On Saturday, February 2, 2019 at 2:59:30 PM UTC-7, [email protected] wrote: > > > > On Saturday, February 2, 2019 at 1:40:29 AM UTC-7, Bruno Marchal wrote: >> >> >> On 1 Feb 2019, at 21:29, [email protected] wrote: >> >> >> >> On Friday, February 1, 2019 at 5:55:30 AM UTC-7, Bruno Marchal wrote: >>> >>> >>> On 31 Jan 2019, at 21:10, [email protected] wrote: >>> >>> >>> >>> On Thursday, January 31, 2019 at 6:47:12 AM UTC-7, Bruno Marchal wrote: >>>> >>>> >>>> On 31 Jan 2019, at 01:28, [email protected] wrote: >>>> >>>> >>>> >>>> On Wednesday, January 30, 2019 at 2:38:58 PM UTC-7, [email protected] >>>> wrote: >>>>> >>>>> >>>>> >>>>> On Wednesday, January 30, 2019 at 5:16:05 AM UTC-7, Bruno Marchal >>>>> wrote: >>>>>> >>>>>> >>>>>> On 30 Jan 2019, at 02:59, [email protected] wrote: >>>>>> >>>>>> >>>>>> >>>>>> On Tuesday, January 29, 2019 at 4:37:34 AM UTC-7, Bruno Marchal wrote: >>>>>>> >>>>>>> >>>>>>> On 28 Jan 2019, at 22:50, [email protected] wrote: >>>>>>> >>>>>>> >>>>>>> >>>>>>> On Friday, January 25, 2019 at 7:33:05 AM UTC-7, Bruno Marchal wrote: >>>>>>>> >>>>>>>> >>>>>>>> On 24 Jan 2019, at 09:29, [email protected] wrote: >>>>>>>> >>>>>>>> >>>>>>>> >>>>>>>> On Sunday, January 20, 2019 at 11:54:43 AM UTC, [email protected] >>>>>>>> wrote: >>>>>>>>> >>>>>>>>> >>>>>>>>> >>>>>>>>> On Sunday, January 20, 2019 at 9:56:17 AM UTC, Bruno Marchal wrote: >>>>>>>>>> >>>>>>>>>> >>>>>>>>>> On 18 Jan 2019, at 18:50, [email protected] wrote: >>>>>>>>>> >>>>>>>>>> >>>>>>>>>> >>>>>>>>>> On Friday, January 18, 2019 at 12:09:58 PM UTC, Bruno Marchal >>>>>>>>>> wrote: >>>>>>>>>>> >>>>>>>>>>> >>>>>>>>>>> On 17 Jan 2019, at 14:48, [email protected] wrote: >>>>>>>>>>> >>>>>>>>>>> >>>>>>>>>>> >>>>>>>>>>> On Thursday, January 17, 2019 at 12:36:07 PM UTC, Bruno Marchal >>>>>>>>>>> wrote: >>>>>>>>>>>> >>>>>>>>>>>> >>>>>>>>>>>> On 17 Jan 2019, at 09:33, [email protected] wrote: >>>>>>>>>>>> >>>>>>>>>>>> >>>>>>>>>>>> >>>>>>>>>>>> On Thursday, January 17, 2019 at 3:58:48 AM UTC, Brent wrote: >>>>>>>>>>>>> >>>>>>>>>>>>> >>>>>>>>>>>>> >>>>>>>>>>>>> On 1/16/2019 7:25 PM, [email protected] wrote: >>>>>>>>>>>>> >>>>>>>>>>>>> >>>>>>>>>>>>> >>>>>>>>>>>>> On Monday, January 14, 2019 at 6:12:43 AM UTC, Brent wrote: >>>>>>>>>>>>>> >>>>>>>>>>>>>> >>>>>>>>>>>>>> >>>>>>>>>>>>>> On 1/13/2019 9:51 PM, [email protected] wrote: >>>>>>>>>>>>>> >>>>>>>>>>>>>> This means, to me, that the arbitrary phase angles have >>>>>>>>>>>>>> absolutely no effect on the resultant interference pattern which >>>>>>>>>>>>>> is >>>>>>>>>>>>>> observed. But isn't this what the phase angles are supposed to >>>>>>>>>>>>>> effect? AG >>>>>>>>>>>>>> >>>>>>>>>>>>>> >>>>>>>>>>>>>> The screen pattern is determined by *relative phase angles >>>>>>>>>>>>>> for the different paths that reach the same point on the >>>>>>>>>>>>>> screen*. >>>>>>>>>>>>>> The relative angles only depend on different path lengths, so >>>>>>>>>>>>>> the overall >>>>>>>>>>>>>> phase angle is irrelevant. >>>>>>>>>>>>>> >>>>>>>>>>>>>> Brent >>>>>>>>>>>>>> >>>>>>>>>>>>> >>>>>>>>>>>>> >>>>>>>>>>>>> *Sure, except there areTWO forms of phase interference in Wave >>>>>>>>>>>>> Mechanics; the one you refer to above, and another discussed in >>>>>>>>>>>>> the >>>>>>>>>>>>> Stackexchange links I previously posted. In the latter case, the >>>>>>>>>>>>> wf is >>>>>>>>>>>>> expressed as a superposition, say of two states, where we >>>>>>>>>>>>> consider two >>>>>>>>>>>>> cases; a multiplicative complex phase shift is included prior to >>>>>>>>>>>>> the sum, >>>>>>>>>>>>> and different complex phase shifts multiplying each component, >>>>>>>>>>>>> all of the >>>>>>>>>>>>> form e^i (theta). Easy to show that interference exists in the >>>>>>>>>>>>> latter case, >>>>>>>>>>>>> but not the former. Now suppose we take the