This algebra describes EVERYTHING.

Transcript

before delving into this video's topic I'd like to make a few disclaimers the title of it is a bit misting the geometric algebra I'm covering the algebra of physical space doesn't describe everything in fact it's only a subalgebra of the SpaceTime algebra however I chose the title for a good reason the algebra of physical space is able to elegantly describe classical mechanics special relativity and many aspects of quantum mechanics it is one mathematical framework that essentially unifies all these seemingly sep seate physical theories the general awareness of the algebra of physical space is in my opinion quite lacking so I'm making this video to attempt to raise awareness yes many of its uses are superseded by the SpaceTime algebra but the algebra of physical space is much more conceptually and mathematically simple due to this I think the algebra holds a lot of importance I'll be presenting this algebra and its uses in five main sections not including the introduction the first will be on both ukian and hyperbolic rotations the second on the differential operator the Third on Einstein's theory of special relativity the fourth on electromagnetism and the fifth on the dur equation in this video I'll be using the poly representation of the algebra of physical space which I will call the po algebra of space furthermore this video is to serve as an introduction to the applications of the P algebra space so I will not be extrapolating upon the basics of geometric Algebra I have two series which will cover this information and I recommend checking those out there are also many good YouTubers who teach geometric algebra let's return to the PO representation for a second essentially the main difference between the algebra of physical space and the PO algebra space is notation the vectors and the multiplicative identity one are replaced with the PO matrices on a surface level this does seem to only be a notational difference but truthfully the difference is quite profound while in the algebra of physical space the vectors and the identity are just abstract elements in the PO algebra of space these abstract elements are simultaneously 2x two matrices the entire algebra now has a natural Matrix representation note the B vector and tri Vector matrices before fully continuing on I'd like to present the conjugations of the algebra they're pretty important as we'll see suppose we have a general element of the algebra called a multiv vector the first conjugation is called inversion and inverts all odd grade elements of the multiv vector G the second conjugation is called reversion and reverses all Vector products within the multiv vector G this is equivalent to inverting all pseudo elements recall that pseudo elements are the pseudo vectors bi vectors in this case and the pseudo scaler the tri Vector in this case the third conjugation is simply the compound of both conjugations and is called the Clifford conjugate this is a equivalent to inverting all vector and bi Vector parts and since the PO algebra of space is simultaneously a matrix algebra these conjugations directly correspond to Matrix operations the inversion of a multiv vector is the same as the adjoint of its hermiss conjugate the reversion the same as its hermiss conjugate and the Clifford conjugate the same as the adjoint of the multiv vector because the unit Tri Vector called the unit pseudo scaler of the space serves as the imaginary unit rotations are naturally encoded within the PO algebra of space if we consider some Unit B Vector B which is equal to the unit pseudoscalar times the unit Vector normal to the B Vector then the square of the B Vector behaves similarly to the imaginary unit remember we're in the PO representation so the multiplicative identity one is replaced by Sigma T taking a leap of mathematical intuition a rotation can be constructed using using Oiler formula this equation will rotate any Vector that is in the plane of the B Vector B by Theta / 2 the reason it's defined to be half the angle is because to work on an arbitrary Vector the equation must be applied to both sides this general form will rotate any vector by Theta in the plane of the bi Vector B this formula is how rotors are defined if you've ever worked with quaternion this might look familiar to you I discussed it in a previous video but the subalgebra of rotors in the PO algebra of space is actually isomorphic to the algebra of quaternion a mapping can be seen as such Loren boost can also be easily represented in the algebra since the boosts are nothing more than hyperbolic rotations it follows that a general boost is of the following form where V is the unit Vector in the direction of motion and the hyperbolic angle Fe is called the rapidity defined as the arc tangent of relative velocity this boost will transform into the rest frame of whatever is moving with velocity V integrating a differential operator into the PO algebra of space is actually very straightforward it is equivalent to the gradient operator from traditional Vector mechanics note that for the rest of this video I'll be using the shorthand partial derivative notation this differential oper Ator functions like the traditional gradient