joigus

Yes. On the one hand stating more clearly that what you're saying is that every \( \left| q \right\rangle) \) in the space can be written as \( \left| v \right\rangle + \left| v_{\perp} \right\rangle \) (what I've written as \( v=v_{\parallel}+v_{\perp} \) ), and on the other hand, noticing that taking the action of \( P \) from the second to the first factor in the scalar product is not simply "looking at it as acting on its left", but also complex conjugating. In matrix notation: q⁺Pr=(P⁺q)⁺r where "+" as a superindex means complex conjugate and transpose. It's only when P⁺=P that you can write, q⁺P=(P⁺q)⁺=(Pq)⁺ Otherwise, you can just say that q⁺P=(P⁺q)⁺, which is always true, because it's a definition. This you cannot say clearly in Dirac notation, as Dirac notation automatically assumes that any operator "sandwiched" between both factors is Hermitian. Otherwise the notation is ambiguous. That's why someone as careful as Weinberg sometimes drops it in his proofs. Weinberg proved most everything he said, so he was very careful about these questions. The point of the exercise being that the eigenvalues of any projector are always real. Something that is not true for any linear operator. Another way of seeing it is by investigating the eigenvalue equation. As P²=P, the eigenvalues must satisfy p²=p, so either p=0 or 1.

You're most welcome. The rest of the expansions seem right to me. As to proof 4.3, I see what you're doing there, and it's correct too, AFAICS. The only glitch is for these kind of proofs is that it's perhaps best to drop Dirac's notation, because it kind of stands in the way of distinguishing the vector, the operator, and the action of the inner product more clearly, so you wouldn't have to use the --somewhat awkward, IMO-- double parenthesis on your last line. So, for example, I would write something like, for every \( w \), \( v \) in \( \mathscr{H} \) (the Hilbert space of states), \[ \left( w, P\right) = \left( P^{\dagger}w,v \right) \] (That is just a definition of \( P^{\dagger} \), of course) I would also write, \[ v=v_{perp}+v_{parallel} \] etc, with, \[ Pv_{perp}=0 \] \[ Pv_{parallel}=v_{parallel} \] for an arbitrary vector \( v \). And then, as you say, \[ \left( w, Pv \right) = \left( w,w_{\parallel} \right) = \left( w_{\parallel}, v_{\parallel} \right) = \left( w_{\parallel}, v \right) = \left( Pw, v \right) \] So indeed \( P^{\dagger}=P. To me, it's a bit more transparent with this notation, but I understood what you meant, and if you think about it we're saying the same thing. The devil is in the details, as they say. In infinite dimension one would have to be much more careful than this, but I don't have the chops for it. 😊 Ok. Something got messed up in the LaTeX rendering, I'm afraid... I hope you can see what it is. I always have to be very careful that my text is not interpreted as rich text at some point (eg, when the editor refreshes) so the rendering is messed up. It might have to do with the software at my end. I dunno.

Give me some time to check conventions. They differ from QC to "atoms-and-molecules" QM... But arent you missing 1/sqrt(2) factors there? Plus and minus kets should be normalised. Didn't get the time to check everything else yet. One of the most common mistakes is when taking the adjoint of |0> + i |1> which should be <0|-i<1| but I'm guessing you didn't make that mistake.

How do you know? You haven't done any calculations. You really don't know any of that. As Markus has just pointed out, you're using relabeling of coordinates to claim some physical effect, which is deeply mislead. You're confusing active and passive transformations. I made other points about continuous vs discrete transformations, as well as discrete transformations not being possible to interpret kinematically, which you have chosen not to address, even though they came from comments you made and I read quite carefully. At least much more than their internal logic --or lack thereof-- deserves.

Yes, it's almost as if it's all a matter of how much you perceive the other side is clutching at straws really. I have to say I do not really like this kind of physics of impossibility theorems, of what is possible and what isn't. In pure mathematics, everything is crystal clear --technical difficulties apart--. Your premises are what you say they are. In physics, on the other hand, it seems as though it were always possible to relax the hypotheses some way or another, even at the cost of making extremely unnatural or strangely contrived assumptions. IMO, people who are working on SD are living dangerously, while people who prefer to think in terms of multiple realities are too narrow-minded. Some synthesis will appear eventually and it will feel like "how could we not have thought of this before?" It's obvious to me that's dropping some implicit or hidden assumption that has been invisible to us so far.

Well, I cannot be sure 100% of what they mean. But I would say that they must be implying that all information that determines what polarisation direction the experimenters are going to measure is somehow "diluted" in the dynamical information that's been going around for billions of years, and that information can be "tapped" locally at the moment of the measurement actually being performed. I cannot conceive of any other sense in which they can be speaking. IOW, information can propagate strictly locally, and yet have had plenty of time to reach every corner of the universe, so to speak.

Again, no. We won't get past this. I see how you would think breaking of a continuous symmetry transformation, like time translations, would affect energy. But T as in "CPT" is a discrete transformation. So its breaking has no implication whatsoever on the sign of energy, or on anything about energy for that matter. You need to understand physical principles better. It's amazing how you ignore what you quote yourself (highlighted by myself above.) Here is the complete quote: In other words: to avoid negative energies, we choose a time inversion operator that is antilinear and antiunitary. In no way do we accept negative energies, as that would lead to an unstable vacuum. If you actually read Weinberg's paragraph that you quote it cannot be made any more obvious. The fact itself that we're forced to choose antilinear operators to represent these "reshuffling" of states says very clearly that it's a very different beast. In particular, not amenable to a kinematical interpretation, as you previously claimed.

Oh I see. It's not a sky prince above the clouds. It's a professional up there above the glass. That changes everything. Disprove. Redraw. Otherwise enjoy your radicalisation.

