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Natural Science Forum / Physics / Relativity / June 2005the basis of relativity
relativity is based on the equivalent principle, then the developed relativity shows that the equivalent principle is wrong > relativity is based on the equivalent principle, > then the developed relativity shows that the equivalent > principle is wrong You misunderstand. The equivalence principle states that given a dynamic gravitational field ("real force") in a given geometry you can alter the geometry and alter the dynamic force to yield an equivalent predictive theory. Given this then you can *by convention* choose a geometry in which the dynamic force goes away and in that choice of geometry the gravitational force is just geodesic evolution. It is similar to saying you can set "zero electrostatic potential" to be at any point you like. When doing problems you set the ground of your device to be at zero volts. That is another "relativity principle" namely that voltage is relative and thus it is only meaningful to speak of voltage differences. Setting the ground to be zero volts does not mean the original relativity is wrong, it rather relies implicitly on the relativity principle being right. Otherwise you'd have to worry about whether the ground is "really at zero volts". These choices of convention are loosly refered to as "gauge conditions". There is a deep connection between "equivalence principles" and gauge theories. One is effectively considering a whole class of equivalent models with an explicit group of equivalence transformations (the gauge group). One then insists that physical phenomena which one may predict be independent of the choice of model (choice of gauge). This is how Einstein formulated his field equations. Note however that in the case of gravitation the purely geometric formulation leads some to take geometry too seriously as a physical quality instead of a feature of the formal language. Hence attempts to quantize gravitation by "quantizing geometry". This I believe to be the major flaw of quantum string and 'brane' models inaccurately refered to as "theories".  SignatureRegards, James Baugh > > relativity is based on the equivalent principle, > > then the developed relativity shows that the equivalent [quoted text clipped - 4 lines] > a given geometry you can alter the geometry and alter the > dynamic force to yield an equivalent predictive theory. are the "real forces" considered "dynamic gravitational fields"? I thought that there was a huge difference between gravitational and a Newtonian force > Given this then you can *by convention* choose a geometry > in which the dynamic force goes away and in that choice of > geometry the gravitational force is just geodesic evolution. "geodesic evolution" means no Newtonian forces? > It is similar to saying you can set "zero electrostatic potential" > to be at any point you like. When doing problems you set the ground > of your device to be at zero volts. That is another "relativity > principle" namely that voltage is relative and thus it is only > meaningful to speak of voltage differences. thanks, but I still can't see the connection between the two forces > Setting the ground to be zero volts does not mean the original > relativity is wrong, it rather relies implicitly on the relativity > principle being right. Otherwise you'd have to worry about whether > the ground is "really at zero volts". you sounds convincing, but I still can't understand > These choices of convention are loosly refered to as "gauge conditions". > There is a deep connection between "equivalence principles" and gauge > theories. One is effectively considering a whole class of equivalent > models with an explicit group of equivalence transformations (the gauge > group). One then insists that physical phenomena which one may predict > be independent of the choice of model (choice of gauge). I think I begin to understand, thanks > This is how Einstein formulated his field equations. > [quoted text clipped - 8 lines] > Regards, > James Baugh I understand now, so the "forces" are actually the same type of forces depending on one's point of view, thanks Let me clearify further. My analogy with regard to electrostatic potential was aimed at the point of the relativity being broken by fixing an aspect of the theory by convention. This was in response to your claim that the theory contradicted itself in its practice. With regard to the theory of gravity as I described it. The point is that the equivalence principle states that you can either treat gravity as a "real" force or as a pseudo-force or as a hybrad of the two. You can't distinguish between a "real" gravitational force and a pseudo-force due to curvature. (curving time coordinates is equivalent to accelerating the frame.) It is not completely correct to say gravity is "just geometry" rather one should say gravity is indistinguishable from geometry. It is a subtile but possibly important distinction. Take some solution to Einstein's equations, then perturb the geometry but at the same time "add by hand" an additional field of forces in such a way that the combination predicts particles will follow the original paths. You have the same theory with slightly changed metaphysical interpertation. Since it is redundant it is just as well to only work with purely geometric form. But it is by no means an affirmation of metaphysical facts. You can look at the perturbative analysis of gravity waves as an example of a hybrid description of both geometric and dynamic components to the gravitational field. You can also look at it as simply "all geometry" but treated perturbatively which is the usual "interpretation". The point is that neither "interpretation" is a true interpretation. The true interpretation is that test particles will go "that-a-way" in the presence of matter distributions as predicted by the theory. >>>relativity is based on the equivalent principle, >>>then the developed relativity shows that the equivalent [quoted text clipped - 68 lines] > I understand now, so the "forces" are actually the same type of forces > depending on one's point of view, thanks SignatureRegards, James Baugh > Let me clearify further. My analogy with regard to electrostatic > potential was aimed at the point of the relativity being broken by [quoted text clipped - 19 lines] > it is just as well to only work with purely geometric form. > But it is by no means an affirmation of metaphysical facts. Just a quick input James, been following your posts and I think you're quite smart! > You can look at the perturbative analysis of gravity waves as an example > of a hybrid description of both geometric and dynamic components to the [quoted text clipped - 3 lines] > The true interpretation is that test particles will go "that-a-way" > in the presence of matter distributions as predicted by the theory. S Weinberg's writes similiar to James about geometrization of gravitation, and I rarely disagree with SW. however, we have extreme experimental evidence that only 3 spatial dimensions (by testing freedom of movement) exist. Also, that movement requires a real time. I respect James and SW's open mindness but I regard that as a dangerous philosophy. It is dismissive of operations in curved 4D, as being real. We've worked hard to define spacetime and we've measured carefully the effects of gravity on light, like deflection, Shapiro, Pound-Rebka, etc...where light defines our viewpoint. For those reasons, there is no way I'll reconsider the idea of spacetime being an imaginary frame for solving physics problems, spacetime is real. Personally, I don't buy the idea of a slow divorce from reality to suck up some math, on the contrary I would have the logic of math confirmed by Nature, and not the other way around to fit our fantasies...keep that going and we're back to the idiot Catholics who decided creation happened in 4004 BC, and a lot more dummy poop the pope sells to flockies. I meant that paragraph to be severe, because science must retain a firm foot in measureable reality. Everyone reading this post has access to a clock and ruler, and thus we all share spacetime, that non-negotiable. Regards Ken S. Tucker PS: Once again I think Mr. Baugh posts well. >>Let me clearify further. My analogy with regard to electrostatic >>potential was aimed at the point of the relativity being broken by [quoted text clipped - 36 lines] > (by testing freedom of movement) exist. Also, that movement > requires a real time. You've neglected some other degrees of freedom, namely rotation and Lorentz boosts, phase translation, boson-number etc... My point being that the dimensionality you so carefully verify from experiment is that of the group acting on the object. Only in the singularity of the non-semi-simple Poincare group do we see a distinction between translational and rotational-boost transformations. This singularity is lost say in a deSitter model and one man's translation is another man's boost. I see the space-time dimensions as simply the dimensions of a normal subgroup of the group of observable transformations when you choose a highly singular perspective. One may always add to this group and subtract from it. The essential question is how we classify physical systems and how the groups we choose transform between them. > I respect James and SW's open mindness but I regard that as a > dangerous philosophy. It is dismissive of operations in curved > 4D, as being real. I don't dismiss the operations I dismiss the *fundamental* necessity of a specific geometric model in which the operations must be embedded. Rather the operations are the primary elements of actuality. How we describe their relationships to each other gives us a topological dimension. But this is a dimension of parameters of description. On a pragmatic level a four dimensional space-time model is highly useful and descriptive. It however invokes aprior assumptions which may cease to be valid when for example we consider the interior of the nucleii of atoms where quarks and gluons are said to swim. > We've worked hard to define spacetime and > we've measured carefully the effects of gravity on light, like > deflection, Shapiro, Pound-Rebka, etc...where light defines our > viewpoint. Yes but these careful measurements needn't rely on a specific choice of space-time geometry (or connection). The question is begged as to whether the hard work in defining a space-time itself is fruitful. > For those reasons, there is no way I'll reconsider the idea of > spacetime being an imaginary frame for solving physics problems, > spacetime is real. As a personal opinion that's fine. As a scientific debate it is moot. One does not observe space-time points, one observes events occuring to objects to which we assign space-time coordinates. > Personally, I don't buy the idea of a slow > divorce from reality to suck up some math, on the contrary I [quoted text clipped - 9 lines] > Regards > Ken S. Tucker I don't totally disagree with your intent and severity. I think it is slightly off aim. I've picked up the language of my thesis advisor and mentor of distinguishing between *actuality* which consists of the events of measurement and interaction i.e. what some call "phenomenological reality" and the meta-physical objective *reality* we may imagine underneath. Empiricism cannot see beyond the *actuality*, we must build a *reality* from our imagination. That's