inner product of the >>>>>>>>>>>>> wf with >>>>>>>>>>>>> the ith eigenstate of the superposition, in order to calculate >>>>>>>>>>>>> the >>>>>>>>>>>>> probability of measuring the eigenvalue of the ith eigenstate, >>>>>>>>>>>>> applying one >>>>>>>>>>>>> of the postulates of QM, keeping in mind that each eigenstate is >>>>>>>>>>>>> multiplied >>>>>>>>>>>>> by a DIFFERENT complex phase shift. If we further assume the >>>>>>>>>>>>> eigenstates >>>>>>>>>>>>> are mutually orthogonal, the probability of measuring each >>>>>>>>>>>>> eigenvalue does >>>>>>>>>>>>> NOT depend on the different phase shifts. What happened to the >>>>>>>>>>>>> interference >>>>>>>>>>>>> demonstrated by the Stackexchange links? TIA, AG * >>>>>>>>>>>>> >>>>>>>>>>>>> Your measurement projected it out. It's like measuring which >>>>>>>>>>>>> slit the photon goes through...it eliminates the interference. >>>>>>>>>>>>> >>>>>>>>>>>>> Brent >>>>>>>>>>>>> >>>>>>>>>>>> >>>>>>>>>>>> *That's what I suspected; that going to an orthogonal basis, I >>>>>>>>>>>> departed from the examples in Stackexchange where an arbitrary >>>>>>>>>>>> superposition is used in the analysis of interference. >>>>>>>>>>>> Nevertheless, isn't >>>>>>>>>>>> it possible to transform from an arbitrary superposition to one >>>>>>>>>>>> using an >>>>>>>>>>>> orthogonal basis? And aren't all bases equivalent from a linear >>>>>>>>>>>> algebra >>>>>>>>>>>> pov? If all bases are equivalent, why would transforming to an >>>>>>>>>>>> orthogonal >>>>>>>>>>>> basis lose interference, whereas a general superposition does not? >>>>>>>>>>>> TIA, AG* >>>>>>>>>>>> >>>>>>>>>>>> >>>>>>>>>>>> I don’t understand this. All the bases we have used all the >>>>>>>>>>>> time are supposed to be orthonormal bases. We suppose that the >>>>>>>>>>>> scalar >>>>>>>>>>>> product (e_i e_j) = delta_i_j, when presenting the Born rule, and >>>>>>>>>>>> the >>>>>>>>>>>> quantum formalism. >>>>>>>>>>>> >>>>>>>>>>>> Bruno >>>>>>>>>>>> >>>>>>>>>>> >>>>>>>>>>> *Generally, bases in a vector space are NOT orthonormal. * >>>>>>>>>>> >>>>>>>>>>> >>>>>>>>>>> Right. But we can always build an orthonormal base with a decent >>>>>>>>>>> scalar product, like in Hilbert space, >>>>>>>>>>> >>>>>>>>>>> >>>>>>>>>>> >>>>>>>>>>> *For example, in the vector space of vectors in the plane, any >>>>>>>>>>> pair of non-parallel vectors form a basis. Same for any general >>>>>>>>>>> superposition of states in QM. HOWEVER, eigenfunctions with >>>>>>>>>>> distinct >>>>>>>>>>> eigenvalues ARE orthogonal.* >>>>>>>>>>> >>>>>>>>>>> >>>>>>>>>>> Absolutely. And when choosing a non degenerate >>>>>>>>>>> observable/measuring-device, we work in the base of its >>>>>>>>>>> eigenvectors. A >>>>>>>>>>> superposition is better seen as a sum of some eigenvectors of some >>>>>>>>>>> observable. That is the crazy thing in QM. The same particle can be >>>>>>>>>>> superposed in the state of being here and there. Two different >>>>>>>>>>> positions of >>>>>>>>>>> one particle can be superposed. >>>>>>>>>>> >>>>>>>>>> >>>>>>>>>> *This is a common misinterpretation. Just because a wf can be >>>>>>>>>> expressed in different ways (as a vector in the plane can be >>>>>>>>>> expressed in >>>>>>>>>> uncountably many different bases), doesn't mean a particle can exist >>>>>>>>>> in >>>>>>>>>> different positions in space at the same time. AG* >>>>>>>>>> >>>>>>>>>> >>>>>>>>>> It has a non null amplitude of probability of being here and >>>>>>>>>> there at the same time, like having a non null amplitude of >>>>>>>>>> probability of >>>>>>>>>> going through each slit in the two slits experience. >>>>>>>>>> >>>>>>>>>> If not, you can’t explain the inference patterns, especially in >>>>>>>>>> the photon self-interference. >>>>>>>>>> >>>>>>>>>> >>>>>>>>>> >>>>>>>>>> >>>>>>>>>> >>>>>>>>>> Using