Why would that be? You're using the term "symmetry breaking", which means something different from what you think it means. If by T symmetry being broken you mean there is no exact T symmetry, I don't see why that implies the gravitational interaction changes sign. In fact, we know of a (slight) violation of T symmetry in elementary particle interactions. But both gravitational mass and inertia seem to be unaffected in the way you seem to claim. I don't think you mean gravitational interaction changing sign (that would be because of gravitational mass changing sign). I think you mean inertial mass changing sign, as studied by Bonnor et. al. From the looks of it. The whole thing looks like a rehashing of dead thread: by muruep00, your old self. After a somewhat lengthy discussion, you were told, Doesn't look like you've come up with any maths supporting these intuitions. Copying and pasting the Schwarzschild solution and the relativistic EM equation really doesn't do it, does it?

Anti-orthocronous Lorentz transformations, parity transformations and the like cannot be understood kinematically. They do not relate inertial frames. It's always possible to shift continuously from a kinematical state to another. It's not possible to invert time in a continuous way. It is designed to violate CP, as that's what's shown in B meson decays. As CPT is an exact symmetry of Nature, violation of CP is tantamount to violation of T. There are no negative energy particles really nor does the theory say so. The Hamiltonian is positive definite in QFT. "negative energy particles" is some kind of slang for "antiparticles" that might have crept in here and there due to certain criteria in the Feynman propagator. Actual antiparticles always have positive energy, and so it appears in calorimetric etc experiments.

There is absolutely no problem in extending GR to include anti-orthocronous Lorentz transformations. GR is invariant under the whole group, not only the connected subgroup. It's the standard model that's in conflict with T, P, CP, not GR.

All words and no maths make Jack's a dull toy

Yes, @Eise. You've just kind of voiced my concerns here. I don't know in detail about those critical voices, but I'm sure they must sound something like this: So we have to accept that, even though everything in the universe thermalises very quickly, somehow this information, which is dynamical in nature, must be protected from thermalising so that one day in a laboratory in, say, Vienna, a physicists chooses a polarisation direction and the universe conjures up that information? While not impossible, it rings totally wrong. It goes against everything else we know about entropy, the arrow of time, etc. Degrees of freedom thermalise, mix, get blurred out with time. What magical DoF's are these?

OK. So here's what my friend tells me. Keep in mind it's not a technical explanation..., <Translation> Yes, sure. They do it with their uncertainties. They actually see how electrons go from one atom to another to form a molecule, as I understand. It's kind of my topic. Well.., the attosecond pulses themselves, not their utility concerning interactions with matter and watching electrons. </end of translation> Maybe the "see" and "watch" had better be put in quotation marks, or something like that.

Thanks, Eise. I'll have to look into it more deeply, but for the time beeing it checks with my understanding that superdeterminism is similar to the observation that bipartite, tripartite etc entangled states are correlated (strangely, non-realistically, unintuitively, so on) from the beginning. It's that statement but on steroids. Namely: Everything is entangled from the very beginning (or perhaps non-beginning, as the Mahayana teachers say) of time. Something like that, but I would like to read more about it.

I'm guessing that's a piece of journalistic lingo. Sorry, it's not my claim, it's my friend's. He seems to have picked it up from the newsreel. It's not like he wrote it trying to be rigorous. It does sound akin to some kind of stroboscopic view, like @geordief suggests.

I'm asking my friend, see what he tells me. He's a university professor and a researcher in non-linear optics, so he should know... BTW, @geordief, a clarification on what I meant before: A femtosecond would be 10-15 seconds. People in the field already call a 0.999 fs short pulse an "attosecond", even though it's almost 1000 attos (a femto). In the meantime, looking up: https://en.wikipedia.org/wiki/Ultrashort_pulse You find, Which, I think, must have the answer to our question.

I've just received an email from a friend who's working on the theoretical part of the same stuff these people are doing. These are the people who've made attosecond-short (10-18 seconds) pulses of light a reality. The record is in 25 attoseconds. People already call "attoseconds" a fraction-of-a-femtosecond-short pulse of light. The one 25-attoseconds-short already allows you to see electrons moving. Amazing.

Yes, they'll probably still be here after we're long gone.

Not true. Not accurately enough for certain cases (perihelion of Mercury) and for certain purposes (high precision location in GPS systems). In those cases, it's general relativity that does the job. And don't forget that GR reduces to special relativity at every point. There are very many ways in which we know SR to be right.

These are actually the significant ones, the ones that are dimensionless. Including, of course, the Planck scale.

Sorry, I meant theologians. This reminds me of comments made by @Eise on the late thread about local realism in the sense that we've reached a point where nothing but endless model-building, extrapolation and back-consistency checks --so to speak-- is possible.

I'm not aware of having pointed out that the idea of a multiverse is diferent from a multiverse itself. But it's obvious, isn't it? Same goes for the idea of anything compared to the thing itself. Thinking otherwise is known in philosophy as the use-mention error, which consists in ignoring the use-mention distinction. Theologists do it all the time. History of God = History of the concept of God. Not the same thing!! I'm sure Daniel Dennett has dealt with this question somewhere. The idea of a multiverse is what our minds handle when we speculate wich such possibility. The multiverse would be what we would experience if we could travel through time and space quite freely and experience those domains directly, which we never will. I don't think we will.

Exactly. I didn't want to open that can of worms, but I agree. How can you get any picture at all of a sample space (and its odds) that you've never probed, and never will?

I would rephrase "worlds apart in many senses". Maybe "in almost every sense conceivable" is too much. My point is: It took about 20% of the age of the Earth to get to something like eukaryotes that was sure-footed enough in evolutionary terms when prokaryots already had more than a foothold. Which must mean higher-organization is not easy to come by.