Complete?"
Why, if Einstein supported (or tried to claim) that this "spooky action existed", did EPR conclude: "Previously we proved that either (1) the quantum-mechanical description of reality given by the wave function is not complete or (2) when the operators corresponding to two physical quantities do not commute the two quantities cannot have simultaneous reality. Starting then with the assumption that the wave function does give a complete description of the physical reality, we arrived at the conclusion that two physical quantities, with noncommuting operators, can have simultaneous reality. Thus the negation of (1) leads to the negation of the only other alternative (2). We are thus forced to conclude that the quantum-mechanical description of the physical reality given by wave functions is not complete."

Now, clearly I can't understand anything about the above quote. After all, it seems to me that EPR tried to show that if quantum mechanics (in which the wave function completely describes/characterizes the quantum system) is complete, then it must be that "two physical quantities...can have simultaneous reality".

Having "proved" this, the authors then claim not that we should accept this (i.e., nonlocality), but rather that the problem is with quantum mechanics?

And if Aspect et al. proved there were no "spooky actions" and Einstein was wrong, why does their 1982 study in Physical Review Letters "Experimental realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment: a new violation of Bell's inequalities", conclude as follows:
"In conclusion, our experiment yields the strongest violation of Bell's inequalities ever achieved, and excellent agreement with quantum mechanics. Since it is a straightforword transposition of the ideal Einstein-Rosen-Bohm scheme, the experimental procedure is very simple, and needs no auxiliary measurements as in previous experiments with single-channel polarizers. We are thus led to the rejection of realistic local theories"?

Why does "Violation of Bell inequalities by photons more than 10 km apart" (another Gisin study), state: "Quantum theory is nonlocal. Indeed, quantum theory predicts correlations among distant measurement outcomes that cannot be explained by any theory which involves only local variables. This was anticipated by Einstein, Podolsky, and Rosen [1] and by Schrödinger [2], among others, and first demonstrated by Bell in 1964 with his now famous inequality [3]."?





So no links to articles. Got it. Are monographs, volumes, graduate textbooks, and similar physics literature I have (and clearly either not read, or read without ever understanding a single word) off the table?

I'm not going to go over what Einstein said, as his words are a mass of confusion. But what Aspect established is that Bell's inequalities are violated, meaning that classical logic doesn't apply to quantum system.

Here's what's at stake -

Take the example of tossing a fair coin, with probability 1/2 for heads, and 1/2 for tails.

Classically:

1. the word "and" means multiply. What's the probability of tossing a heads and tossing a heads again? Answer (1/2 ) * (1/2) = 1/4
2. the word "or" means add. What's the probability of tossing heads or tails? Answer 1/2 + 1/2 = 1

In QM: we do not perform these operations on the probabilities but on the "probability amplitude", then we square the amplitude to get the probability. Example, If a particle is in a spin state (up or down), we write:

|ψ>= a|+1> + b|-1>, with a^2 + b^2 = 1

If we want to know what's the probability it is in one state say, up, then we calculate the probability amplitute <&#968;|+1>, after which the probabilty will be P = <&#968;|+1>^2. In doing so, you will get mix terms. And it's with that we can explain the double-slit experiment, and many other features that classical physics can't.

 

[Mr Spinkles]

Yes, Einstein was against the notion of spooky action at a distance, or at least abhored it, and believed wrongly that's what QM was suggesting.

Again: are you claiming this is what standard physics says, or are you claiming this based on your own personal view? Legion and I have already established that according to most physicists, QM does indeed suggest action at a distance. Einstein wasn't wrong that QM suggests this, Einstein was wrong that QM is wrong (or rather, incomplete).

An entangled system is one in which you preparte one particle in a state such that you know the state of the second particle. There's nothing personal here. It's basic stuff.

Yes, it is basic stuff. And you seem to not realize that you are confused about it. Can you tell me, what is the spin (up or down) of an electron which is prepared with another electron in the entangled state known as the singlet state? (Hint: it's a trick question.) For your reference, here is the singlet state:

 

[LegionOnomaMoi]

How am I supposed to interpret that when quantum states don't live in 4D space? (they live in Hilbert space)

By reading what the author writes in the first real chapter (i.e., after the intro):

"Instead of describing the state of a physical system by means of intuitive symbols that corresponded directly to the “objectively existing” physical properties of our experience, in quantum mechanics we have at our disposal only an abstract quantum state that is defined as a vector (or, more generally, as a ray) in a similarly abstract Hilbert vector space.
The conceptual leap associated with this abstraction is hard to overestimate. In fact, the discussions regarding the “interpretation of quantum mechanics” that have occupied countless physicists and philosophers since the early years of quantum theory are to a large part rooted precisely in the question of how to relate the abstract quantum state to the “physical reality out there.”...

