Particle Physics is Founded on This Principle!

the standard model of particle physics is the most spectacularly predictive theory of nature that human beings have come up with since we started doing science and the whole construction is built on the foundation of symmetry so much so that when we speak of the theory we typically denote it simply by its symmetry group su3 times su 2 times u1 more precisely this is called the gauge symmetry of the standard model which is a special kind of field theory known as a gauge theory as we've learned in past videos here on the channel we can't understand

symmetries in physics without appreciating their profound connection to conservation laws the two go hand in hand and in this video i want to start you out on the road toward understanding gage theory by studying the most basic symmetry and conservation law at the heart of the standard model the symmetry of electromagnetism which is called u1 and its associated conserved quantity electric charge this connection between symmetries and conservation laws is captured by noder's theorem translation symmetry in space for example is tied to momentum conservation meaning that if you can pick up your system and slide it

over without changing anything about the physics then the total momentum of the system in that direction is a constant rotational symmetry likewise leads to angular momentum conservation and energy conservation follows from translation symmetry in time meaning if the dynamics of your system looked the same yesterday as they will tomorrow what node's theorem says is that if a system of particles has a symmetry then you're guaranteed to find a corresponding conserved quantity q in the last mini lesson though we started discussing field theory where we're not only interested in how the coordinates of a bunch of

particles move around but in fluctuating fields that permeate space and time interacting with particles and potentially with each other the most intuitive examples to keep in mind are the electric and magnetic fields that propagate the electromagnetic forces between charged particles and that are bouncing off your eyeballs at this very moment in a field theory like electromagnetism the connection between symmetries and conservation laws becomes even deeper you're no doubt familiar with the conservation of electric charge for example but the conservation of charge isn't simply the statement that the total amount of electric charge in the universe

is a constant for example if an electron disappeared in tokyo at the same moment a muon appeared next to tau ceti the total amount of charge would not have changed but a conservation law in field theory is stronger than that charge is conserved locally meaning that the only way the amount of charge can change inside any box large or small is if a current continuously carries charges in or out but if conservation laws result from symmetries of nature then what symmetry is responsible for the conservation of electric charge we'll discover the answer in this video

in the first part we'll see how local conservation laws arise in field theory and how they're captured by the continuity equation which constrains the charge density rho and the current density j then in the second part we'll identify the associated symmetry known as u1 in a theory like electromagnetism that's tied to the conservation of electric charge by node's theorem and in part three we'll put the pieces together and write down the simplest example of a gauge theory by the way i should point out that the u1 of electromagnetism is not literally the u1 in the

symmetry that labels the standard model the su-3 factor stands for the theory of the strong force the su-2 stands for the weak force and the third u1 is called hypercharge the electromagnetic force is a different but related u1 that sits inside here and electromagnetism falls out after the famous higgs field does its work we'll better understand what all these symbols mean as we go along though the higgs mechanism we'll have to wait for another video let me know in the comments if that's something you'd be interested in we're getting into an advanced subject here so

this lesson isn't going to be easy but considering we're talking about some of the deepest principles of nature it's definitely going to be worth the effort to understand and as always you can get the notes that i've written at the link in the description to take your time understanding all of these ideas before we get too deep into the physics though let me take a quick second to tell you about this video's sponsor blinkist blinkist is a really cool app that lets you take away the key insights from a huge library of nonfiction books with

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where it might take us in the future and hawking's last book brief answers to the big questions which is about his thoughts about where the universe came from and where we as a species are going hawking was working on it right up until he passed away in 2018. one key thing he argues for is how important it is that we begin to expand beyond earth he thinks that it's within reach to set up a moon base or send a manned mission to mars in the next 50 years right now if you use the link that

i've shared down in the description you can start a free 7-day trial of blinkist and then get 25 off a premium membership if you decide to sign up so go try it out it's a really nice app that lets you understand and grow from big ideas in 15 minutes and thank you so much to blinkist for sponsoring this video the first thing we need to do to understand the u1 symmetry of the standard model is to sort out what it means for electric charge to be locally conserved which you may or may not have learned