well and good provided we recognize the source. The crucial point I think you want to make is not to mistake mathematics for physics. But I would point out that this is more often done by presupposing that the mathematics represents a metaphysical construct e.g. a manifold or a "super-string" which is out. One gets into silly arguments about trans-finite cardinalities when the actuality we experience in any given experiment is finite in nature. What is measured and observed are the behaviors and phenomena which we associate with that object in a reality model. It is important to put the phenomena first so that we don't inadvertently cross the line of scale or scope where the objective reality assumptions cease to be valid. This is precisely what is happening in the so called paradox of Schrodinger's Cat. One mistakes superpositions of mathematical description for superpositions of "realities". > PS: Once again I think Mr. Baugh posts well. Thank you. SignatureRegards, James Baugh Studied J. Baugh's post, and stretched a few neurons:-). > >>Let me clearify further. My analogy with regard to electrostatic > >>potential was aimed at the point of the relativity being broken by [quoted text clipped - 61 lines] > describe their relationships to each other gives us a topological > dimension. But this is a dimension of parameters of description. Ok, I think we're using different definitions of the word *dimension*, ugh, a possible sematic misunderstanding. Coming from a tensor background, I generally understand the summation of the indexs as the dimesionality, eg. ds^2 = g_uv dx^u dx^v where u,v sum over {0,1,2,3}, and the 0,1,2,3 are the dimensions, (I do respect Kaluza's 5D too). That's a fairly clear definition of dimension. But I think you (James) have a different definition, if so could you define it? I find the context alone a little to difficult to see the re-definition. > On a pragmatic level a four dimensional space-time model is highly > useful and descriptive. It however invokes aprior assumptions which [quoted text clipped - 10 lines] > begged as to whether the hard work in defining a space-time itself > is fruitful. The unification of Spacetime by the Lorentz transform and better defined by Minkowski, was the basis of the unification of the Mass-Energy Conservation Law. GR went further and Energy-Mass and Spacetime using G_uv=T_uv were unified, though not as thoroughly as many would like to see. From that, theoreticians and experimentalists are able to compare results, so it has been fruitful. > > For those reasons, there is no way I'll reconsider the idea of > > spacetime being an imaginary frame for solving physics problems, [quoted text clipped - 26 lines] > the *actuality*, we must build a *reality* from our imagination. > That's well and good provided we recognize the source. Interesting paragraph. I sometimes study "Occurance" in place of "events", by defining a change in an "Occurance" as, DO = Dx Dy Dz Ds DE , where Ds is a proper time increment and DE is D(Energy) corresponding to and Energy increment like a photon. > The crucial point I think you want to make is not to mistake mathematics > for physics. But I would point out that this is more often done by [quoted text clipped - 12 lines] > One mistakes superpositions of mathematical description for > superpositions of "realities". Yes to the above. Thanks for your time. ((my brain hurts :-)) Ken S. Tucker > Studied J. Baugh's post, and stretched a few neurons:-). > [quoted text clipped - 70 lines] > > ds^2 = g_uv dx^u dx^v where u,v sum over {0,1,2,3}, Fine in which case the dimension of dx is 4 being as it is a 4 dimensional representation of Lorentz (and of SL(4,R)) while g_{uv} is a 10 dimensional object. (9+1 dimensional reducible rep of SO(3,1) or SL(4,R).) But when you write down g_{uv}(x) you are really dealing with an infinite dimensional representation of the infinite dimensional diffeomorphism. You can view (x) as another index ranging over infinite index values. My point being that refering to the index is simply refering to the dimension of a specific representation of a specific group. > and the 0,1,2,3 are the dimensions, (I do respect Kaluza's > 5D too). That's a fairly clear definition of dimension. My point being that dimension is a mathematical quality not a physical one. The reason we "see" three spatial dimensions is due to our small scale relative to the (hypothesized) radius of the spatial universe which is to say our extreme representation under the group SO(4) of spatial translations-rotations for a closed spatial universe. We are condensed so that three of the six parameters (R_wx R_wy and R_wz) of SO(4) appear to be "straight" (P_x,P_y,P_z) and at this scale the universe appears flat e.g. SO(4) ~ ISO(3). Where R is rotation generator and P is \epsilon R the infinitesimalized rotations which commute to order \epsilon. Just ignoring time for a moment, consider a Kaluza-Klein type E-M theory, we may also assert that the real spatial group is SO(5) with two scale parameters separating spatial translations (P_k = R_{k4}) and spatial-phase translation rates (A_k = R_{k5}) and spatial rotations. Or maybe its SO(4,1) with pseudo-rotations representing A_k. Or maybe we must also include additional terms for the other gauge groups... or maybe these other gauge groups represent strange combinations of prior transformations but occuring at more than one point at a time (i.e. weak and strong forces are emergent from composite structures). Then of course no particle isolated at a point. It invokes a field in its environment. The number of parameters necessary to give a complete state of an electron is huge (usually taken to be infinite). If you write its vector down you will need an index summing over all momenta or all positions. > But I think you (James) have a different definition, > if so could you define it? I find the context alone > a little to difficult to see the re-definition. The dimension of space is 3 the dimension of space-time is 4, etc. But space and space-time are model elements not physical entities. The dimension of an electron? I don't know? An electron has many properties which can range over different ranges of values. An electron can also be considered as a specific case of more general system...i.e. a lepton in which case the number of properties with variable values goes up. Likewise an electron can be considered as a special case of an ensemble of leptons in which case the variable values go up even more... We define dimension by defining our system. Once you consider quantized properties you loose the topology which defines the dimension. You can range over the discrete values with one index of five or as many as there are values. But I think arguments over "how many dimensions are there really" are not fruitful. Identify the appropriate transformations groups of your theory identify the representations of your system and interpert it all in terms of operational laboritory actions and measurements. See if your theory is better than the last at predicting phenomena and as you seek new theories don't get hung up on metaphysical questions. >>On a pragmatic level a four dimensional space-time model is highly >>useful and descriptive. It however invokes aprior assumptions which [quoted text clipped - 19 lines] > From that, theoreticians and experimentalists are > able to compare results, so it has been fruitful. Yes. Possibly I'm pushing the point too far. Models are useful, and even if there is no space-time manifold "out there" there is necessarily a mathematical "space-time manifold". Just as it is fruitful to construct the real number system it is fruitful to build a space-time manifold. But I still assert its utility is as a mathematical construct (with specific physical application) but not as a metaphysical reality. To the contrary the reification of space-time may trap us into a class of theories which are not the most predictively accurate. It leads to field theories which don't quantize without serous "massaging" of the formal language to remove the divergences and in the ultimate case of a QFT for gravitation we can't. >>>For those reasons, there is no way I'll reconsider the idea of >>>spacetime being an imaginary frame for solving physics problems, [quoted text clipped - 34 lines] > where Ds is a proper time increment and DE is D(Energy) > corresponding to and Energy increment like a photon. Interesting... but I don't think analogous. I considered "Euclideanizing" SR by working with (x,y,z,s) but of course one must be careful in that two particles may follow different paths starting at (x0,y0,z0,0) then later correspond in all four (x,y,z,s) coordinates, they however do not come close to each other physically unless the path-lengths t1 and t2 are close at this "occurance". I.e. locality is not manifest in these coordinates. >>The crucial point I think you want to make is not to mistake mathematics >>for physics. But I would point out that this is more often done by [quoted text clipped - 18 lines] > ((my brain hurts :-)) > Ken S. Tucker SignatureRegards, James Baugh Studied J. Baugh's post, and stretched a few more neurons:-). [snip old stuff] > > Ok, I think we're using different definitions of the word > > *dimension*, ugh, a possible sematic misunderstanding. [quoted text clipped - 6 lines] > dimensional representation of Lorentz (and of SL(4,R)) while g_{uv} is a > 10 dimensional object. That seems to be a unique view of dimension, I've never heard g_uv being called a 10D object. > (9+1 dimensional reducible rep of SO(3,1) > or SL(4,R).) [quoted text clipped - 28 lines] > rotations. Or maybe its SO(4,1) with pseudo-rotations representing > A_k. > Or maybe we must also include additional terms for the other gauge > groups... or maybe these other gauge groups represent strange [quoted text clipped - 7 lines] > to be infinite). If you write its vector down you will need an index > summing over all momenta or all positions. > > But I think you (James) have a different definition, > > if so could you define it? I find the context alone [quoted text clipped - 4 lines] > The dimension of an electron? I don't know? An electron has many > properties which can range over different ranges of values. What I like about this (James) approach is it reminds me of a computer. Having programed complex sims I would ARRAY(a,b,c,d,e,f...) as I please, as a,b... represent different characteristics of a cyber object. Ultimately we would like to make a computer program that sims the universes physical laws, and provide correct answers. That computer running the program will have no need of conscious, but only provide the correct result. It would unburdened by our prejudiced traditions for solving problems. > An electron can also be considered as a specific case of more general > system...i.e. a lepton in which case the number of properties with [quoted text clipped - 15 lines] > and as you seek new theories don't get hung up on metaphysical > questions. Fair enough. [...] > > The unification of Spacetime by the Lorentz transform > > and better defined by Minkowski, was the basis of the [quoted text clipped - 14 lines] > of space-time may trap us into a class of theories which are not > the most predictively accurate. Ok. > It leads to field theories which > don't quantize without serous "massaging" of the formal language > to remove the divergences and in the ultimate case of a QFT for > gravitation we can't. That might be matter of interpretation. [...] > > Interesting paragraph. I sometimes study "Occurance" in place > > of "events", by defining a change in an "Occurance" as, [quoted text clipped - 12 lines] > close at this "occurance". I.e. locality is not manifest > in these coordinates. Well, an event is defined by a *point* in 4 spacetime coordinates, however that is idealized. What really happens is a measureable Occurance. That Occurance requires some change in Energy in a finite Volume in a finite time, respecting QT. > Regards, > James Baugh Thanks, I hope the snips were ok... Ken S. Tucker > [big snip] > > Thanks, I hope the snips were ok... > Ken S. Tucker Yea, I a bit talked out on the subject but you've gotten me to thinking about some things I put aside to finish the GDT (Gawd Damned Thesis ;-) Thanks for the neural stimulation. If I can get something fruitful from it I'll post...