In view of the properties of quantum states discussed in the preceding section, it has often been argued that these states represent only “potentialities” for the various observed “classical” states. At the same time, however, it is important to emphasize that (according to our current knowledge) quantum states represent a complete description of a quantum system, i.e., the quantum state encapsulates all there is to say about the physical state of the system. Yet, in general quantum states do not tell us which particular outcome will be obtained in a measurement but only the probabilities of the various possible outcomes. This seemingly intrinsic probabilistic character of quantum mechanics is one of the central features distinguishing this theory from classical physics. In an experimental situation, the probabilistic aspect is represented by the fact that, if we measure the same physical quantity on a collection of systems all prepared in exactly the same quantum state, we will in general obtain a set of different outcomes."

 

[LegionOnomaMoi]

I'm not going to go over what Einstein said, as his words are a mass of confusion.

1) They aren't just his words. EPR isn't an abbreviation for some procedure or a serious neurological condition, it's the standard way of referring to the 1935 paper that Einstein, Podolsky, and Rosen published (Phys. Rev. 47, 777-780).
2) Somehow, this mass of confusion is one of the most cited papers in physics literature. It's cited by Aspect et al. in both the 1981 and 1982 studies.
3) If EPR is too confusing, I provided you with numerous quotes, with links, including that to the Tittel, Brendel, Zbinden, & Gisin experiment ("Violation of Bell inequalities by photons more than 10 km apart").

How is this: "Quantum theory is nonlocal. Indeed, quantum theory predicts correlations among distant measurement outcomes that cannot be explained by any theory which involves only local variables. This was anticipated by Einstein, Podolsky, and Rosen and by Schrödinger, among others, and first demonstrated by Bellin 1964 with his now famous inequality."

unclear?

But what Aspect established is that Bell's inequalities are violated, meaning that classical logic doesn't apply to quantum system.

Guess what Bell based his work on? Einstein, Podolsky, and Rosen (EPR). Guess what he hoped to accomplish? The same thing Einstein wanted: a quantum physics in which there exists a one-to-one correspondence between every element in the theory and every element in reality. Neither he, nor Einstein, liked nonlocality. EPR was designed to show that quantum physics had to be complete, because the only other option was (to Einstein and others) so obviously wrong nobody would accept it. EPR, and therefore Einstein, were absolutely correct that QM entailed "spooky action at a distance". Einstein was wrong, however, in thinking that this meant QM was incomplete.

Bell's theorem was an attempt to resolve this, but instead he ended up providing a perfect theoretical framework to test whether or not Einstein was correct (not in terms of what Einstein wanted, but what he and co-authors showed QM entailed: non-locality). Violating Bell's inequalities is a confirmation of "spooky action at a distance" and of EPR.

Here's what's at stake -

The last stanza of Carroll's "Father William" keeps running through my head for some reason.
"I have answered three questions, and that is enough,'
said his father; `don't give yourself airs!
Do you think I can listen all day to such stuff?
Be off, or I'll kick you down stairs!'"

Take the example of tossing a fair coin, with probability 1/2 for heads, and 1/2 for tails.

As much as I appreciate the lesson in basic quantum logic, I'd rather you just take the time to understand why it is you are wrong. It's not that hard, and it's probably easier than explaining why physicists are the only people on the planet who interpret the mod square of some complex number as a probability function, which (despite your fascinating explanation useing coins and complete with bras and kets), is a waste of everybody's time.

 

[zaybu]

Again: are you claiming this is what standard physics says, or are you claiming this based on your own personal view? Legion and I have already established that according to most physicists, QM does indeed suggest action at a distance. Einstein wasn't wrong that QM suggests this, Einstein was wrong that QM is wrong (or rather, incomplete).
Yes, it is basic stuff. And you seem to not realize that you are confused about it. Can you tell me, what is the spin (up or down) of an electron which is prepared with another electron in the entangled state known as the singlet state? (Hint: it's a trick question.) For your reference, here is the singlet state:

It is a singlet state but if you want to deal with entanglement, it is in the wrong format. Entanglement means that is you know the state of one particle, you know automatically the state of the other.