about before in a class on e m this is going to be essential for understanding how conservation laws emerge from symmetries in field theory so it's important to take a few minutes to get it straight say we have some region of space v to measure the amount of charge inside the box we start from the charge density row which represents the amount of charge per volume at any point in the box at any time t in other words if we look at any infinitesimally tiny box the amount of charge inside it is its little volume

dx dy dz times the charge per volume rho i'll write that volume as d3r for short to find the total charge inside our whole box we just dice it up into lots of little pieces like this and then we add them all up by integrating over the region this is the total amount of charge q in our big box v at any time t what charge conservation means is that the only way q can change with time is if some of the charges inside the box move outside through the surface or if additional charged particles

from outside make their way in these moving charges would constitute an electric current the amount of charge per second leaving through the surface of the box let's call that boundary surface b and what we need to know is given some current how much charge is flowing through this boundary b at any moment similar to rho which measured the charge density per volume of space we measure the amount of current by the current density j but there are some important differences between row and j note first of all that j is a vector with components j

x j y and j z because a current can flow in any direction in space also whereas rho was the amount of charge per unit volume which we used to find the total amount of charge inside the box with j we want to find the amount of current flowing out through the boundary surface we therefore define j to be the current per unit area rather than per unit volume let's look at the top surface of our box for example take a little patch of the surface of width dx and length dy since we want to

know how much current is flowing out of the box what we care about in this case is the z component of j at that point j x and j y just measure the current flowing parallel to the surface then the amount of current passing through that little patch is given by its area d x d y times j z the current per area in the z direction we get the total current passing through the whole top surface by integrating over it at any instant in time this is the amount of charge per second leaving through

the top surface of the box of course now we need to do the same to find the current passing through the other sides of the box and then add them all up to get the total current going out through the surface and in general our volume v need to be a neat box and its boundary need to be a cubical surface it could be some misshapen blob instead but the idea is the same we slice up the surface into many little patches each of area da then we multiply that by the current per area flowing

out to the patch that's given by j at that point but again we need to pick out the component that points perpendicular to the surface in order to obtain the amount of current going out we can do that by finding the unit vector n hat that's perpendicular to the surface at that point for example on the top surface of the box n hat equals z hat was the unit vector pointing up in the z direction on the right surface it will be y hat pointing to the right in the y direction and so on then

we can pick out the perpendicular component of j by taking the dot product j n finally we add it all up by integrating over the surface and that gives us the total amount of charge leaving the volume per second which brings us back to charge conservation local conservation of charge is the statement that if charge i per unit time flows out through the boundary surface and the amount of charge inside the volume of the box goes down at that same rate dq by dt equals minus i this is the mathematical statement of charge conservation again

the minus sign is there because we defined positive i to mean that current is flowing out through the boundary in which case the amount of charge inside the box is decreasing at that same rate spelled out the charge conservation equation looks like this if in particular we took v to encompass all of space so that the boundary b is going to infinity the current density had better go to zero there in any physically reasonable setup since there's nowhere left for the current to flow out to then the right hand side vanishes and this equation says

that the total charge in all of space is a constant that's the statement of global charge conservation but again local conservation of charge is a stronger statement that this equation must hold for any volume v that we like that's why a charge can't disappear from tokyo and reappear at tau ceti instead of choosing v to fill all of space just build a box around tokyo the total charge inside can only change if a current continuously carries charge in or out through the surface on the flip side we can alternatively take our box to be an

infinitesimally small cube whose dimensions delta x delta y and delta z are going to zero that lets us turn this integral equation into a differential equation and that's the continuity equation that i mentioned at the beginning which will play an essential role in formulating node's theorem for field theory let's first of all bring the d by dt inside the integral on the left the only change is that it turns into a partial derivative because rho is a function of both time and space now when we shrink our volume down to a teeny tiny cube these