a non orthonormal base makes only things more complex. >>>>>>>>>>> >>>>>>>>>> *I posted a link to this proof a few months ago. IIRC, it was on >>>>>>>>>>> its specifically named thread. AG* >>>>>>>>>>> >>>>>>>>>>> >>>>>>>>>>> But all this makes my point. A vector by itself cannot be >>>>>>>>>>> superposed, but can be seen as the superposition of two other >>>>>>>>>>> vectors, and >>>>>>>>>>> if those are orthonormal, that gives by the Born rule the >>>>>>>>>>> probability to >>>>>>>>>>> obtain the "Eigen result” corresponding to the measuring apparatus >>>>>>>>>>> with >>>>>>>>>>> Eigen vectors given by that orthonormal base. >>>>>>>>>>> >>>>>>>>>>> I’m still not sure about what you would be missing. >>>>>>>>>>> >>>>>>>>>> >>>>>>>>>> *You would be missing the interference! Do the math. Calculate >>>>>>>>>> the probability density of a wf expressed as a superposition of >>>>>>>>>> orthonormal >>>>>>>>>> eigenstates, where each component state has a different phase angle. >>>>>>>>>> All >>>>>>>>>> cross terms cancel out due to orthogonality,* >>>>>>>>>> >>>>>>>>>> >>>>>>>>>> ? Sin(alpha) up + cos(alpha) down has sin^2(alpha) probability >>>>>>>>>> to be fin up, and cos^2(alpha) probability to be found down, but has >>>>>>>>>> probability one being found in the Sin(alpha) up + cos(alpha) down >>>>>>>>>> state, >>>>>>>>>> which would not be the case with a mixture of sin^2(alpha) >>>>>>>>>> proportion of up >>>>>>>>>> with cos^2(alpha) down particles. >>>>>>>>>> Si, I don’t see what we would loss the interference terms. >>>>>>>>>> >>>>>>>>>> >>>>>>>>>> >>>>>>>>>> *and the probability density does not depend on the phase >>>>>>>>>> differences. What you get seems to be the classical probability >>>>>>>>>> density. >>>>>>>>>> AG * >>>>>>>>>> >>>>>>>>>> >>>>>>>>>> >>>>>>>>>> I miss something here. I don’t understand your argument. It seems >>>>>>>>>> to contradict basic QM (the Born rule). >>>>>>>>>> >>>>>>>>> >>>>>>>>> *Suppose we want to calculate the probability density of a >>>>>>>>> superposition consisting of orthonormal eigenfunctions, * >>>>>>>>> >>>>>>>> >>>>>>>> Distinct eigenvalue correspond to orthonormal vector, so I tend to >>>>>>>> always superpose only orthonormal functions, related to those >>>>>>>> eigenvalue. >>>>>>>> >>>>>>>> >>>>>>>> >>>>>>>> >>>>>>>> >>>>>>>> *each multiplied by some amplitude and some arbitrary phase shift. * >>>>>>>>> >>>>>>>> >>>>>>>> like (a up + b down), but of course we need a^2 + b^2 = 1. You >>>>>>>> need to be sure that you have normalised the superposition to be able >>>>>>>> to >>>>>>>> apply the Born rule. >>>>>>>> >>>>>>>> >>>>>>>> >>>>>>>> >>>>>>>> *If we take the norm squared using Born's Rule, don't all the cross >>>>>>>>> terms zero out due to orthonormality? * >>>>>>>>> >>>>>>>> >>>>>>>> ? >>>>>>>> >>>>>>>> The Born rule tell you that you will find up with probability a^2, >>>>>>>> and down with probability b^2 >>>>>>>> >>>>>>>> >>>>>>>> >>>>>>>> *Aren't we just left with the SUM OF NORM SQUARES of each component >>>>>>>>> of the superposition? YES or NO?* >>>>>>>>> >>>>>>>> >>>>>>>> If you measure in the base (a up + b down, a up -b down). In that >>>>>>>> case you get the probability 1 for the state above. >>>>>>>> >>>>>>>> >>>>>>>> >>>>>>>> * If YES, the resultant probability density doesn't depend on any >>>>>>>>> of the phase angles. AG* >>>>>>>>> >>>>>>>> >>>>>>>> *YES or NO? AG * >>>>>>>> >>>>>>>> >>>>>>>> >>>>>>>> Yes, if you measure if the state is a up + b down or a up - b down. >>>>>>>> No, if you measure the if the state is just up or down >>>>>>>> >>>>>>>> Bruno >>>>>>>> >>>>>>> >>>>>>> *I assume orthNORMAL eigenfunctions. I assume the probability >>>>>>> densities sum to unity. Then, using Born's rule, I have shown