(or publish). SignatureRegards, James Baugh Ok James... > > [big snip] > > [quoted text clipped - 10 lines] > Regards, > James Baugh I've studied "Partial Interdimensial transformations". In that case the summations of indices are not whole numbers. Look at it this way James, you crashed what dimensionality means from your PoV, (Point of View), and I've questioned something similiar. By convention we order and indice "u" it be an integer, in a tensor equation like, A = e_u A^u where u=0,1,2,3. (spacetime) Parallel to your questions of dimensionality, is it reasonable to insist "u" must be an integer series?! Who decided that rule/law of mathematics?! (?! == rhetorical). The reason I developed *partial interdimensional transformations* is based on physics experiments. Watch this...to obtain the perihelion shift (43" per century) of Mercury's orbit we can modify Newtons Law by, f = GMm/r^(2.000 000 2). See that?! that cabouse on the exponent varies the orbit into a precession. Furthermore, we replace the fixed dimensionality of r^2 by a fluid, varing dimensionity by noting the following, r^2 == Area, r^3=Volume , what's something between those two, like r^(2.000 000 2) ?! See that, we're doing a *partial inter-dimensional transformation* as we get closer to the gravitating body. The significance of that is the ripping of the spacetime field into nonorthogonality. Specifically, the metric components, time g_00 and radius g_11 deviate in proportion to the induction of a nonorthogonal spacetime field, that's equally the same as a *partial interdimensional tranny*. In math that easy to see, i.e. g_00 => 1 - 2m/r , g_11 => 1 + 2m/r where the mass "m" rips spacetime. Suppose I asked, what is the dimensionality of r^2.5 ?! That's sensible given that the above has been experimentally verified, (precession). There is no implied mystery. My best description is using the g= det|g_uv| to descibe the above. Regards and best wishes, Ken S. Tucker PUN warning: you use the word "dimension" in at leeast three different ways, without mentioning that fact. While I suspect you are aware of the differences, some readers clearly are not. Unacknowledged puns like this are not conducive to good communication. >>> I see the space-time dimensions as simply the dimensions of a normal >>> subgroup of the group of observable transformations when you choose a >>> highly singular perspective. That is different from everyone else. The standard definition of "dimension of spacetime" is the number of coordinates required to specify a unique point in the spacetime manifold. In GR this is 4. Spacetime _is_ a manifold, and the dimension of the manifold is necessarily the dimension of spacetime. Note, please, that "the group of observable transformations" on spacetime is infinite dimensional. I don't know what you mean by "a highly singular perspective", but I suspect there is a pun on "singular" in there.... > the dimension of dx is 4 being as it is a 4 > dimensional representation of Lorentz (and of SL(4,R)) while g_{uv} is a > 10 dimensional object. Here is your pun -- nobody else considers g_{uv} to be "10 dimensional". Instead we say that the {g_uv} has 10 algebraically-independent values (at each point). Using the word "dimension" here introduces needless confusion. Because you made an error: > (9+1 dimensional reducible rep of SO(3,1) > or SL(4,R).) g_{uv} is not at all any representation of any transformation group. It is the components of a tensor field on spacetime. > But when you write down g_{uv}(x) you are really dealing > with an infinite dimensional representation of the infinite > dimensional diffeomorphism. WHOA! There is no diffeomorphism here. A diffeomorphism is a map, but the g_{uv}(x) are simply the components of a tensor field on the manifold. Those are completely different concepts, and the metric or its components are in no way, shape, or form any sort of "representation" of any diffeomorphism. > You can view (x) as another > index ranging over infinite index values. No. x (which is really 4 values) specifies a specific point in the manifold, which is a completely different concept than an index of tensor components. Yes, the real value g_uv(x) depends on u, on v, and on x (point in the manifold), but these dependencies are quite different: in particular x is differentiable but u and v are integers. > My point being that dimension is a mathematical quality not a physical > one. The reason we "see" three spatial dimensions is due to our > small scale relative to the (hypothesized) radius of the spatial > universe which is to say our extreme representation under the > group SO(4) of spatial translations-rotations for a closed spatial > universe. No. We "see" 3 spatial dimensions because it requires 3 real numbers to specify a unique point in the spatial manifold. This has nothing to do with "small scale" relative to anything, nor any group of transformations (except that there is a direct relationship to the cardinality of the group of translations). > Then of course no particle isolated at a point. It invokes a > field in its environment. The number of parameters necessary > to give a complete state of an electron is huge (usually taken > to be infinite). If you write its vector down you will need an index > summing over all momenta or all positions. This depends on your theoretical context: Classically the state of an electron (or any other pointlike particle) at a given time can be expressed in 6 real numbers ({x,y,z} and {Px,Py,Pz}) plus whatever internal state it may have (classically, none); if you include the EM field then _the_field_ is infinite dimensional[#] (not the electron itself). But quantum mechanically you are correct, in that the state of an electron involves a wave function on the manifold, which is of course infinite dimensional[#], plus internal state (spin). [#] Beware - this "dimension" is not the spacetime dimension, but rather the dimension of a Hilbert space. > The dimension of space is 3 the dimension of space-time is 4, etc. > But space and space-time are model elements not physical entities. Yes. This is the common usage of "dimension". > The dimension of an electron? This is _NOT_ the standard usage of "dimension", unless you mean "size" (which you clearly do not). Trying to apply that word to a particle does not make sense, unless you specifically describe to what you are applying the word (e.g. to the Hilbert space of its wave function). > Models are useful, > and even if there is no space-time manifold "out there" there is > necessarily a mathematical "space-time manifold". Yes. All we can do in physics is discuss _models_ of the world. At present, every fundamental physical theory starts with a spacetime manifold. > To the contrary the reification > of space-time may trap us into a class of theories which are not > the most predictively accurate. Spacetime need not be reified to be useful as a _model_ of the world. IN fact the usual interpretation is that the manifold is not itself "real", but the spatial and temporal relationships between objects are (and they are what is summarized by the spacetime manifold and its metric). > It leads to field theories which > don't quantize without serous "massaging" of the formal language > to remove the divergences and in the ultimate case of a QFT for > gravitation we can't. Perhaps QFT is not the right context for "quantum gravity". Certainly the currently most promising-looking approaches to QG do not involve QFT.... QFT is a low-energy limit of all of them, becuase QFT works so well at the energies we can probe with current experiments. Tom Roberts tjroberts@lucent.com > PUN warning: you use the word "dimension" in at leeast three different > ways, without mentioning that fact. While I suspect you are aware of the [quoted text clipped - 11 lines] > manifold. In GR this is 4. Spacetime _is_ a manifold, and the dimension > of the manifold is necessarily the dimension of spacetime. Hence the part of my post you didn't quote: "The dimension of space is 3 the dimension of space-time is 4, etc." > Note, please, that "the group of observable transformations" on > spacetime is infinite dimensional. I don't know what you mean by "a > highly singular perspective", but I suspect there is a pun on "singular" > in there.... In considering deformations of Lie groups the semi-simple Lie groups are stable. The non-semi-simple Lie groups (except a few small dimensional cases) are not stable. In my thesis research we referred to these cases as regular and singular respectively. Hence as you walk upon the earth and your scale becomes infinitesimal relative to the earths you mistake the regular group SO(3) for a singular group ISO(2) due to your singular (infinitesimally small) perspective. In the extreme case a Lie algebra is the abelianized form of the Lie group under addition. This again due to the singular scale of the generators relative to the group elements. You are looking at the group only in a tiny neighborhood of the origin = identity. The process of deformation of a stable Lie group such as SO(3) yields again SO(3) unless you deform all the way to the boundary of the stable region. This boundary is singular in the specific mathematical sense, there is a reduction of dimension in the manifold of deformations. Thus any infinitesimal deformation from the boundary of ISO(2) groups will yield with unit probability either again an SO(3) group or SO(2,1) which resides on the other side of that boundary. This is the sense in which I used the term *singular*. It also emerges in the Killing form (metric) of the Lie group which is singular for the non-simple case and non-singular (regular) for the simple case. >> the dimension of dx is 4 being as it is a 4 dimensional representation >> of Lorentz (and of SL(4,R)) while g_{uv} is a 10 dimensional object. > > Here is your pun -- nobody else considers g_{uv} to be "10 dimensional". > Instead we say that the {g_uv} has 10 algebraically-independent values > (at each point). Note I was comparing g_{uv} with dx_{u}. The tangent space *at a point* which has dimension 4 with the space of possible metrics at a point which is larger. Using the word "dimension" here introduces needless > confusion. Because you made an error: > [quoted text clipped - 3 lines] > g_{uv} is not at all any representation of any transformation group. It > is the components of a tensor field on spacetime. A *tensor* is specifically and precisely a representation of a linear group (and thus of its subgroups). When