integrals become pretty boring on the left we just get d row by dt times the volume of the box delta x delta y delta z the reason being that d row by dt is essentially constant over this infinitesimally small region the right hand side is slightly more interesting take the top surface again for example the outward pointing perpendicular direction is going up so we get j dot n equals j z evaluated at the top of the box and the area is delta x delta y so the top surface contributes delta x delta y times jz

at the top to the integral for the bottom surface on the other hand the outward direction is pointing down so for that piece of the integral we get j dot n equals minus jz together we're going to get delta x delta y times the change in jz between the top and bottom of the box we had a factor of the volume on the left hand side of our equation that we're going to want to cancel out so let me go ahead and multiply by delta z over delta z on the right then we can cross

out these common factors and in the limit that the box becomes infinitesimal we just get the derivative of j z with respect to z of course we also have to include the front and back and the right and left surfaces of the box as well those give us the derivatives of j x in the x direction and j y in the y direction altogether our charge conservation equation when we shrink the region down to be infinitesimally small becomes d rho by d t equals minus d x j x plus d y j y plus d

z jz this is the continuity equation it's the most direct local statement of electric charge conservation and it's the prototype what it means to have a conservation law in any field theory we usually shorten it by defining a three component object called del with the x y and z derivatives denoted by this upside down triangle then this sum of derivatives is just the dot product of del and j and we can express the continuity equation more compactly as d rho by dt plus del dot j equals zero what we're going to discover is that a

local conservation law like this falls out anytime we have a continuous symmetry of a field theory and this relationship is instrumental to our understanding of particle physics the symmetry leads to a conservation law and those charge and current densities become the sources for the fields just like electric charges and currents produce electric and magnetic fields by the way what we basically did in our argument here was discover the divergence theorem which you'll learn about in your math classes and which lets us turn an integral over a boundary surface like this into the integral of the

derivatives of j over the volume v inside it's called the divergence theorem because del dot j is called the divergence of j since our volume v was arbitrary we can shrink it down and conclude that the integrands d rho by dt and minus delta j on the two sides have to be equal point by point again giving us the continuity equation okay now we've understood what it means for electric charge to be conserved in electromagnetism next we want to understand how all of this extends to a more general field theory defined by its action given

by integrating the lagrangian density curly l over space and time and most of all we want to understand how these conservation laws are related to symmetries by nother's theorem in particular i hope you're really curious at this point to discover what symmetry is responsible for the conservation of electric charge in the last mini-lesson video we studied the simplest example of a field theory called the klein gordon theory it consists of a single free field phi that assigns a number to each point in space at each time and it's a great example for learning the fundamentals

of field theory it's defined by this lagrangian density one over two c squared d5 by dt squared minus half divided by the x squared minus half kappa squared phi squared plus the y and z derivative terms which i haven't written out c here is the speed of light and kappa is a parameter that we saw is related to the mass of the particles that you get when you turn this into a quantum theory by applying the principle of least action to this theory we found that the equation of motion for phi is this klein gordon

equation it says that the second derivatives of the field with respect to time and space equals kappa squared times phi which is a generalization of the wave equation and we talked about how we can write the general solution of this equation as a sum of plane waves i also showed you last time how to write all this much more compactly using relativistic notation it makes all the formulas involved in this subject much more neat and concise but on the other hand if it's new to you then i think it might backfire and make the equation

seem mysterious so first i'll work things through with all the t's and x's spelled out and then afterwards we'll see how much simpler things look with a better notation the catch is that we're going to have some very long formulas to deal with using the non-relativistic notation don't let them scare you off when we talked about node's theorem for regular old particle mechanics what we discovered was that whenever we had a symmetry of the lagrangian meaning an infinitesimal transformation that left it invariant there would be a corresponding conserved quantity for example we looked at a

problem with a block sitting on top of a flat frictionless table attached to a spring that's pinned down at the other end this setup does not have translation symmetry if you pick up the block and slide it over to the right say the spring gets stretched so you've changed the system on the other hand the system does have rotational symmetry because you can pick up the block and rotate it around without changing the length of the spring the potential energy stored in the spring one half k times r minus l squared only cares about how