that >>>>>>> multiplying each component by e^i(theta) where theta is arbitrarily >>>>>>> different for each component, disappears when the probability density >>>>>>> is >>>>>>> calculated, due to orthonormality. * >>>>>>> >>>>>>> >>>>>>> >>>>>>> That seems to violate elementary quantum mechanics. If e^I(theta) is >>>>>>> different for each components, Born rule have to give different >>>>>>> probabilities for each components---indeed given by the square of >>>>>>> e^I(theta). >>>>>>> >>>>>> >>>>>> *The norm squared of e^i(thetai) is unity, except for the cross terms >>>>>> which is zero due to orthonormality. AG * >>>>>> >>>>>>> >>>>>>> *What you've done, if I understand correctly, is measure the >>>>>>> probability density using different bases, and getting different >>>>>>> values. * >>>>>>> >>>>>>> >>>>>>> The value of the relative probabilities do not depend on the choice >>>>>>> of the base used to describe the wave. Only of the base corresponding >>>>>>> to >>>>>>> what you decide to measure. >>>>>>> >>>>>>> >>>>>>> >>>>>>> *This cannot be correct since the probability density is an >>>>>>> objective value, and doesn't depend on which basis is chosen. AG* >>>>>>> >>>>>>> >>>>>>> Just do the math. Or read textbook. >>>>>>> >>>>>> >>>>>> *Why don't YOU do the math ! It's really simple. Just take the norm >>>>>> squared of a superposition of component eigenfunctions, each multiplied >>>>>> by >>>>>> a probability amplitude, and see what you get ! No need to multiply >>>>>> each >>>>>> component by e^i(thetai). Each amplitude has a phase angle implied. >>>>>> This >>>>>> is Born's rule and the result doesn't depend on phase angles, >>>>>> contracting >>>>>> what Bruce wrote IIUC. If you would just do the simple calculation you >>>>>> will >>>>>> see what I am referring to! AG* >>>>>> >>>>>> >>>>>> >>>>>> Bruce is right. Let us do the computation in the simple case where >>>>>> e^i(theta) = -1. (Theta = Pi) >>>>>> >>>>>> Take the superposition (up - down), conveniently renormalised. If I >>>>>> multiply the whole wave (up - down) by (-1), that changes really >>>>>> nothing. >>>>>> But if I multiply only the second term, I get the orthogonal state up + >>>>>> down, which changes everything. (up +down) is orthogonal to (up - down). >>>>>> >>>>>> Bruno >>>>>> >>>>> >>>>> *Fuck it. You refuse to do the simple math to show me exactly where >>>>> I have made an error, IF I have made an error. You talk a lot about >>>>> Born's rule but I seriously doubt you know how to use it for simple >>>>> superposition. AG * >>>>> >>>> >>>> *If you take the inner product squared (Born's rule) using an >>>> orthonormal set of eigenfunctions, you get a sum of the form (a_j)^2 + >>>> (b_j) ^2 where A_j is the complex probability amplitude for the jth >>>> component, A_j = a_j + i * b_j. The cross terms drop out due to >>>> orthonormality, and the phase angles are implicitly determined by the >>>> relative values of a_j and b_j for each j. * >>>> >>>> >>>> If you have prepared the state, so that you know that the state of your >>>> object is given by >>>> >>>> phi = A_1 up + A_2 down, say, then, if you decide to measure the >>>> up/down state, and use the device doing that, you do not need to make the >>>> inner product between phi and phi, but between the base state up and/or >>>> down to get the probability given by the square of phi * up (to get the >>>> probability of up) and the square of phi*down, to get the probability of >>>> down. They will both depend on the value of A_1 and A_2. They are >>>> respectively (A_1)^2 and (A_2)^2. Of course, we suppose that we have >>>> renormalised the state so that (A_1)^2 + (A_2)^2 = 1 (which makes them >>>> into >>>> probability of getting up and down). >>>> >>>> >>>> >>>> >>>> >>>> >>>> >>>> *The