you transform your tangent basis you invoke SL(4,R). The meaning of *covariance* is the representation of the respective field variable for the said same group. >> But when you write down g_{uv}(x) you are really dealing >> with an infinite dimensional representation of the infinite [quoted text clipped - 5 lines] > components are in no way, shape, or form any sort of "representation" of > any diffeomorphism. Pardon, I meant to say infinite dimensional diffeomorphism group, i.e. the group of diffeomorphisms of the space-time manifold. This group may also be defined as the group of analytically connected ISL(4) transformations at each coordinate point which is why we enumerate tensor field components at each point. >> You can view (x) as another >> index ranging over infinite index values. > > No. x (which is really 4 values) . . . Your saying there are only 4 points in all of space-time or only 1? The "index" x has 4 *components* each of which range over an infinite continuum of values. > specifies a specific point in the manifold, > which is a completely different concept than an index of > tensor components. Yes, the real value g_uv(x) depends on u, on v, and > on x (point in the manifold), but these dependencies are quite > different: in particular x is differentiable but u and v are integers. Yes but you impose analyticity conditions on the functions of x. You can resolve f(x) into a countable infinite basis with discrete index. You are representing the same information. >> My point being that dimension is a mathematical quality not a physical >> one. The reason we "see" three spatial dimensions is due to our [quoted text clipped - 8 lines] > transformations (except that there is a direct relationship to the > cardinality of the group of translations). But we don't "see" points we "see" objects and to uniquely identify said objects we need more than 3 parameters. The objects have orientation as well as location, they may also have additional parameters. This is where we mustn't confuse the mathematics for the physics. Mathematically you are absolutely correct. The spatial manifold we invent to describe point particles with no other degrees of freedom is 3 dimensional. But you have (a) by hypothesis chosen the dimension by specifying "no other degrees of freedom" and (b) selected a hypothetical object we simply do not see in nature. It is an idealized limit of objects we do see in nature which have additional degrees of freedom which due to the scales of the interactions we are considering may be ignored but which are there nonetheless. We by this intentional ignorance of other degrees of freedom thereby *choose* the dimension, not observe it. >> Then of course no particle isolated at a point. It invokes a >> field in its environment. The number of parameters necessary [quoted text clipped - 3 lines] > > This depends on your theoretical context: Of course, this is exactly my point, the dimension depends on the theoretical context, it is a quality of the model not of "reality". > Classically the state of an > electron (or any other pointlike particle) at a given time can be [quoted text clipped - 7 lines] > [#] Beware - this "dimension" is not the spacetime > dimension, but rather the dimension of a Hilbert space. Yes and especially beware of the fact that we do not work with points in Hilbert space but rather 1-dim subspaces. Thus when we speak of dimension you need to subtract one. Further beware that the so called "state" vector of the Hilbert space does not describe the physical object but the mode of preparation for the class of physical objects. >> The dimension of space is 3 the dimension of space-time is 4, etc. >> But space and space-time are model elements not physical entities. [quoted text clipped - 15 lines] > present, every fundamental physical theory starts with a spacetime > manifold. I disagree, we can predict actual events, or at least their likelihoods. Sometimes it helps to work in a model when doing this, sometimes it hinders. >> To the contrary the reification >> of space-time may trap us into a class of theories which are not >> the most predictively accurate. > > Spacetime need not be reified to be useful as a _model_ of the world. Precisely my point. I am not proposing that we abandon the space-time model in its proper context. I am proposing that we not confuse the model with "reality". Thus when I point out that you may consider space-time to be a sub-manifold of a group's parameter manifold, I am expanding the model not proposing that "the universe is a Lie group". > IN fact the usual interpretation is that the manifold is not itself "real", You and I understand this point, many do not. > but the spatial and temporal relationships between objects are (and they > are what is summarized by the spacetime manifold and its metric). And I summarize these relationships by specifying the group element necessary to bring the two objects into correspondence. In doing this I am trying to move away from the fiber-bundle structures of the conventional models which impose implicit and unjustified constraints on the generated theories. Field theories and string theories and M-theories are inherently "singular" in the sense I described above. This singularity ... >> It leads to field theories which >> don't quantize without serous "massaging" of the formal language [quoted text clipped - 5 lines] > QFT.... QFT is a low-energy limit of all of them, becuase QFT works so > well at the energies we can probe with current experiments. Yes and the flat earth model works so well at the distances we can walk. I assert that although the current most funded approaches to QG do not invoke QFT they invoke the same pathologies which exist in QFT. The point of string theory done in some specific dimension (11 or what ever) was to hope that these pathologies cancel themselves out. I assert we need to remove them completely. I and some others are working toward that goal. Part of removing these pathologies is to abandon questions of "how many dimensions are there really" and ask questions of "what are the transformations we can effect on elementary particles" and "what is their approximate group structure?" By acknowledging the imprecision of our determination of that structure we cannot make such strong assumptions as that the groups involved reside on the singular boundaries between stable groups. This is precisely the assumption which gives hbar = 0 and classical commuting observables, that gives 1/c = 0 and Galilean relativity with commuting boosts, that gives 1/r = 0 and the flat earth geography with commuting surface translations. In each case a non-semi-simple Lie group and/or Lie algebra with its singular Killing metric is replaced by a simple Lie group/algebra with non-singular Killing metric. SignatureRegards, James Baugh > Let me clearify further. My analogy with regard to electrostatic > potential was aimed at the point of the relativity being broken by [quoted text clipped - 16 lines] > such a way that the combination predicts particles > will follow the original paths. Baugh I must admit I find your responses fascinating, a bit different, and insightful. Often I read them and think I am not so sure about that - then I think a bit more and say he has a point. But with regard to the above even after thinking about it a bit I am not so sure. I am thinking of Kretchmans objections to the principle of general covariance - he showed it had no physical basis - any law can be put in covariant form Which is why we need to add in the requirement of invariance - namely all the absolute terms appearing in the equations remain unchanged (Ohanian - Gravitation and Space-Time - page 374). Doing what you suggest would seem to violate this requirement - or am I missing something? Thanks Bill >You have the same theory with > slightly changed metaphysical interpertation. Since it is redundant [quoted text clipped - 81 lines] > > I understand now, so the "forces" are actually the same type of forces > > depending on one's point of view, thanks >>Let me clearify further. My analogy with regard to electrostatic >>potential was aimed at the point of the relativity being broken by [quoted text clipped - 30 lines] > Thanks > Bill Covariance and invariance imply an underlying group. The "principle of covariance" can be understood simply as a principle of good theory formulation. You should be explicit in how the defined quatities transform with respect to the various transformation groups. In the case of canonical mechanics the invocation of "general covariance" is usually implied to be with respect to diffeomorphisms of either time or space-time. Then of course there is invariance which we must assume for groups of "non-physical transformations", e.g. gauge transformations. These I would say are alterations of the "model" which do not change the "theory". The invarance business thus is a formal way of saying we are not taking the reality of the model too seriously. We pay attention to only those quantities which are "model independent". SignatureRegards, James Baugh Baugh: >Then of course there is invariance which we must assume for groups of >"non-physical transformations", e.g. gauge transformations. We do not ``have'' to assume them. We are perfectly free to to create theories which are not gauge invariant. We assume gauge invariance, because there is a great deal of _physics_ in that assumption. >These >I would say are alterations of the "model" which do not change the >"theory". The invarance business thus is a formal way of saying we >are not taking the reality of the model too seriously. The entire standard model follows from gauge invariance. (I don't simply mean that gauge invariance is an additional assumption which doen't change the theory. Gauge invariance _IS_ the standard model. The gauge fields are called gauge fields because they are the fields required for a gauge invariant theory. The gauge field you get from U(1) invariance is the electromagnetic field, A_u. It's not an additional assumption in qed. It's the assumption responsible for qed. >We pay attention to only those quantities which are "model >independent". I don't think I understand that statement. There's nothing model independent about gauge invariance. The gauge invariance _is_ the model. The gauge degrees of freedom have immediate physical sgnificance. It's the freedom to make a gauge transformation that has no physical significance. For example, the photon is the gauge field, A^u, in E&M. The fact that it's massless implies that there are only two indepemdent degrees of freedom. That in turn, insures that the time component of the four-polarization can always be made to vanish, thus giving us an electromagnetic interaction that conserves charge. A gauge transformation has no physical significance for the simple reason that the time component and longitudinal polarization will either be zero separately or will cancel if p^2 = 0. The other two degrees of freedom are readily observable as the two transverse polarization states of the photon. You can't make all four degrees of freedom vanish. > Baugh: > [quoted text clipped - 5 lines] > gauge invariance, because there is a great deal of _physics_ > in that assumption. For the quantity to be meaningful it *must* be independent of those choices we make in the description which are not physical, i.e. choice of basis, choice of zero phase, etc. Otherwise we "change reality" by