What you need is

|&#968;>=|+1,u> + |-1,d>.

The +1,-1 would indicate that one particle is going to the right, the other to the left. The u and d would indicate that one has an up-spin, the other, a down-spin.

 

[zaybu]

EPR, and therefore Einstein, were absolutely correct that QM entailed "spooky action at a distance". Einstein was wrong, however, in thinking that this meant QM was incomplete.

Sorry, that's a lot of nonsense. Einstein can think whatever. He might have believed that there is a spooky action at a distance, and as I said, Einstein misunderstood QM. So what he thought is irrelevant. But people seem to have a vested interest in resuscitating his arcane arguments.

Violating Bell's inequalities is a confirmation of "spooky action at a distance" and of EPR.

Wrong, you don't seem to get. It proves that classical logic doesn't apply to quantum system. Reading anything else will only bring you troubles.

As much as I appreciate the lesson in basic quantum logic, I'd rather you just take the time to understand why it is you are wrong. It's not that hard, and it's probably easier than explaining why physicists are the only people on the planet who interpret the mod square of some complex number as a probability function, which (despite your fascinating explanation useing coins and complete with bras and kets), is a waste of everybody's time.

That you consider this difference between classical and quantum mechanics as a waste of times when it is at the core of QM speaks loudly of your incompetence in this matter.

.

 

[Mr Spinkles]

It is a singlet state but if you want to deal with entanglement, it is in the wrong format. ...

No, you're wrong. Furthermore, I smell a troll.

I think the rest of us should move on and not let zaybu derail this thread.

 

[LegionOnomaMoi]

Sorry, that's a lot of nonsense. Einstein can think whatever. He might have believed that there is a spooky action at a distance, and as I said, Einstein misunderstood QM.

Yes you've said that. And this too:

Sorry but that is totally wrong. There is no spooky action at a distance. There is no need of hidden parameters. And no signal was sent faster than the speed of light. It simply means that if two particles are "entangled", it means that they were prepared in a given quantum state, and unless there is an interaction, they will continue to stay in that quantum state.

And my personal favorite:

This shows you absolutely got it wrong. Aspect's experiment showed that Bell was right, and what Bell had claimed was that there are no hidden local parameters. IOW, what Einstein had claimed - spooky action at a distance - is wrong. Get it. What all this demonstrates is that quantum logic is different than classical logic.

Now the first problem is what Aspect et al. actually said in their study "Experimental realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment: a new violation of Bell's inequalities"

"In conclusion, our experiment yields the strongest violation of Bell's inequalities ever achieved, and excellent agreement with quantum mechanics. Since it is a straightforword transposition of the ideal Einstein-Rosen-Bohm scheme, the experimental procedure is very simple, and needs no auxiliary measurements as in previous experiments with single-channel polarizers. We are thus led to the rejection of realistic local theories".

Even better, while Aspect's study showed correlations over a fairly small spatial seperation, Gisin and colleagues did it over several kilometers in e.g., "Violation of Bell inequalities by photons more than 10 km apart"

And better still, they are even more clear about what Einstein said: "Quantum theory is nonlocal. Indeed, quantum theory predicts correlations among distant measurement outcomes that cannot be explained by any theory which involves only local variables. This was anticipated by Einstein, Podolsky, and Rosen."

You continue to say there's no "spooky action at a distance" and that Einstein was wrong, but you've also said:

I'm not going to go over what Einstein said, as his words are a mass of confusion.

So you know what Einstein claimed, but his words are a mass of confusion you can't go over. I don't find them confusing at all. It's very clear what this means:
"Starting then with the assumption that the wave function does give a complete description of the physical reality, we arrived at the conclusion that two physical quantities, with noncommuting operators, can have simultaneous reality. Thus the negation of (1) leads to the negation of the only other alternative (2). We are thus forced to conclude that the quantum-mechanical description of the physical reality given by wave functions is not complete."

So what he thought is irrelevant. But people seem to have a vested interest in resuscitating his arcane arguments.

Yes, people seem to keep on going back to EPR. Of course, these people happen to be (among others) the greatest physicists that ever lived, along with the rest of the physics community, e.g., Aspect, Gisen, Hawking, Penrose, Stapp, Wheeler, Tegmark, Smolin, Barrow, Vilenkin, Susskind, Hartle, Deutsch, Bohm, Heisenberg, and not to mention the hundreds of studies published already in 2013 alone in journals like Physical Review Letters (which published, this past April, "Einstein-Podolsky-Rosen Spatial Entanglement in Ordered and Anderson Photonic Lattices"), Nature Physics, Science, PNAS, and on and on.

