APS Talk: Quaternion Quantum Field Theory

Transcript

hello I'm Doug sweetzer this talk is entitled querian Quantum field Theory one method to calculate it all it was supposed to be presented at the fall APAP meeting at Brown University but it turned out there was no room in the Inn or at least on the schedule and so I'll probably do it ining spring 2011 but since I prepared the talk anyway I'll go ahead and record it and put it up here on YouTube so I wanted to give you a bit of the backstory I've been having a lot of really productive chats about my unified standard model work with a uh graduate physics graduate student in uh Charlotte North Carolina uh L and he's actually the lead salesman for uh the unified field Theory uh t-shirts in all of North America uh but we usually actually talk about more technical sorts of things and he said he that he read this book uh Quantum field Theory demystified a self-teaching guide uh by David McMahon and he thought it would be good if I read it too and so I took him up on the challenge and took me about a week to get through the whole thing and he said now the reason he recommended it was because he thought that I would think about issues differently now I I take this as a compliment I mean I hope it's a compliment right um thing is though I mean I know how to embrace the standard approach uh but I also know how to come at it at a different angle and when I went back to reread it coming at it from this different angle it took me literally a month to get through three pages because I had to resynthesize everything that was there and this book is um sh uh is is very direct as it were uh lots of equations lot of uh nice succinct descriptions of what's going on um not a lot of proofs so it was the kind of way I like to work and in those three pages he done the correspondence principle uh the Schrodinger equation the Klein Gordon equation uh the D rack equation so it was it may not have been long but there was a lot of content um and this talk really is about those three pages and New Perspectives I have on Quantum field theory that came out of L's suggestion that I get this book and think about it so let's get into a little bit of history history that uh new methods basically rewrite everything done in the past take for example special relativity lots of equations that are only about time or only about space well those have to be reformulated into equations of SpaceTime that use four vectors and when you square that you get that Loren and variant interval and lots of equations about energy lots of equations about momentum but now with special relativity you have to think about four momentum and when you square for momentum you get a Lorent invariant mass now in Quant mechanics there is the correspondence principle that says hey if you got a classical equation I bet we can formulate a nice Quantum mechanic one and they have have these complex amplitudes all over the place that you have to square in order to get real observables but although you could rewrite every equation uh so that it works with special relativity what physicists do in practice is they say well you know if and only if things are really going fast am I going to bother with special relativity and for quantum mechanics they say well if and only if those things are really small and am I going to bother with quum mechanics otherwise uh that just just just won't bother well what I want to do is write equations so that those extremes are always taken into account I don't have to go I wonder if stuff is moving fast or if this is small I just want it to work okay to have one accounting system that will be true you know so what I have done is come up with one method with four rules to rewrite all equations in field Theory so the rules are number one to keep keep the four vectors together using querian now I know a good number of you are going to say but I don't even know what a quitan is okay if you've had enough exposure to math to learn about scalers and vectors A little history those who were born in the same house in fact the house was called the house of querian um but but the folks who worked with querian over promised they didn't deliver and so you don't learn the history at all but you know in special relativity it's not this that four numbers go together it's that you have a scaler and a vector going together okay they're more complicated structures than that too but uh you can actually usually back down into a combination of four vectors but it's usually going to be a scaler and a three Vector get that have to hang out together and I don't think that's an accident because that's exactly the same kind of structure as what a querian has and you say Well then why haven't physicists done this before it's because if you can't deliver things like uh the schinger equation the kle Gordon equation Loren group all the these technical kinds of equations if you can't do that there's no reason to to bother with them and that's essentially the core of my research is bringing these really Central equations into a qu Quan form all right rule number two there are no factors of I well actually there are they're just implicit it into a querian you know how I said it's a scaler and a vector that Vector has is not only a vector it's a three vector and it's got an i it's got a j it's got a k guaranteed okay so you don't have to you know waste any penciled on it rule number three is to write out all constants uh the the folks who are professional well they use uh natural units cuz they are so good they don't want to be