doing a little math. You are correct in that we needn't do physics. One can create theories which are not self consistent or unambiguously interpreted. > >These > >I would say are alterations of the "model" which do not change the [quoted text clipped - 4 lines] > simply mean that gauge invariance is an additional assumption which > doen't change the theory. Gauge invariance _IS_ the standard model. Gauge invariance "under a particular group structure". The Model part being the assumption of that specific group structure beyond transformations of observables, e.g. SU(2) isospin gauge and SU(3) color gauge, when we as yet have no mechanism for observing isospin (except the one component contributing to E-M charge) or color directly. But in addition there are ad hoc assumptions which break the gauge symmetries. The model aspect is the assumption of an underlying dynamic structure where these symmetries are restored. It's of course not a bad assumption given the success of the resulting theory. > The gauge fields are called gauge fields because they are the fields > required for a gauge invariant theory. The gauge field you get from > U(1) invariance is the electromagnetic field, A_u. It's not an > additional assumption in qed. It's the assumption responsible for > qed. Yes, good point. I'm thinking more in terms of GR wherein one has recognized the need to impose gauge invariance within a prior theory. (Though the term "gauge" wasn't used.) > >We pay attention to only those quantities which are "model > >independent". > > I don't think I understand that statement. There's nothing model > independent about gauge invariance. The gauge invariance _is_ the model. Yes, I'm playing a bit loose with the term "model" which is why I put it in quotes. The "models" to which I refer are the underlying fiber-bundles of e.g. phases over space-time. The "change of model" to which I refer is the shifting of the fibers at each base-point i.e. the resetting of the zero phase at each space-time point which is the action effected by gauge transformations. This in the classical formulation of electro-dynamics. Or if you prefer consider the fiber-bundle of A fields over space-time. I would also say that in the modern treatment of both classical and quantum electro-dynamics these "models" are ignored almost entirely which is the point of the formal language of gauge theory. My reason for pointing to these "models" is to emphasize their analogue in the geometric formulation of GR. If you unify the fiber-bundle of phases over space-time as in Kaluza-Klein's unification of EM and gravitation you then have in the gauge transformations of standard EM a variation of the geometry of the 5 dimensional K-K manifold. In the K-K theory this is changed to a variation of which submanifold one projects onto to define space-time (at zero phase). Essentially one re parametrizes the this 5-d manifold so that the x_mu in one case is x'_\mu, a mixture of x_mu and theta in the other. It is when we assume that both x and x' are coordinates in a common manifold with unchanged geometry that we must then introduce an additional dynamic force, the E-M field. Alternatively we can as is done in the K-K theory recognize it as a change in the geometry via a change in the choice of sub-manifold defined by theta=0 vs theta'=0. One can always add enough dimensions so that the geometry is fixed and the dynamic forces are merely the non-geodesic components of the curvature of the sub-manifold we identify as space-time. This is what M-Theory is all about. But I assert that this is introducing too many artifacts into the theory when we quantize. I could be wrong and quantum M-theory may result in the final big theory of quantum gravitation-color-isospin-hypercharge-matter, or TOE for short. I'm suggesting that the equivalence principle is not fully integrated into such theories. I'm suggesting that reifying space-time as a physical 'brane' in some larger dimensional space is going to lead to non-renormalizable divergences pretty-much no matter how cleverly it is quantized. I don't have a proof. I simply have an intuition based on my belief that we shouldn't "quantize" the mathematical artifacts of our exposition but rather the physical observables of the theories. > The gauge degrees of freedom have immediate physical sgnificance. Yes but not as physical degrees of freedom. Rather as appended degrees of freedom so that we may embed the physical variables in a canonical structure. The canonical structure is not semi-simple. The Heisenberg relations of p and q do not generate a stable Lie algebra. It is the combination of this artificial format along with the gauge degrees of freedom which together "have immediate physical significance." As a simple example consider a periodic physical degree of freedom. Now force that into a flat model by describing it as a rotation in two space. You must then append the radial scaling as a gauge degree of freedom and you have a gauge constraint R = 1 e.g. by which the point object is rotating on a unit circle in your model. The periodicity is physically significant hence to get your point to be periodic it is necessary to impose the gauge condition which requires you address that gauge degree of freedom. This is a simplistic model but when we consider canonical quantization we embed the particle states in a 2n dimensional extended phase space and then impose constraints to identify a submanifold which corresponds to state space. One in fact has a whole continuum of state spaces parameterized by the gauge variables. This is precisely how and why we effect gauge degrees of freedom when we seek to quantize these classical systems. It all rests on the insistence on P's and Q's as observables obeying the canonical Heisenberg relations and from which we construct the Hamiltonian. But at its heart we are simply identifying the physical transformation group (non-commuting translations over classical state manifold). The canonical method does this within a specific class of (singular) groups. That choice has to do with the singularities of the classical limit not the structure of the underlying quantum physics. If we drop the canonical formalism and simply observe the specific group structure for which the Hamiltonian is a variable generator then the gauge part goes away. You still get the U(1) or SO(2) "gauge" group of EM but it is no longer a *gauge* group (i.e. a set of U(1) groups at each physical point.) but rather simply a (variable) subgroup of the large unitary group of transition probability conserving frame transformations on a given quantum system. > It's the freedom to make a gauge transformation that has no physical significance. Right. > For example, the photon is the gauge field, A^u, in E&M. The fact that > it's massless implies that there are only two indepemdent degrees of [quoted text clipped - 6 lines] > as the two transverse polarization states of the photon. You can't make > all four degrees of freedom vanish. Implicit in your exposition is an underlying canonical structure. [p,q]=i, and the underlying inhomogeneous group ISO(3,1) wherein p^2 is an invariant. Mind you I don't have a replacement theory in which to demonstrate a counter example (yet) but I cannot consider the one instance as a demonstration that no such theory is possible. At the level of the group there is no gauge transformation, only a choice of frame. It is when we project the group representations into the infinite dimensional representation of arbitrary fields over a manifold that we get the infinity of choices which constitute choices of gauge. By excising this model of fields over a base manifold we remove those non-physical gauge components without loosing any of the physics. SignatureRegards, James Baugh Baugh: >> Baugh: >> [quoted text clipped - 9 lines] >those choices we make in the description which are not physical, >i.e. choice of basis, choice of zero phase, etc. Once again, I said nothing to the contrary. You seem to missing the point, entirely. >Otherwise we "change reality" by doing a little math. You are >correct in that we needn't do physics. One can create theories >which are not self consistent or unambiguously interpreted. If you believe that, then why are you having so much difficulty attributing physical significance to the mathematical quantities that make the theory consistent? >> The entire standard model follows from gauge invariance. (I don't >> simply mean that gauge invariance is an additional assumption which >> doen't change the theory. Gauge invariance _IS_ the standard model.) > >Gauge invariance "under a particular group structure". The point being what? Try to find a different group structure that corresponds to physical measurements. If it were that simple, minimal SU(5) would have been guaranteed to work, since the standard model _is_ a _particular_ decomposition of SU(5). However, it did not work. >The Model part being the assumption of that specific group structure >beyond transformations of observables, e.g. SU(2) isospin gauge and SU(3) >color gauge, when we as yet have no mechanism for observing isospin >(except the one component contributing to E-M charge) or color directly. Nonsense. SU(3) color is believed to be an exact symmetry. You can ``observe'' the color degrees of freedom as directly as most anything else observed (I've never ``seen'' an electron either, beyond the predicted behavior of a signal in counter.) The color degrees of freedom can be observed in deep inelastic scattering and electron positron scattering. For example, the branching ratio for e+ + e- -> hadrons vs. e+ e- -> \mu+\mu- differs for a model with and without color degrees of freedom. By a factor of 3. The existence of weak isospin also has direct physical consequencea. (In the futre, please qualify isospin with the proper qualifier, i.e., weak or nuclear. Nuclear isospin is very commony used in nuclear physics and without context, it would not be obvious to which you refer.) The conserved vector current hypothesis (CVC) os probably one of the most aspects of the standard model. It predates the standard model by more than 20 years. It says that the weak interaction is a triplet of vector currents which differ by a rotation of weak isospin. The classic test is a measurement of ft_values for superallowed fermi decays (0+ -> 0+). In fact, if the weak coupling constant G_f is a constant, you've proven the same thing, since you have 3 different currents all with the same coupling. That defines a symmety such as weak isospin. >But in addition there are ad hoc assumptions which break the gauge >symmetries. So what? Everything is ``ad hoc'' until it's either shown to be incorrect or explained by something less ``ad hoc.'' The standard model and general relativity explain a great deal more with a great many fewer ad hoc assumptions. By an yardstick ever used in any physical theory, the quantities in the standard model are justifiably considered physical. >The model aspect is the assumption of an underlying dynamic structure >where these symmetries are restored. Again, so what? (You have that backwards by the way - the dynamical structure is the broken symmetry. The underlying structure is the fundamental part.) >It's of course not a bad assumption given the success of the >resulting theory. On that basis, one could say it's the best assumption in history. >Yes, good point. I'm thinking more in terms of GR wherein