far away from the origin the block is measured by r equals the square root of x squared plus y squared it doesn't care about the angle theta that it makes in the x y plane we showed that this rotational symmetry implies by node's theorem that the angular momentum of the block is conserved we're going to discover a similar and even stronger relationship in field theory between symmetries and conservation laws in fact the simplest symmetry at the foundation of the standard model that's tied to electric charge conservation is closely analogous to the symmetry of the block

on a spring it's a rotational symmetry but now in field space to see how this works it'll actually be more interesting and more closely analogous to the field theory that describes the electron and electromagnetic force the study a slight generalization of the klein gordon theory instead of the real field phi that assigns a real number to each point in space at each time let's consider a complex field that assigns a complex number to each point meaning a number phi equals a plus ib with a real part and an imaginary part write 5 bar for the

complex conjugate of phi meaning a minus ib then we define the lagrangian for the complex field like this one over c squared dt of phi bar dt of phi minus dx phi bar dx phi minus kappa squared phi bar phi where it's conventional to leave out the factors of half for the complex version despite the complex numbers the lagrangian is real because when you multiply a number by its complex conjugate like in the last term here you get a real number a squared plus b squared because the imaginary cross terms cancel out and likewise for

the other terms in the lagrangian if you split up phi into a real and imaginary part like this then you can see that this new theory is actually just two copies of our old real klein gordon theory one for the real part and one for the imaginary part in the quantum version we'll therefore get two kinds of particles corresponding to a particle and its anti-particle now what symmetries does this theory have like i mentioned the one i want to focus on today is closely analogous to our block on a spring example from a minute ago

the real and imaginary parts of phi equals a plus ib give us a point in a 2d plane with coordinates a and b in other words we can think of the complex number phi like an arrow that goes over to the right by a in the real direction and up in the imaginary direction by b the length of the arrow is the square root of a squared plus b squared which i'll write as the absolute value of phi also called mod phi and it makes an angle theta with the horizontal axis so just like in

polar coordinates we can write the lengths of the two sides of the triangle as a equals mod phi cosine theta and b equals mod phi sine theta but just like the block on a spring our cline gordon lagrangian only depends on the length of the arrow not on the angle theta that it makes in this plane the reason why is that phi and phi bar always showed up paired together in each term of lagrangian the last term for example is just 5 bar phi equals a squared plus b squared the squared length of the arrow

the same goes for the terms with the derivatives because again phi and phi bar always appear together we therefore learn that this lagrangian has rotational symmetry in the sense of the complex phi plane this is the same kind of symmetry that leads to electric charge conservation in electromagnetism another way to say the same thing is to use the fact that a complex number phi equals a plus ib can equivalently be written as mod phi e to the i theta that's thanks to euler's identity which lets us expand out e to the i theta as cosine

theta plus i sine theta but modify cosine theta and mod phi sine theta are just the horizontal and vertical components a and b of our arrow so this is the same thing as writing a plus i b the reason why writing things this new way is convenient is that it makes it very simple to rotate phi to a new angle just multiply it by e to the i alpha for whatever angle alpha you want to rotate by the things in the exponents just add and so after we multiply by e to the i alpha phi

gets rotated around to a new angle theta plus alpha but its magnitude doesn't change the rotational symmetry of our lagrangian is therefore simply the transformation phi goes to e to the i alpha phi and phi bar goes to e to the minus i alpha phi bar and in this notation it's even easier to see why the lagrangian is invariant since each term has a phi and a phi bar when we make the transformation one picks up a factor of e to the i alpha and the other e to the minus i alpha and when they're

multiplied together they cancel out this kind of symmetry is called u1 where the u stands for unitary the terminology comes from the definition of a unitary matrix which is a matrix m that satisfies the property that if you take its complex conjugate and then its transpose you should get back the inverse of the matrix you started with the space of n by n matrices satisfying this property is called u of n for n equals one though the matrix is just a single number and so the transpose doesn't do anything at all in this condition it