question then becomes how do we calculate the probability density >>>> with the phase angles undetermined. Are we assuming they are known given >>>> the way the system is prepared? AG* >>>> >>>> >>>> >>>> Yes. The Born rule, written simply, is only that if phi = A_1 up + A_2 >>>> down, (so the state has been prepared in advance) then if you measure if >>>> the object is in up or down, you will find up with a probability given >>>> respectively by (A_1)^2 and (A_2)^2. >>>> All probabilities are relative to the state of the object and the >>>> choice of what you decide to measure. It is always simpler to write the >>>> state in the base corresponding to the measurement, so that the “simple” >>>> Born rule above can be applied immediately. >>>> >>>> Bruno >>>> >>> >>> *For reference I repeat my last comment and add a significant point:* >>> >>> If you take the inner product squared (Born's rule) using an >>> orthonormal set of eigenfunctions, you get a sum of the form (a_j)^2 + >>> (b_j) ^2 where A_j is the complex probability amplitude for the jth >>> component, A_j = a_j + i * b_j. The cross terms drop out due to >>> orthonormality, and the phase angles are implicitly determined by the >>> relative values of a_j and b_j for each j. The question then becomes how do >>> we calculate the probability density with the phase angles undetermined. >>> Are we assuming they are known given the way the system is prepared? AG >>> >>> The question for me is how the phase angles are related to interference. >>> >>> >>> But that is explained by may calculation above. You calculation does not >>> make sense to me. You compute an inner product of the wave to itself? I >>> don’t see the relation with your problem. >>> >> >> *Obviously, you don't know how to apply the rule you speak so highly of, >> Born's rule. To calculate the probability density of wf function psi, you >> must calculate <psi, psi>. Do you dispute this? * >> >> >> Yes, you need to put some projection operator (corresponding to some >> eigenvalue you intend to measure) in between. >> <psi,psi> is the amplitude of probability to go from the psi state to the >> psi state, and should be equal to one (psi being normalised). >> > > *Wrong! Not equal to one. I see you like to talk the talk, but refuse to > walk the walk. Just read the 4th paragraph of the Wiki link. I correctly > calculated the probability density for orthonormal eigenstates! AG * > >> >> >> >> *How the phase angles relate to interference is another issue, which I >> think Phil explained. AG* >> >>> >>> The calculation above shows that the cross terms drop out due to >>> orthonormality. >>> >>> >>> Do it again, explicitly. Take the simple state phi = A_1 up + A_2 down. >>> Up and down are orthonormal, >>> >> >> *Up and Dn are NOT orhonormal. AG* >> >> >>> but phi is not orthonormal with either up or down. If “up” means go to >>> the left hole, and “down” is go the right hole, the amplitude A_1 and A_2, >>> if not null, will interfere, even if only one photon is sent.The wave go >>> through both silts, and interfere constructively along some direction and >>> destructively along other direction, making it impossible for that photon >>> to lend on those last place, like anyway, by the laws of addition of >>> sinus/wave. >>> >>> But IIUC these are the terms which account for interference. >>> >>> >>> I am not sure what you say here. The interferences comes only from the >>> fact that we have a superposition of two orthogonal state, and that >>> superposition is a new state, which is not orthogonal to either up or down. >>> >>> Thus, applying Born's rule to a superposition of states where the >>> components are orthonormal, leaves open the question of interference. >>> >>> >>> That does no