one has >recognized the need to impose gauge invariance within a prior theory. >(Though the term "gauge" wasn't used.) But general relativity is a classical theory. The gauge symmetry is evident in maxwell's equations, but in the context of maxwell's equations cnnot be used to predict any physics. That's why the potentials in classical theory are artifacts. In quantum theory, the potentials become interactions coupled to fields. I mean, look, this is about the aharanov-bohm effect. If you can demonstrate how to obtain the shift in the interference pattern using just the E and B field, please do. [...] >I would also say that in the modern treatment of both classical and >quantum electro-dynamics these "models" are ignored almost entirely >which is the point of the formal language of gauge theory. I beg to differ. Allow me to quote a line from the first volume of ``Quantum Fields and Strings: A Course for Mathematicians'', Witten, E., et al, Chapter 1, Classical Field Theory: ``Rather, we can couple [a relativistic particle] to fields, specifically to an electromagnetic field (abelian connection) and to a gravitational field (variable lorentz metric).'' The electromagnetic field here is A_u. By ``potential function,'' they refer to the type of potential one would write in a galilean invariant theory, e.g., U(r) = e^2/r. [...] >gravitation you then have in the gauge transformations of standard EM >a variation of the geometry of the 5 dimensional K-K manifold. Sure. So what? >In the K-K theory this is changed to a variation of which submanifold >one projects onto to define space-time (at zero phase). I think you're getting carried away. Kaluza klein theory is simply a 5-d wave equation that has _exactly_ the same solution as the 4-d wave equation x a phase. The phase is not observable. Moreover, the radius of the compactified dimension was (is) considered to be physically significant. >Essentially one re parametrizes the this 5-d manifold so that the >x_mu in one case is x'_\mu, a mixture of x_mu and theta in the other. In which case, it can't ``project'' out at ``zero phase.'' The phase transforms. It's unobservable. That's not the same as being zero. [...] >One can always add enough dimensions so that the geometry is fixed >and the dynamic forces are merely the non-geodesic components of >the curvature of the sub-manifold we identify as space-time. That's the point of invariance. One is seeking a theory in which there are _no_ forces, per se. >This is what M-Theory is all about. But I assert that this is >introducing too many artifacts into the theory when we quantize. I would assert that no one knows what are artifacts and what deserve status as a genuine part of theory until experiments tell you which is which. In the case of E&M, the four potential _is_ an artifact in _classical_ theory. It would be an artifact in quantum theory, too, were it not for experiments that cannot be explained using E and B. [...] >I'm suggesting that the equivalence principle is not fully >integrated into such theories. Sure they are. The standard model treats all mass as inertial mass. The entire point of creating a single model which is poincare invariant is to obtain poincare invariants. One could possibly define different ``kinds'' of mass, i.e., electromagnetic mass, weak mass and strong mass. If one uses the same definition of spacetime to formulate each of those theories, mass must have the same meaning in all three cases so that mass is just inertial mass. The equivalence principe equates inertial mass with gravitational mass. There is an additional assumption involved: no curvature coupling. Maxwell's equations, for example, differ by a term proportional to the ricci tensor if curvature coupling is allowed. >I'm suggesting that reifying space-time as a physical 'brane' in >some larger dimensional space is going to lead to non-renormalizable >divergences pretty-much no matter how cleverly it is quantized. I >don't have a proof. Either way, it wouldn't matter, since I'm not going to cheerlead for string theory. I'm perfectly willing to believe it, if there is experimental evidence that agrees with predictions unique to string theory, but so far there aren't any (with the possible exception of some qcd sum rule, which has been validated, but as far as I know, hasn't been derived from quantum field theory). Since ed witten knows a great deal more about string theory than I do, I'll leave the believing up to him for the time being. >I simply have an intuition based on my belief that we shouldn't >"quantize" the mathematical artifacts of our exposition but rather >the physical observables of the theories. The aharanov-bohm experiment demonstrates that A is not an artifact. So far, you've said a lot of stuff that is rather straigt forward and uncontoversial, but have not explained the aharanov-bohm effect using only E and B, which you seem to think is possible. If you want to join in such a discussion, you might help out eugene stefanovich on sci.physics.research, who has similar ideas and took his argument there after being shot down here. The two people arguing with him on sci.physics.research seem to have more patience. >> The gauge degrees of freedom have immediate physical sgnificance. > >Yes but not as physical degrees of freedom. I think you should compare theory to experiment a bit more. A four-vector has four degrees of freedom. The photon has two. How does one account for the two missing degrees of freedom? The condition d_u A^u = 0, reduces the number to three. That makes the photon a spin 1. The remaining three degrees of freedom are the three possible polarizations. In order to conserve charge, it must be possible to insure that the time component vanishes. That imposes the additional constraint which eliminates the longitudinal polarization. The absence of a longitudinal polarization means the photon is massless. The two degrees of freedom which remain are the two transverse polarizations. >It is the combination of this artificial format along with the gauge >degrees of freedom which together "have immediate physical significance." I have no idea what that means. Are you saying the electromagnetic field is not a vector field? If it is, then I just described the two degrees of freedom. >As a simple example consider a periodic physical degree of freedom. Now >force that into a flat model by describing it as a rotation in two [quoted text clipped - 4 lines] >necessary to impose the gauge condition which requires you address that >gauge degree of freedom. That's entirely different. What you are describing is hidden variable theory. There is nothing hidden about a gauge field. [...] >> It's the freedom to make a gauge transformation that has no >> physical significance. > >Right. So, what's the problem? Do you also think the lack of an absolute position in spacetime implies the measurement of a distance is an artifact? [...] >Implicit in your exposition is an underlying canonical structure. And? >[p,q]=i, and the underlying inhomogeneous group ISO(3,1) wherein >p^2 is an invariant. No, it doesn't. The mass is casimir operator of the poincare group, just like the spin. >Mind you I don't have a replacement theory in which to demonstrate >a counter example (yet) but I cannot consider the one instance as >a demonstration that no such theory is possible. Quite honestly, I can't figure out what _kind_ of a possible theory you could be talking about, or even the degree to which it would be funda- mental. What you seem to be saying is that you can find a different representation for the same group structure. But that's irrelevant. Physics is represetation independent. One representation can be transformed into a different one. Reality might look different to a lightcone observer than to us, but who cares? >At the level of the group there is no gauge transformation, only a >choice of frame. Nonsense. Gauge transformas are unitary transforms. Explain to me how a unitary transformation is not ``at the level of the group.'' >It is when we project the group representations into >the infinite dimensional representation of arbitrary fields over a >manifold that we get the infinity of choices which constitute choices of >gauge. By excising this model of fields over a base manifold >we remove those non-physical gauge components without loosing any of >the physics. One ``removes those non-physical gauge components'' by insisting the physical results are covariant and finite. That doesn't imply you remove the physical gauge components. > The point being what? Try to find a different group structure that > corresponds to physical measurements. If it were that simple, minimal > SU(5) would have been guaranteed to work, since the standard model > _is_ a _particular_ decomposition of SU(5). However, it did not > work. Minimal SU(5) GUT may only be minimally wrong, proton decay experiments and the resulting data analysis may be the more wrong part. > But general relativity is a classical theory. The gauge symmetry > is evident in maxwell's equations, but in the context of maxwell's > equations cnnot be used to predict any physics. That's why the > potentials in classical theory are artifacts. In quantum theory, > the potentials become interactions coupled to fields. I think you can do gravity in a minimal SU(5)-like way as shown by the following from Tony Smith's website: The physical 4-dimensional SpaceTime of the D4-D5-E6-E7-E8 VoDou Physics model is a 4-dimensional HyperDiamond lattice SpaceTime that is continuously approximated globally by RP1 x S3 and locally by Minkowski SpaceTime, with Gravity coming from the 15-dimensional Conformal Group Spin(2,4) by the MacDowell-Mansouri mechanism. The curved SpaceTime of General Relativity is not considered fundamental, but is produced by by starting with a linear spin-2 field theory of massless gravitons in flat spacetime, and then adding higher-order terms to get Einstein-Hilbert gravity (without a cosmological constant - to get a cosmological constant, use massive spin-2 gravitons). The observed curved spacetime is therefore based on an unobservable flat spacetime. M. Botta Cantcheff, in gr-qc/0010080, says: "... MacDowell and Mansouri proposed a gauge theory of gravity based on the group SO(3,2) ... we focus our attention on the equations of motion ... of a Y-M's [Yang-Mills's] theory ... the real difference between GR [General Relativity with cosmological constant] and an YM-theory ... is ... a single [YM] constraint which has an extremely simple interpretation: the torsion-free condition. ... Torsion appears in a natural way in modern formulations of the gravitational theories [ I. L. Shapiro, "Physical Aspects of the Space-Time Torsion", hep-th/0103093 ] ... by relaxing the constraint, we are naturally led to a particularly elegant theory of gravity with torsion, whihc remarkably enough turns out to be an ordinary ...