bothered by these details but B NOA said the be's in the be details so I'm going to keep them I I know Fineman was someone who thought you should always write your units out and he didn't like it in his day uh when natural units were kind of catching on and believe me they've they've really won over um the at least the theory people um but I'm I'm kind of go going retro I'm going with a fan on that one um and the fourth idea is to make all equations dimensionless now I used to actually write that if possible and I was like well who gets to decide if it's possible you know if you're going to come up with rules make them strict and then follow them like all the time uh because that way you there's no question it's like I got these strict rules I follow the strict rules because you know nature seems to be like that you know you don't catch nature sometimes like uh breaking conservation of energy it just she just doesn't and it's like how can you be that consistent man I I I know I personally am not you know so uh but nature is so anyway I I I I like the idea of of these rules because like none of them are like it all radical I don't think I mean keeping four vectors together well that's the message of special relativity um no factors of I well I I I guess some people in trained in quantum mechanics might get upset until they said well actually they're they're always there um and writing out constants now I know a lot of people are going to say I'm not going to bother it's like good then you won't join my research team but but at least for now I'm going to do that and I'm going to make things dimensionally so so let's just apply all this stuff um by writing Newton's second law with the method um seeing that it actually uses zeros all over the place so we have the standard way of of writing it out and that's just fals Ma and then if we use this querian Quantum field Theory way you would go oh please don't show that to the children they'll be scared okay uh because you got wow you've got H bars you have G's you have C to the five there is like uh but but also notice that we have two scalar operators that basically the time derivatives and and what those are multiply uh sorry uh yeah no there's one time derivative sorry one time derivative there um that has the the plon time it's like classical physics you're using plon time how absurd that makes no sense it's like no no no this is dimensional analysis this isn't about plunk it's about making your time derivative operator dimensionless that's all it means okay um and then you've got uh Mass uh divided by a plank mass and again I'm not talking about anything other than making this expression dimensionless so that's got no units and then um yes we do have a DT of of a Dr but that's a velocity so we divide it by C and that's also dimensionless so the resulting expression um it has the the ma term in there uh it also has the rocket science term um which of course uh is um sheltered usually from uh people starting out in physics but rocket science is that second term there and it's this factor of G over C 4th that makes uh this force dimensionless and you'll notice actually that the H uh kind of the calling card of quantum mechanics is not there and that's just the way it is so what let's just do an application here and think about uh gravity on the earth and you do this calculation and you go oh we involve G uh small number H bar uh small number and we divide by C to the 7th which is a big number it's like whoa you're really trying to make the acceleration I feel on this planet look ridiculously small it's uh 4.42 * 10 to the 51 minus 51 okay I actually went around for a week going I I'm not comfortable with this idea I I'm so used to hearing 9.8 m/s squared well actually earlier in my education I think I I learned it was 32 feet perss squared and now you're telling me or I'm telling myself actually um it's got no units it's like that's okay all right and it's like but how does that fit in with everything else that I know that's what I was struggling with I mean if I can't use my meter stick well I can use your meter you can use your meter stick to make the measurement you can use a watch with second hands to make the measurement but you just report it um in a way that it just has no units and it's like a really small number now think about this in terms of the message we're getting from cosmologists about the role that the planet Earth plays within the entire span of the universe and you go God it is such a small role uh it's it's ridiculous how incredibly insignificant and actually words fail here to say how stupidly small the Earth is okay and so actually this looks much more consistent with that cosmal logical message I mean a good 25 30% of the global economy is based on moving stuff around in in in our our gravitational field here but in terms of the cosmos our gravitational field is like just just really really weeny and this number is really really weeny and so it's it's consistent uh uh with that message and the other thing I really like about it is that all those zeros are kind of the the the signal that we're dealing with something that's totally classical if somebody says is this equation totally classical I mean I don't know how to tell unless I kind of see it you know broken into its generators and if I see my the generators every one of got a zero in it I'm go you're doing some classical physics there now it's perfectly fine thing to do I mean I'm not no not disparaging classical physics in any way it's just that it's nice to have a system to be able