operator but also as a distinct Vector Operator by this I mean that if it operates on a scaler it is equivalent to the gradient and when it operates on a vector it combines both the inner and outer products of the differential with the vector this inner product differential is just the familiar Divergence of a vector likewise the outer product differential is mathematically dual to the curl of a vector I've highlighted the bi vectors and tri vectors in purple to demonstrate this Duality between the tradition I curl vectors and the outer differentials resulting by vectors the best way to motivate a description of special relativity is to consider the determinant of a general multiv Vector in the algebra as I said earlier a multiv vector is a general element of the algebra thus in the poly algebra of space a multiv vector has a scalar part in blue a vector part in Orange a bi Vector part in purple and a tri Vector part in yellow because of the natural Matrix representation of the algebra this multiv Vector is equal to a matrix then the determinant of the multiv VOR is easy to find note that Big C is a complex constant that for the moment is unimportant but from this determinant form it's clear that a multiv vector with only a scalar and Vector part has a determinant that gives them inowy metric of space time but some of you might have caught on that we technically leave the algebra by defining the determinant in this fashion this would be due to the fact that there's no PO element attached to the resulting value of the determinant there's an easy fix to this though a result of linear algebra is that the determinant of any Matrix and therefore in the PO algebra a multiv vector is the Matrix itself times its ad joint in the algebra we're using the adjoint is just the Clifford conjugate so the determinant is just a multiv vector times its Clifford conjugate then the determinant stays within the algebra in a more rigorous way going back to the Natural space SpaceTime element being a scalar and a vector we Define what's called a par Vector then the magnitude of a power Vector follows from the determinant suppose we have another Power Vector B then the space time inner product can be defined the same can be done to define the SpaceTime outer product therefore a well-defined product between parir vectors exist called the space-time product where the inner product gives a scalar and the outer product gives what's called a bipar vector a bipar vector is a multiv vector with only a vector and a bi Vector component it is convenient to interpret bipar vectors as either complex vectors or complex bi vectors and to further interpret them as operators on space-time Elements which are the power vectors A Quirk of the PO algebra of space is that power vectors and bip power vectors transform under Lorent Transformations differently from each other note that a Lorent transformation is a combination of a ukian rotation and a hyperbolic rotation power vectors transform under the reversion of the Lorent transformation whereas bipar vectors transform using the Clifford conjugate of the Lorent transformation if this is confusing I encourage you to work through the math yourself there's not much benefit in me deriving these results for a video but doing it by hand helps a lot with one's understanding the explicit forms of these are quite complicated and messy so I'll restrict my presentation to the special case of only hyperbolic rotations otherwise called the Rin boosts notice that the result of a boost on a par Vector is indeed a par Vector recall that vat is the unit Vector in the direction of motion all the boosted Vector components are the components from the formula for a lorence boost in an arbitrary Direction now for boosts applied to bipar vectors again notice that the result of a boost on a bipar vector is still a bipar vector the boosted Vector components are the components from the formula for a lorence boost applied to the vector component of a bipy vector in an arbitrary Direction while the boosted bi Vector components are from a similar formula but for a bi Vector boosted in an arbitrary Direction the differential operator is capable of being represented in the algebra but this three-dimensional operator that we've previously defined cannot provide the entire description of differentiation with SpaceTime after after all there is no derivative involving time therefore in SI units of course the space-time differential can be defined from this the SpaceTime Divergence and SpaceTime curl can be derived thus the SpaceTime differential of a power Vector is well defined note that the operations for cliffed conjugation are well defined this then implies that the dalan is defined everything shown allows for the full use of the PO algebra space for the physics of special relativity the mathematics are not particularly difficult and are in fact simple logical extensions of traditional Vector mechanics in a linear algebra but not only does the algebra allow for a simple yet rigorous application of special relativity it reveals a new approach to physics this new approach called the igen spinner approach formulates the description of relativistic mechanics based upon the Lorent transformation that transforms a lab