says z bar z equals one for a complex number z which is satisfied precisely by our rotation e to the i alpha the other symbols in the standard model symmetry group su-3 and su-2 stand for similar larger symmetries with n equals 2 and n equals 3. the s just means that in addition to being unitary these rotation matrices are required to have determinant equal to one but we'll focus on understanding this u1 symmetry for this video now that we've identified it we want to see how it leads to a conservation law by notice theorem remember

the basic way that node's theorem worked when we studied it before in particle mechanics the point was that under an arbitrary transformation of the coordinates x goes to x epsilon the change in the lagrangian always takes the form dl equals the equation of motion times epsilon plus d by dt of something call it q eom is the thing that vanishes when the particle is following its physical trajectory if we choose a specific epsilon that gives us a symmetry of the lagrangian then dl is equal to zero on the left hand side then on the physical

trajectory where eom equals zero this equation tells us that whatever quantity q comes out for the particular transformation we picked will be conserved dq by dt equals zero so when we had a translation symmetry q came out to be the momentum when we had a rotation symmetry it was the angular momentum and so on we'll discover a very similar relationship in field theory the main difference is that because our fields now depend on both time and space when we compute the change in the lagrangian density we'll get both time and space derivatives on the right

hand side for some quantities row and j x j y and j z that'll depend on the transformation we're making the argument is again that if we choose a specific symmetry transformation for which dl is equal to zero then when the field is a solution of the equation of motion we discover a continuity equation d rho by dt plus del dot j equals zero a local conservation law this is the way node's theorem guarantees that symmetries lead to conservation laws in field theories now we're going to work it out for the rotation symmetry of our

klein gordon theory we'll be running into some long equations for all the different ingredients here if all this is new to you then the most important thing on a first pass is to understand the shape of the argument not to sweat all the details when you are ready to iron them out you can pull up the notes from the link in the description to take it step by step but for now just try to keep track of the basic idea the change in the lagrangian density always amounts to the equation of motion plus some time

and space derivatives and for a symmetry that gives us a local conservation law so for our klein gordon lagrangian we want to see how it changes when we make an arbitrary transformation of the field phi goes to phi plus epsilon where epsilon is an infinitesimal shift it can be anything here including a function of time and space and even phi itself for any random choice of epsilon lagrangian certainly isn't going to be invariant if we make the substitution leaving phi bar alone for the moment we get this new lagrangian these three terms with the phi's

just give us back the lagrangian we started with the new pieces are these terms with the epsilons then the change in the lagrangian under this transformation is dl equals one over c squared dt phi bar dt epsilon minus dx phi bar dx epsilon minus kappa squared phi bar epsilon next up just like when we learn to apply the principle of least action we want to integrate by parts on the first two terms to grab the derivatives on epsilon and pull them over to the other factor at the cost of a minus sign so for the

first term we get minus the second derivative of phi bar with respect to time times epsilon plus the time derivative of the whole thing that's just an identity that we get by expanding out the derivative with the product rule we can do the same with the x derivative term and together we find that the leading change in the lagrangian when we make this tiny variation of phi is given by this big expression this looks like a lot i can barely fit the thing on the screen but remember the idea is pretty simple this is just

what we predicted for dl from a moment ago the first big quantity in parentheses is just the thing that vanishes when phi bar satisfies its klein gordon equation and then on top of that we get some time and space derivative terms indeed this was almost exactly the procedure we followed last time to derive the klein gordon equation by applying the principle of least action the only difference was that in that case we required epsilon to vanish at the boundaries of the action integral but our identity here for dl holds for any transformation phi goes to

phi plus epsilon and of course phi bar will also transform as well in general by phi bar goes to phi bar plus epsilon bar that works out in a totally analogous way and when we add it all up to find the total change in the lagrangian we get an equation of the form we expected the l equals terms with the equations of motion plus d rho by dt plus dxjx dyj and dz jz where i have to find rho to be 1 over c squared epsilon dt phi bar plus epsilon bar dt phi and jx