to spot it at work all right so now we're going to switch over to quantum mechanics and we're going to try to write Schrodinger's equation using a full four derivative and a three derivative uh specifically a spatial derivative so here's the standard way of writing the Schrodinger equation that you'll find in Wikipedia and um and good sources of information um and you'll say oh well the spatial uh derivative is on one side side and the time derivative is on the other well that is not consistent with the rules as I have laid them out um so I'm going to have to bring that over going to put them together and there's that factor of I so we can just cross that one out um but the you see that's a second spatial derivative but only a first derivative of time and that's been know since Ringer's di that there's that that first order versus second order uh thing going on now if you go and you make the querian operators to pull this thing off the dimensionless ones okay so that um harar T um with a Time derivative has units of energy and so that g Square < TK of G over H Bar C 5 is actually one over uh plun energy and so that thing has units of nothing no units okay um and that's a full four derivative because we've got that Dell operator there but then the next term well it's got Unity okay as as a scaler we're not taking No Time derivative okay so it's just a one um and then we've got this Dell thing and to make the Dell thing dimensionless okay we have to come up with um well there there are different ways to do it but this is one way throw in um an H over MC and if you do that you'll realize oh the M's cancel oh the times cancel oh I only have a length and since the Dell is going to get me a one over length that's going to be dimensionless very good um and then you go hold it and then you just kind of repeat it again well I'm repeating it again uh not quite I throw in an extra conjugate operator what's that going to do well that's going to wipe out a um a three vector and as you can see in this second line there um there is a three Vector there that's got uh that's just a zero sort of thing and you see this scalar operator that looks like it's going to be able to do the job of the schinger equation because you've got a single um time derivative with a single harar you've got an har squared going on um now the one thing I know people are going to say well oh but do you have your factors of I all taken care of in the correct kind of way because the dell the the the the spatial derivative doesn't have an I factor um the time does maybe you're you're messing up in that way and I think the only real solid uh answer to that question is uh let's just wait I am through page three of the book okay and it is my full intention to go through the I don't know 180 190 uh pages of the book and do everything you know I like to make really rigid rules it's like if I can't do something then the hypo ois is wrong okay I can't pull off this this transition to a querian Quantum field Theory um so if you don't want to bet uh on my work that that's fine um but I have always on this project it's just kind of like wow that's a really really hard thing to do and if I give it enough time and Imagination it it it seems to turn out um and I think it's just it just speaks of of of more elegance and more potential information than standard physics because I mean that zero doesn't look good to me to my eye um perhaps you could say Well it has to do with this the schinger being kind of classical quantum mechanics uh I'm a little suspicious I I think that we might be able to get more information about uh Quantum systems if we just didn't uh pretend like it wasn't there and this is actually a recurring theme um that phases are are often just ignored I mean so many people have written the Schrodinger equation like that and there's just nothing wrong with also having a phase factor in there uh but it ain't there oh and believe me huge volumes of work are done um on uh with Schrodinger's equation as is and when I look at it I see the zeros and I get a little concerned all right but that is classical uh field Theory which applies to a particle so now we're going to make the uh switch over up as it were um to the Klein G an equation which actually keeps everything even the phase so this is uh the way it's written in the standard form you've got a second time derivative you've got your Dell operator you've got this um this Mass sort of uh thing squared everybody's squared everybody X on fee and people say isn't that beautiful and I say well actually it's not beautiful enough because you should keep your time and space derivatives together I'm sorry if I'm sounding like a broken record uh I'm a consistent broken record okay um what turns out to be really cool to me anyway is that you're basically calculating e SAR minus um p^2 uh which as we know is a relativistic uh um no which is the luren and variant Mass we're doing this using operators um but then we're kind of dividing through by the mass that that's what the mc^ squ is and we've got Unity over there and unity is so important in Quantum field Theory because uh you think of the groups like U1 and su2 and su3 and they're always saying I'm one I'm one I'm one in different ways you know uh but that that comes up time and again and um of course somebody being observant I hope you're observant um out there they go uhoh uhoh you got the sign of mess wrong therefore you're wrong you're an idiot you're complete fool um because we've dealt with this signs of mass uh before and it's supposed to be um it's