make sense. The Born rule just say that if you measure >>> (up/down) on phi = A_1 up + A_2 down, you get up with probability (A_1)^2 >>> and down with probability (A_2)^2. But if you do any measurement, the state >>> beg-have like a wave, and the amplitudes add up, constructively or >>> destructively. >>> >>> If you don’t understand that, it means you begin to understand quantum >>> mechanics, as nobody understand this, except perhaps the Mechanist >>> Philosophers …(which predicts something at least as weird and >>> counter-intuitive). >>> >>> Bruce wrote that the phase angles are responsible for interference. I >>> doubt that result. Am I mistaken? AG >>> >>> >>> Yes, I’m afford you are. The relative phase (in a superposition) angles >>> are responsible for the interference. A global phase angle changes nothing. >>> >> >> *If I am wrong, it's just because I assumed all interference comes from >> the interactions due to the cross terms -- which cancel out for orthonormal >> component states. Also, I never introduced a global phase angle in my >> calculation. If you would do my calculation, or at least understand it, >> you'd understand Born's rule. I don't need to read Albert's book to >> understand Born's rule. AG* >> >> >> >> Once you say that up and down are not orthonormal, I am not sure you have >> studied the QM formalism correctly. Any two distinguishable eigenstates of >> any observable are orthogonal (and normalised). >> > > *Right. I was mistaken. AG * > >> >> I have no clue what you don’t understand in my use of the Born rule. You >> definitely need to study Albert’s book, I think. >> > > *Wiki shows I correctly calculated the probability density. Also I agree > with Phil, and noted the error I made (not in any calculation, but in > interpretation). Didn't you read it? AG * > > In your other post you mention wikipedia. No problem there? Actually you >> can see that they do put the projection operator at the right place. You >> can help yourself with a dictionary, but books and papers are better. >> > *If you put in the projection operator, you're calculating the probability of getting some eigenvalue, not the probability density of the position. AG *
> > > *You could help yourself by reading plain English. SEE PARAGRAPH 4 OF WIKI > LINK. THEY CALCULATED THE PROBABILITY DENSITY AND DIDN'T PUT IN THE > PROJECTION OPERATOR! AG* > >> >> Bruno >> >> >> >> >> >> >>> I really wish you to read the first 60 pages of David Albert’s book. Its >>> exposition of the functioning of the interferometer is crystal clear. I am >>> still not sure if you have a problem with the formalism or with the >>> weirdness related to it. Read that piece of explanation by Albert, and if >>> you still have problem, we can discuss it, but it would be too long (here >>> and now) to do that here. >>> >>> Bruno >>> >>> >>> >>> >>> >>> >>> >>> -- >>> You received this message because you are subscribed to the Google >>> Groups "Everything List" group. >>> To unsubscribe from this group and stop receiving emails from it, send >>> an email to [email protected]. >>> To post to this group, send email to [email protected]. >>> Visit this group at https://groups.google.com/group/everything-list. >>> For more options, visit https://groups.google.com/d/optout. >>> >>> >>> >> -- >> You received this message because you are subscribed to the Google Groups >> "Everything List" group. >> To unsubscribe from this group and stop receiving emails from it, send an >> email to [email protected]. >> To post to this group, send email to [email protected]. >> Visit this group at https://groups.google.com/group/everything-list. >> For more options, visit https://groups.google.com/d/optout. >> >> >> -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