What is it about these idiot physicists who don't understand what you so clearly do?

Wrong, you don't seem to get. It proves that classical logic doesn't apply to quantum system.

1) That's what Einstein said
2) That's what everybody knows Einstein said
3) That's what Aspect and Gisen and Bell all stated (and I quoteed for you) that Einstein said,
4) That's why studies to this day continue to cite EPR when discussing entanglement.
5) That's why active physicists say things like
"The constructive answers to all these questions are prepared by providing detailed background information and the important answer to one of these fundamental questions is that experiments confirm that &#8220;spooky action at a distance&#8221; takes place in the K-meson system"
Brning, E., & Petruccione, F. (2009). Theoretical Foundations of Quantum Information Processing and Communication: Selected Topics. Springer

That you consider this difference between classical and quantum mechanics as a waste of times when it is at the core of QM speaks loudly of your incompetence in this matter.

Ok. I think it is pretty clear to anybody who can read English that your view of just about everything is contradicted by real physicists publishing in current, peer-reviewed scientific literature. If you insist that e.g., Proceedings of the National Academy of Science would publish a study in which the authors state "spooky action at a distance must be accepted as a fact of nature" when (as you claim) "spooky action at a distance" doesn't happen, or if you wish to simply ignore the numerous quotes from actual scientific literature which are extremely clear on this point, go ahead. I'm not interested in arguing with someone who refuses to address very clear statements and I've already derailed this thread going over another pointless discussion.

In the word's of Shakespeare's Benedick, "keep your way, i' God's name; I have done."
EDIT:

I think the rest of us should move on and not let zaybu derail this thread.

Sorry about that. Your post wasn't there when I started writing.

 

[zaybu]

No, you're wrong. Furthermore, I smell a troll.

I think the rest of us should move on and not let zaybu derail this thread.

No, if there is any troll, it is you as you are totally incompetent and can't deliver the proper answers. BTW, providing links that are beyond your comprehension will only fool the amateurs. The problem is I'm no amateur. I have a PHD in physics, plus 25 years of teaching that stuff. There's very little you can teach me. OTOH, I can teach you Gauge Theory, Spontaneous Symmetry Breaking, the Higgs Mechanism, Wilson Loop, Feynman path integral, QED, QCD and QFTCST. If you want to test me on any of those topics, try.

 

[LegionOnomaMoi]

No, if there is any troll, it is you as you are totally incompetent and can't deliver the proper answers. BTW, providing links that are beyond your comprehension will only fool the amateurs. The problem is I'm no amateur. I have a PHD in physics, plus 25 years of teaching that stuff. There's very little you can teach me. OTOH, I can teach you Gauge Theory, Spontaneous Symmetry Breaking, the Higgs Mechanism, Wilson Loop, Feynman path integral, QED, QCD and QFTCST. If you want to test me on any of those topics, try.

Sure. I'll do that. I'll start a new thread, though. I'm setting it up now, and it will be in the general debates section under the title "Einstein and "spooky actions"". That way when you are demonstrating how ignorant I am at least I won't derail this thread again.

 

[zaybu]

Sure. I'll do that. I'll start a new thread, though. I'm setting it up now, and it will be in the general debates section under the title "Einstein and "spooky actions"". That way when you are demonstrating how ignorant I am at least I won't derail this thread again.

Sure, if you want to make a fool of yourself, I'll oblige. Don't ever say I didn't warn you. Just to give a taste, from my personal notes:
Gauge Theory and Higgs Mechanism



HERE'S A GENERAL RUNDOWN OF THE THEORY:

(1) in QM: x &#8594; operator
But to satisfy Relativity, in which time is on an equal footing with space, in QFT: x &#8594; parameter, and &#934;(x) &#8594; operator. Now &#934;(x), a function of x, is called the &#8220;field&#8221;.

(2) L = T &#8211; V. The Lagrangian plays an important role. From Noether&#8217;s theorem, we know that if the Lagragian is invariant under a symmetry, this symmetry points to a conservation law.

Corresponding to L there is a Hamiltonian, H = T + V. The Hamiltonian is known to measure the energy of a system.

(3)In classical mechanics, let v = dx/dt, then L = ½ mv2 &#8211; V(x). The corresponding Hamiltonian is, H = ½ mv2 + V(x). Quantizing this, (&#8463; =1),we get the Schroedinger equation:

i&#8706;&#936;(x)/&#8706;t =( -½m&#8710; 2 + V(x))&#936;(x).