supposed to be a minus sign um on the other side so that when you bring it over to this side it would be a plus sign and you have zero um again I'm going to use the same um I don't call it a defense I I I I mean I'm I'm going to just say I got to see if I can solve actual problems uh with this as it is written this seems the most natural to me when I now look at the Klein Gordon equation and I see you I mean it just doesn't make sense to my eye you put scalers with three vectors okay okay that's what DT andell's about but you can't just tack on something else as it done that that goes on the other side of the darn equation you know um this equation just I mean the the phase of this just looks gorgeous I mean 2 Beta gamma squared I mean there's something cool about that there's something cool about this time space face derivative having to do with the phas you know it's not completely about changing in space it's not completely about changing in time it's the two working together to tell you something about the phase of the wave function I mean I want to know what that can tell me about systems and it could be that my education is not brought enough but I don't know that many people discuss that sort of thing and the reason I feel compelled to discuss it is because it comes up so naturally with this sort of formulation all right so that is my introduction to the method and I'm just going to give you a bonus issue and that is thinking about SpaceTime time graphs because you remember how rigid I said the rules were it's like you don't get a choice about making things dimensionless well think about an event in SpaceTime okay and it's got a time and it's got a space a location and if you divide the time by the plank time and you divide the space by by the plon length then you will have A dimensionless Spacetime number now again these numbers will be really small but if you're just thinking about an event being a finger snap it better be really small okay CU finger Snaps are not important you know Supernova that a whole galaxy super cholester colliding to with something else those are those matter th those should have some significant numbers but the stuff that we can do in the lab they better be vanishingly small and well now they are ordinary everyday activity um on the plank scale no just made dimensionless by using these tools I mean the core reason I do that I think is because what you do with mathematics is you often say well why don't I just take the exponential of this that or the other thing you know and why don't I take the cosine of this well you can actually only do that if what you're dealing with is dimensionless okay because what happens with um with exponentials is you know you look at the tailor series and it's one and then it's got this thing squared and it's got this one you know to the fourth power it's got this one to the eighth power you know I think they're odd ones too whatever um but you're used to seeing that in math okay but if you've got something that's got some units to it you can't do anything you know you can't say and I added the uh this x to this one squared no oh except the units no longer work if everything you do is dimensionless then you can do all this funky math uh to it all this kind of standard math um and I do encourage you to uh take a look at this graph because it's kind of fun um it's got SS cosiness and tangents as we all know those being the ratio of two sides but what's far less known or at least I never really heard anybody discuss it is that hyperbolic signs hyperbolic cosin you know those were always like oh I don't want to learn that that's too hard well doing this graph I go no it's easier it's just like the sign it's just the opposite side of the triangle where the triangle is actually not built on a circle anymore it's yes it's built on a hyperbole okay which is pretty similar to a circle except it's got a minus sign but otherwise it's the same and the cosine is going to just be the the close side the tangent is just going to be the the hypotenuse it's like wow that's that's kind of concise that's kind of nice I kind of like it all right so um and I usually end up uh with a little slide um about myself and say goodbye I would just like to point out though that this is a big research project in terms of how how much stuff uh has to be done in the future in in order to to validate it you know I mean I really have to do calcul ction with my uh Schrodinger equation to prove that you know the factor of I thing doesn't make it just complete gibberish all right I have to show uh that my Klein Gordon equation even though it's got the wrong sign um actually when you go do the nuts and bolts calculation ends up with completely sensible consistent with our standard approach sort of uh um sort of stuff um and that's a lot of work and so I've set up this website visual physics.org um it shows up in bold there with a preprint server so that as I figure out stuff I'm going to post it there cuz I can't post to uh the the official preit server I am not an official physicist and so um they they don't let me go there and you know doesn't bother me we haven't um we don't have enough substance yet at this point uh to say we've got a really valid alternative um but it's going uh pretty well uh with the uh I have moved beyond the F first three pages um but I keep going back to them actually and just making sure that it's just refined as as can be um it's been a lot of fun and I hope you've enjoyed this little introduction uh to querian Quantum field Theory thank you very much