[ SO(2,3) or SO(4,1) Yang-Mills ]... we observe that the cosmological constant must be non-vanishing. ...". If you were to start, not with locally Minkowski SpaceTime, but with the curved SpaceTime of General Relativity, then you would see that the Conformal transformations of Minkowski SpaceTime by the 15-dimensional Conformal Group Spin(2,4) corresponds to the Conformal transfomations of the curved SpaceTime by the infinite-dimensional Conformal subgroup of the group Diff(M4) of General Relativistic coordinate transformations of the 4-dimensional SpaceTime M4 of General Relativity, which Conformal subgroup is defined as those General Relativistic coordinate transformations that preserve conformal structure and which infinite-dimensional Conformal subgroup can be called the Weyl Conformal Group. (See Ward and Wells, Twistor Geometry and Field Theory, Cambridge 1991, p. 261.) jcgonsowski@yahoo.com: >> The point being what? Try to find a different group structure that >> corresponds to physical measurements. If it were that simple, minimal [quoted text clipped - 3 lines] > >Minimal SU(5) GUT may only be minimally wrong, Minimally wrong is still wrong. >proton decay experiments and the resulting data >analysis may be the more wrong part. There are some people attempting to resurrect the model, but personally, I think it's more desparation than anything else. >> But general relativity is a classical theory. The gauge symmetry >> is evident in maxwell's equations, but in the context of maxwell's [quoted text clipped - 4 lines] >I think you can do gravity in a minimal SU(5)-like way as shown by the >following from Tony Smith's website: Minimal SU(5) is only large enough to hold the standard model. >The physical 4-dimensional SpaceTime of the D4-D5-E6-E7-E8 VoDou Uh, that's a few more parameters than SU(5). I've seen tony's web site and have noted that he claims to have predicted the t-quark mass, but unfortunately, I've never seen a clear exposition of the theory he keeps talking about. Basically, all I know for certain is the he likes clifford algebras, since he says that in big letters with a link explaining why he likes them. Perhaps he should invest some time in motivating the physical aspects of his theory and posting it. It's been a long time since I've read the articles he has written, but from what I recall, they weren't very clear. >Physics model is a 4-dimensional HyperDiamond lattice SpaceTime that is >continuously approximated globally by RP1 x S3 and locally by Minkowski [quoted text clipped - 6 lines] >cosmological constant, use massive spin-2 gravitons). The observed >curved spacetime is therefore based on an unobservable flat spacetime. That idea isn't unique to his theory. However, there is one obvious question one would ask. Why is spacetime flat, when there's no particular reason for it to be anything? >M. Botta Cantcheff, in gr-qc/0010080, says: "... MacDowell and Mansouri >proposed a gauge theory of gravity based on the group SO(3,2) ... we [quoted text clipped - 22 lines] >called the Weyl Conformal Group. (See Ward and Wells, Twistor Geometry >and Field Theory, Cambridge 1991, p. 261.) I have tod's book on twistors. I'll look through it. > Baugh: > >Bilge wrote: [quoted text clipped - 14 lines] > Once again, I said nothing to the contrary. You seem to missing > the point, entirely. Very well what then do you mean by "we are perfectly free to ... not gauge invariant." above. > >Otherwise we "change reality" by doing a little math. You are > >correct in that we needn't do physics. One can create theories [quoted text clipped - 3 lines] > attributing physical significance to the mathematical quantities > that make the theory consistent? Because the physical significance is relative to two artificial constructs, in this case space-time and the potential fields. Let x be a physical quantity, make up quantities y and z = x+y. y and z are not physically significant alone but their relation to each other is. Given a theory derived using quantities y and z as elementary then of course you cannot simply drop one. This is what occurs when we insist on the canonical formalism and its underlying fixed symplectic structure to derive quantum gauge theories. > >> The entire standard model follows from gauge invariance. (I don't > >> simply mean that gauge invariance is an additional assumption which [quoted text clipped - 7 lines] > _is_ a _particular_ decomposition of SU(5). However, it did not > work. Your point being what? I assert that the standard model and various extensions makes a major unfounded assumption, that the generators of one the distinct gauge sub-groups, U(1), SU(2) and SU(3) all commute with the the generators of another of these groups and with the Poincare group. There are more ways to vary the total gauge group than just tacking on more generators. And there is no specific reason to assume that weak isospin is independent of color and hyper-charge or say spin and momentum. To the contrary there is obviously serious interplay there since the various gauge charges for the fundamental particles do indeed have strong correlations. > >The Model part being the assumption of that specific group structure > >beyond transformations of observables, e.g. SU(2) isospin gauge and SU(3) [quoted text clipped - 9 lines] > vs. e+ e- -> \mu+\mu- differs for a model with and without color > degrees of freedom. By a factor of 3. Yes but observing that one model fits data better than another is not the same thing as observing the component of that model. You cannot directly measure the color of a particle it is not a physical observable in the theory although it is treated as one in the model. I'm not disagreeing with the model, I am not saying color doesn't exist. I'm saying color is as yet a construct and its existence or non-existence is a moot point. For example, what evidence have we that rotations commute with color transformations? Design an experiment to verify this. > The existence of weak isospin also has direct physical consequencea. > (In the futre, please qualify isospin with the proper qualifier, i.e., > weak or nuclear. Nuclear isospin is very commony used in nuclear physics > and without context, it would not be obvious to which you refer.) Yes, thanks for the qualifier, I'll be more specific in future. But as you say weak isospin has direct physical consequences, I simply say e.g. you cannot observe w2 vs w1 vs w3 where w3 is the component used in the E-M charge formula. We observe w3 in that we observe the distinction between neutrinos and their dual leptons. But this in fact is an observation of charge and mass. I wouldn't replace the weak isospin component of the model without a better one, but it is in this sense ad hoc. Before we get too involved in the question of "why is weak isospin symmetry broken?" we should consider more carefully how its restoration fits in with the other symmetries e.g. translations and rotations and boosts. In short, why are we looking for a Higgs particle? > The conserved vector current hypothesis (CVC) os probably one of the > most aspects of the standard model. It predates the standard model [quoted text clipped - 10 lines] > So what? Everything is ``ad hoc'' until it's either shown to be > incorrect or explained by something less ``ad hoc.'' But my point is that we introduce "ad hoc" assumptions to explain phenomena which are not directly observed but rather are components of the model. It is like asking what is the mechanism by which the aether shrinks measuring rods in the pre-Einsteinian interpretation of the M-M experiment. > The standard > model and general relativity explain a great deal more with a > great many fewer ad hoc assumptions. By an yardstick ever used > in any physical theory, the quantities in the standard model are > justifiably considered physical. Possibly you are correct. I may be drawing a line between "physical" and "conceptual" which must be drawn somewhere and I may simply be drawing it too conservatively for your tastes. The issues get fuzzier the more extreme the experiments in which we "observe" quantities. The point I was making was that the Standard Model is justifiably called a model and not a theory because there are elements which are not directly observable, that thus should not be set in stone before considering alternative models. The level at which the Standard Model predicts phenomena, the theory built upon this model, is the level at which we may directly observe hadrons and leptons. When we then consider quarks, is it meaningful to assume the operator defining momentum commutes with the projection operators which distinguishes a red top quark? Certainly with confinement we cannot readily translate them. > >The model aspect is the assumption of an underlying dynamic structure > >where these symmetries are restored. > > Again, so what? (You have that backwards by the way - the dynamical > structure is the broken symmetry. The underlying structure is the > fundamental part.) Not quite what I meant. I was speaking to dynamic structure at both levels, symmetry and symmetry broken. As for example the dynamics of atoms below the level of the dynamics of crystal vibrations. Hence the "underlying" qualifier. > >It's of course not a bad assumption given the success of the > >resulting theory. [quoted text clipped - 14 lines] > can demonstrate how to obtain the shift in the interference > pattern using just the E and B field, please do. I am by no means arguing that you can ignore the A field in the field theory. (classical or quantum). But in the case of the A-B effect we do not observe phase, we rather observe the shift of interference which we interpret in terms of relative phase shifts for paths. So saying the A field affects phase doesn't quite mean we observe the A field indirectly in the A-B experiment. It is two levels of abstraction deep rather than the one level deep for E and B which we observe by observing forces on charged particles. By the same token, the necessity of including ghost particles in the successful quantum field theories is not an argument for their physicality. The success of the theory argues that they must be included in the mathematical calculations but not that we should take their "reality" seriously and go looking for ghosts. > [...] > >I would also say that in the modern treatment of both classical and [quoted text clipped - 12 lines] > they refer to the type of potential one would write in a galilean > invariant theory, e.g., U(r) = e^2/r. In the modern treatment one works with propagators and Feynman diagrams. The field values at each point in space are replaced by the interactions of various physical and virtual particles. The very distinction between "classic" QFT and "modern" QFT is exactly the abandonment of this picture of field values at every point of a space-time manifold. Rather the notation for creation and annihilation operators, a(x) a*(x) is reinterpreted as a parameterization of the acts of creation and annihilation of quanta. Space-time becomes a *parameter manifold* instead of a physical one. > [...] > >gravitation you then have in the gauge transformations of standard EM > >a variation of the geometry of the 5 dimensional K-K manifold. > > Sure. So what? The point again is the comparison of interpretation for the same quantity in two distinct models of the same theory. > >In the K-K theory this is changed to a variation of which submanifold > >one projects onto to define space-time (at zero phase). > > I think you're getting carried away. Quite possibly. > Kaluza klein theory is simply > a 5-d wave equation that has _exactly_ the same solution as the > 4-d wave equation x a phase. The phase is not observable. Moreover, > the radius of the compactified dimension was (is) considered to be > physically significant. Yes relative to the scale of the other four dimensions and then in that they are comparable via the common metric. In which case the A field is this relationship. The radius is