frame to a particles or distributions rest frame I'll be going over the basics of igen spinners including their basic derivation however I will not go into slightly more complicated derivations like the one for What's called the SpaceTime rotation rate bip Vector first imagine that P is the SpaceTime momentum of a particle as measured in the lab frame then the Lorent transformation that transforms P to its rest frame is as written m is the particle's rest mass and C is the speed of light the transformation from the rest frame to the lab frame is therefore easy to find it's found by multiplying by the inverses of the original transformation on both sides of the equation note that the inverses here are the equivalent to the Clifford conjugations due to the unitary nature of the Loren Transformations the igen spinner of the particle is precisely this transformation by convention the igen spinter is Rewritten as a capital Lambda the igen spinner of a particle can be solved for to determine the particle equation of motion from the following spinorial form where Omega is the SpaceTime rotation rate by par vector and too is the proper time of the particle in its rest frame the general solution of the spinorial form is nearly trivial to derive I'll explore the meaning of this a bit more later in the video when it's used to derive the Lorent Force equation of electromagnetism I'll also demonstrate the Igan Spinner's use in deriving the dur equation without quantization Maxwell's equations of electricity and magnetism are an elegant and beautiful theory of an equally elegant and beautiful phenomenon the four equations listed here are the forms given by the traditional Gibbs heavy side formalism in SI units of course these equations the electric field the magnetic field and the current density are all Vector quantities but these equations are not written in their natural algebraic form as given given by the PO algebra of space within the algebra the electric field is Rewritten so 2 is the magnetic field and also the current density therefore the electric field and magnetic field form a bipar vector field called the electromagnetic field as I mentioned before it's helpful to interpret a bipar vector and therefore the electromagnetic field as a complex vector or by Vector which acts as an operator on a power Vector a simultaneous interpretation which is of extreme geometric and intuitive importance is that of the electromagnetic field containing a vector and bi Vector part the vector part can be thought of as what generates the acceleration or Tim like movement of whatever the field acts on whereas the bi Vector part can be thought of as what generates the gyro motion or space-like movement of whatever the field acts on also the magnetic field being a bi Vector along with the fact that in any charge particles rest frame the field it generates is a pure electric field implies that as a frame electromagnetic field which is purely electric it's rotated through hyperbolic space a portion of it is smeared and sweeps out a two-dimensional object this new two-dimensional object is the magnetic field now using this electromagnetic field it follows that all of Maxwell's equations can be expressed as Maxwell's equation in the PO algebra of space which is the SpaceTime differential of the electromagnetic field equal to C * mu * the Clifford conjugate of the source term all of the traditional equations of electromagnetism can be derived from this I won't spend time showing them off but if you're truly interested or even confused I highly recommend you work through the math yourself it's honestly pretty cool to see for yourself however it's worth showing the wave equations the in homogeneous wave equation that is the equation in the presence of other sources is as follows and the vacuum equation that describes light waves is elegantly simple it's also quite simple to represent Maxwell's equation using the potential formulation defining the potential as the power vector v composed of the scalar electric potential Fe and the vector magnetic potential a the electromagnetic field can be Rewritten note that there's an extra scalar term Lambda this is the Lorent gauge and disappears when f is differentiated in Maxwell's equation this extra constant is typically chosen to be zero but it isn't necessarily zero to learn more about this I'd recommend searching for gauge theory on Wikipedia or other similar mediums and without further Ado the potential formulation of Maxwell's equation now let's return to igen Spinners using the igen spinner approach I'll show the derivation for the Lorent Force equation from dimensional analysis and SI units one can assume the SpaceTime rotation rate by Vector to be equal to the elementary charge divided by the arrest frame momentum all times the electromagnetic field recall the definition of labf frame momentum taking the time derivative of this momentum is the next natural step given we're looking for a force this last form of the equation is not very intuitive but an identity from the POI algebra space says that this form is equal to the projection of the par Vector component of the SpaceTime rotation rate by par Vector times the momentum par Vector it follows that the Loren Force needs