(4) In Relativity, the energy equation is:

E2= p2c2 + m2c4.

Quantizing this, (c =1) yields the K-G equation:

½(&#8706;&#956;&#934(&#8706;&#956;&#934 + ½m&#934;2 = 0.

From this, the Lagrangian can be deduced as:

L = ½ (&#8706;&#956;&#934 2 &#8211; ½m&#934;2.


(5) In QFT, the general Lagrangian is:

L = ½ (&#8706;&#956;&#934 2 &#8211; V(&#934.

(6) Comparing (5) and (4), if V(&#934 contains any terms with &#934; 2, its coefficient is taken to be the mass of the field quanta (particles).

Gauge theory:

From electromagnetism, it was known that Maxwell&#8217;s equations were gauge invariant. In QM, gauge invariance of the Lagragian involves three important steps:

(7) the wave function is transformed as &#934; &#8594; eiqX&#934;
(8) the operator &#8706;&#956; &#8594; &#8706;&#956; + iqA&#956;
(9) the electromagnetic field A&#956;&#8594; A&#956; - &#8706;&#956;X

(10) In QED, in equation (5), V(&#934 &#8594; - ¼ F&#956;&#957;F&#956;&#957;,
where F&#956;&#957;= &#8706;&#956;A&#957; - &#8706;&#957;A&#956;

If you apply, 7,8,9,10 to equation (5), you get the invariance of the Lagrangian under gauge transformation, in which the photon mediates the electromagnetic force. Note that the photon has no mass.

In the weak force, the bosons involved have mass, and one had to figure out how to include a mass term, keeping the Lagrangian gauge invariant.

There is where number (6) comes into play under the notion of SPONTANEOUS SYMMETRY BREAKING.

Higgs Mechanism:

Basically, I will only look at U(1) symmetry. Electroweak interactions need a U(1) x SU(2) symmetry, but SU(2) requires 2 by 2 matrices, and the software on this forum is inadequate to deal with matrices. But you can get the flavor just by doing U(1) symmetry and how mass is introduced in the Lagrangian of equation (5).

I will rewrite this equation as:

(11) L = &#8706;&#956;&#934;&#8224; &#8706;&#956;&#934; - ¼ F&#956;&#957; F&#956;&#957; &#8211; V(&#934;&#8224;&#934.

(12) where V(&#934;&#8224;&#934 = (m2)/(2&#966;2) {&#934;&#8224;&#934; - &#966;2} 2

Three important things to note:

(13) The field &#934; is now a complex number, denoted by (&#934;1, &#934;2) or &#934; = &#934;1 + i&#934;2 ( i being the imaginary number, square root of &#8211; 1), and &#934;&#8224; = &#934;1 &#8211; i&#934;2.

(14) the minimum field energy is obtained when &#934;&#8224;&#934; = &#966;2.

(15) The number of possible vacuum states is infinite. We break this symmetry by requiring that &#934; is real, we take the vacuum state to be (&#966;,0), and expand:

&#934; = &#966; + (½ ½)h

Substituting 7,8,9, 12, and 15 into 11, we get

(17) L = {(&#8706;&#956; - iqA&#956( &#966; + (½ ½)h)}{( &#8706;&#956; + iqA&#956( &#966; + (½ ½)h} - ¼ F&#956;&#957; F&#956;&#957; - (m2)/(2&#966;2) {2½&#966;h + ½h2}2

After calculating the Lagrangian, we separate it into two parts:

(18) L = Lfree + Lint

where

(19) Lfree = ½&#8706;&#956;h&#8706;&#956;h - m2h2 - ¼ F&#956;&#957; F&#956;&#957; + q2&#966;2A&#956;A&#956;

All the remaining terms are lumped into Lint, which offer no interest.

So, we can see that by breaking the symmetry, we end up with two massive particles. In equation 19, the second term refers to a scalar particle with mass equal to 2½m, associated with h (the higgs field) and the fourth term, a vector boson with mass 2½q&#966;, associated with A&#956;( the electromagnetic field).

NOTE: in the Weinberg electroweak theory, with SU(2), equation 19 would have three extra terms for the vector boson instead of a single term, each one was identified with the W+, W -, and Z bosons. This prediction, which was confirmed subsequently in the following years, earned Weinberg, Salam and Glashow the Nobel prize.