significant relative to A and the A field is significant relative to this radius. Neither alone is meaningful. > >Essentially one re parametrizes the this 5-d manifold so that the > >x_mu in one case is x'_\mu, a mixture of x_mu and theta in the other. > > In which case, it can't ``project'' out at ``zero phase.'' The > phase transforms. It's unobservable. That's not the same as being zero. Yes but the "vacuum" case defines the zero phase modulo choices of gauge. Fixing the gauge is precisely the choice of coordinates on this 5-d manifold. The constraint $(x,y,z,t,phi) = (x,y,z,t,0)$ defined the projection to which I referred. > [...] > [quoted text clipped - 4 lines] > That's the point of invariance. One is seeking a theory in > which there are _no_ forces, per se. No, the theory remains unchanged under this alteration of the underlying "model". Rather one is choosing one of many models for the same theory in which there are _no_ forces, per se. This is the meaning in my mind of the equivalence principle, not that the forces go away but that the forces are defined only relative to the geometry et vis versa. > >This is what M-Theory is all about. But I assert that this is > >introducing too many artifacts into the theory when we quantize. [quoted text clipped - 4 lines] > _classical_ theory. It would be an artifact in quantum theory, too, > were it not for experiments that cannot be explained using E and B. Don't confuse the mathematical structure for the theory. The theory as I am using the term is the system of predictions. The E and B fields are part of the theory as the predicted forces on test particles. The probability distribution defining the interference pattern in the A-B experiments is part of the theory. Below this is the A field which is utilized to calculate both of these elements. I wouldn't attempt to guess how you could calculate these elements without utilizing A or some equivalent in a more general theory. I am not arguing for its elimination form the tools we use to "do electrodynamics" I am simply stating that it is a quantity of a distinct type from E B and |psi(x)|^2. > [...] > > >I'm suggesting that the equivalence principle is not fully > >integrated into such theories. > > Sure they are. The standard model treats all mass as inertial mass. That alone is not the E.P. It is the consequence of the E.P. in the geometric formulation of GR. It doesn't matter as much at the classical level but when we attempt to quantize there is a constraint which picks one underlying model from the other i.e. the constraint: Forces := 0. This constraint when we quantize must either commute with the relevant observables or the consequence of its non-commutativity must be reckoned with. It is the same as when one works "off the mass shell" in quantum particle dynamics. And this I assert is part of the reason quantization of GR has been unsuccessful. Of course I can criticize all day at others failures, but I don't expect too much attention unless I can offer something more constructive i.e. a successful theory of quantum gravity. To this I can only say give me some time. > [snip] > >I'm suggesting that reifying space-time as a physical 'brane' in [quoted text clipped - 10 lines] > witten knows a great deal more about string theory than I do, I'll > leave the believing up to him for the time being. I even hesitate to call it a "theory", rather a class of models possibly leading to theories but, again as I assert still incorporating what I see as fundamental conceptual flaws. > >I simply have an intuition based on my belief that we shouldn't > >"quantize" the mathematical artifacts of our exposition but rather > >the physical observables of the theories. > > The aharanov-bohm experiment demonstrates that A is not an artifact. I don't exactly disagree but cannot agree. Rather I view A as being a composite entity representing a mixture of the physical and the artificial. > So far, you've said a lot of stuff that is rather straigt forward and > uncontoversial, but have not explained the aharanov-bohm effect using > only E and B, which you seem to think is possible. Again I do not seek to "explain" the aharanov-bohm effect. It is and empirical effect and I must in any alternative or improved theory be sure it is predicted or my theory dies on the vine. But said hypothetical theory needn't predict or incorporate any feature of the electro-magnetic vector potential provided the A-B effect is predicted accurately. I can't imagine a theory which would not incorporate some analogue of the A field. But I can't see any reason to assume this is necessarily the case. Let me further state that elimination of the A field is not my goal. I'm not trying to "explain the A-B effect using just the F field." I'm not trying to explain anything. I'm specifically trying to quantize gravitation. I don't specifically recall how we got on to this topic. > If you want to > join in such a discussion, you might help out eugene stefanovich > on sci.physics.research, who has similar ideas and took his argument > there after being shot down here. The two people arguing with him > on sci.physics.research seem to have more patience. See previous paragraph. > >> The gauge degrees of freedom have immediate physical sgnificance. > > [quoted text clipped - 3 lines] > A four-vector has four degrees of freedom. The photon has two. > How does one account for the two missing degrees of freedom? By noting that the photon in actuality has a huge number of degrees of freedom which we call infinite for want of an exact number. We pick out two of these for the photon's polarization because we are factoring the photon's irrep of the Poincare group in a specific way. The very fact that we do not see four polarization modes for the photon demonstrates that this factorization: 4-vector x scalar wave function, is not correct and in fact not irreducible. The very reason we factor in this way is because we are working from a model of "fields over a space-time manifold". The very fact that we must impose additional constraints to get the observed two polarization modes is precisely the "kludge" we introduce to fix this less than ideal model. > The condition d_u A^u = 0, reduces the number to three. That > makes the photon a spin 1. The remaining three degrees of [quoted text clipped - 5 lines] > The two degrees of freedom which remain are the two transverse > polarizations. Yes yes yes, I'm well versed in QED though a bit rusty, it's been a while since I parsed through the derivations. But keep in mind I am not talking about alternative field theories but alternatives to field theories. As long as you are working within a field theory framework, all you've said is 100% correct. It none the less is not especially relevant to my points. > >It is the combination of this artificial format along with the gauge > >degrees of freedom which together "have immediate physical significance." > > I have no idea what that means. Are you saying the electromagnetic > field is not a vector field? If it is, then I just described the > two degrees of freedom. I'm saying that as long as you work within the format of fields over a space-time manifold then you must incorporate this additional artifact. I'm saying that field theory itself incorporates an additional artifact, the fixed division of space-time and field. It is comparable to the fixed division of space and time prior to SR. Upon unification we loose the artifact of "absolute universal time". I assert that in a unification of space-time-field you likewise loose the artifact of "absolute vacuum". The A field then will take the same stage as the boost parameters distinguishing different frames, i.e. as a relationship between alternative decompositions of the same physical quantities into components. > >As a simple example consider a periodic physical degree of freedom. Now > >force that into a flat model by describing it as a rotation in two [quoted text clipped - 7 lines] > That's entirely different. What you are describing is hidden > variable theory. There is nothing hidden about a gauge field. No. I am not talking hidden variables. I'm talking appending mathematical variables in order to embed the physical variables in a particular form of mathematical construct. To identify the physical variables you then must impose additional mathematical constraints. That my dear is gauge, not "hidden variables". > [...] > [quoted text clipped - 6 lines] > position in spacetime implies the measurement of a distance is > an artifact? No of course not. But it means the specification of position. The values of the position is not physically meaningful. The difference in positions defines the physical distances. It is the distinction between "real points of space" and space as a construct to express relations between physical objects. > [...] > >Implicit in your exposition is an underlying canonical structure. > > And? See below *** > > >[p,q]=i, and the underlying inhomogeneous group ISO(3,1) wherein > >p^2 is an invariant. > > No, it doesn't. The mass is casimir operator of the poincare > group, just like the spin. Check again. The Casimir invariant is the contraction of the Killing form with the products of pairs of (representations of) generators. I^2_rho = B^{ab} rho(G_a G_b). The Killing form for ISO(3,1) is null for the momentum operators. It is because ISO(3,1) is a "singular" group (not semi-simple) that it has no unique quadratic invariant. The squared mass is in fact the additional quadratic invariant which can be appended to the Casimir invariant. I'^2 = I^2 + \lambda m^2 *** And the non-uniqueness of this invariant is lost if you ever so slightly perturb the defining relations of ISO(3,1). The perturbation yields either SO(4,1) or SO(3,2) in each case of which the full Casimir invariant is a sum of mass and total relativistic spin. The value of \lambda above becomes fixed. The canonical structure goes hand in hand with the singularity of ISO(3,1) and hence the independence of mass from spin. Mass^2 and spin^2 alone will cease to be central elements of the group. Now you can still work within a format of a variably curved space-time manifold. However our choice of a *flat tangent space* instead of say a *tangent pseudo-sphere* is not based on any physical assumptions but rather one of mathematical convention. Replace the tangent space and tangent Poincare group with a tangent pseudo-sphere and tangent SO(4,1) group and you have an entirely different format. In that format we would not define mass but a unified mass-spin. > >Mind you I don't have a replacement theory in which to demonstrate > >a counter example (yet) but I cannot consider the one instance as [quoted text clipped - 4 lines] > mental. What you seem to be saying is that you can find a different > representation for the same group structure. But that's irrelevant. It would be if that is what I was saying. Rather I'm saying one can find a different group structure for the same physical theory (or one very close to the same so as to agree to the level of current experiment) and that said different group structure may in fact remove some of the ad hoc assumptions. This after all is what occurred in SR. It is a matter of looking at what worked in past, why it worked specifically, and applying this to current theory. That is exactly why I'm making issues of "what is" and "what isn't" physical |
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