only a simple substitution then expanding this out the Loren Force becomes more recognizable note that for this work I assumed the momentum par Vector P equals mc+ MV where V is the lab drain velocity I personally think that the igen spinner approach is very elegant and provides great geometric intuition for the mechanics of special relativity and therefore for electromagnetism the direct equation is one of the most important equations in all of physics generalizing the shringer equation within the context of Relativity and allowing for the construction of quantum field theories in tensor component form it's traditionally defined as follows the direct spinner CD is a four component complex function the direct equation is commonly viewed as a purely Quantum result as it was derived for relativistic quantum mechanics however within the lature on the algebra physical space and therefore the PO algebra space it has been shown that the direct equation can be derived naturally from the algebra it can do so with and without quantization meaning that the direct equation is shown to generalize to classical mechanics let us begin the derivation suppose there's some density function row be it charge or mass which is in its rest frame then according to the igen spinner approach the lab frame current density function is Lambda * row * Lambda dagger the the rest frame density is a scalar so it can be split into square roots furthermore Lambda * the square root of the rest frame density can be relabeled sayup C then recall the igen spinner definition of the labra momentum this can be rearranged by right side multiplying both sides of the equation with the inverse of Lambda dagger then if both sides are multiplied by the square root of row the classical direct equation is created the leap from classical physics to quantum physics can be made by the sandwiching OBC on the left hand side between the momentum operator and the vector spin axis it also turns out that the inversion of the entire formula describes the same system very neat side effect of this derivation of the direct equation is that an explicit trigonometric and hyperbolic function representation can be found for the dra spinner namely the coefficients for the multi Vector P can be found as an aside it's possible to rearrange p in into a complex quaternion form it's not necessary for the task at hand but it's nice to know especially because it's already known that the dir spinner in the traditional direct equation is also a function of four complex components the next step in deriving the coefficients of the P multi Vector is recalling the definition of c all that's needed to find the components then is to fully expand this doing so gives the following explicit components for the poly algebra of space formulation of the dur equation well presenting them all like this is a tad overwhelming in Broad Strokes it can be interpreted to mean the following when a rest frame density is transformed by a Lorent transformation it is smeared into a scaler Vector bi vector and tri Vector Parts another Revelation from this which is so unsurprising that it's essentially trivial is that the components are fully relative to The Observer the cool thing here despite the triviality is that this Revelation comes from ukian and hyperbolic trigonometry but maybe I'm excited a bit over nothing now before I end the video I have one last thing to show namely I wish to demonstrate the basic mapping from the dur equation of the PO algebra space to the traditional dur equation so as to show their equivalency an important tool to be used here is the projector it has an important property called the poman property which is the best pun in physics by the way where the unit Vector generating it is completely absorbed by the project Vector the first step in creating the mapping is to multiply both sides of the pair of conjugate direct equations in the algebra this absorbs the spin axis Vector then these two equations must be unified into a matrix equation if you multiply the left hand side out you'll see that the two equations above fall out the components of the Matrix can be expanded a bit from here it's pretty simple as long as you remember the definition definition of Drax gamma matrices therefore assuming that the traditional draax spinner is identical to this projection the traditional direct equation is found furthermore the igen spinner based form of the direct equation within the POI algebra of space maps to the traditional dur equation in these ways well that's all for now I hope that this introduction to the P algebra of space is as interesting to everyone as it is to me comment what you think think and ask any questions you want I enjoy reading through them also sorry for the long time between videos but I hope the wait was worth it for this one this video is a lot of effort requiring a ton of research time and writing I mean the script is 13 pages long and there are over 50 PowerPoint slides with hundreds of animations and I'm not even going to mention the editing time especially because I keep stuttering and I couldn't get all of them out please make sure to also like the video if you liked it and to subscribe for similar content thank you and have a great day