equals minus epsilon dx phi bar plus epsilon bar dx phi and similarly for j y and jz don't worry these aren't supposed to look especially intuitive at this point but node's theorem is now staring us in the face this formula for dl holds for any transformation but if we now choose a specific symmetry transformation for which dl equals zero then when the klein gordon equations are satisfied we learn that the corresponding row and j define a conserved charge and current so now let's plug in our rotation symmetry it's a symmetry for any angle alpha but

to apply nodu's theorem we only care about the infinitesimal version so let's apply the taylor series e to the i alpha equals one plus i alpha plus dot dot but we only need to keep the first interesting term then our infinitesimal rotation symmetry sends phi to phi plus i alpha phi meaning that epsilon is equal to i alpha phi and likewise for five bar when we plug this epsilon in to our formulas for rho and j we find that the rotation symmetry leads to a local conservation law with discharge density and this current density i've

dropped the overall factor of alpha here because that was just an arbitrary constant don't sweat the eyes by the way the things in parentheses are pure imaginary since they're each a complex number minus its complex conjugate then multiplying by the i out front gives us back a real number since this transformation was a symmetry node's theorem implies that rho and j satisfy the continuity equation and we've discovered a conservation law but a conservation law for what what do these formulas mean well remember that the conserved charge is defined by integrating the charge density rho over

space also recall that last time we discussed how the general solution to the klein gordon equation can be written as a sum of plane waves and these turn into wave functions for the particles created by the fields in the quantum theory by plugging that expansion into this integral you can show that what it does is count the number of particles minus the number of antiparticles suppose these particles carry electric charge q and therefore the antiparticles carry electric charge minus q then the total electric charge is q times the number of particles minus the number of

antiparticles in other words it's our conserved quantity here given by integrating rho up to a factor of q so the same conserved quantity coming from our rotation symmetry counts the total electric charge the densities rho and j from the rotation symmetry essentially become electric charge and electric current densities but once we start talking about things with electric charge that means that in addition to our field phi there are electric and magnetic fields floating around as well and our field theory had better incorporate all of them in the last part of the video i want to

show you the lagrangian for the full theory including the electromagnetic fields to do that it's going to be really useful to switch to the relativistic notation we talked a little bit about in the last field theory video i showed you how to write the space-time coordinates as a four-component vector x mu where mu equals 0 1 2 or 3 is an index mu equals 0 is the time component x0 equals ct and mu equals 1 2 3 are the space components x y and z that factor of c is there so that all the entries

have the same dimensions of length then we can define the derivatives with respect to x mu by d by dx mu equals one over c d by dt d by d x d by d y d by d z and we usually write this even more simply as d sub mu we can likewise put the charge density rho and the spatial current j together into a four component space time current j mu the c is again there so that each component has the same units this actually makes a lot of sense the spatial current density

j vector is the amount of current flowing around through space while c times rho is like a current flowing forward through time even if you just set a charged particle down at rest so that the spatial current is zero the particle is still moving forward through time as the clock ticks onward and c rho is measuring the space-time current in that direction in this notation the continuity equation simply says that the sum over mu of d mu j mu is equal to zero because if we expand it out we get d0 j0 which is one

over c d by dt of c rho and the c's cancel plus d x j x d y j y and d z j z just like we had before in fact we usually don't even bother to write out the sum symbol we just adopt the convention that anytime an index like mu appears twice in any given term like it does here then we sum over all of its values the spatial divergence del dot j measures how much j spreads outward from a given point in space like a water sprinkler spraying water out of a

spigot the continuity equation says that j mu can't have any divergence in space time meaning you're not allowed to simply pop out new electric charges at any spacetime point or at least no net charge now i want to use this powerful notation to write the lagrangian for the full theory including the electromagnetic fields really getting into all the details here would deserve a video of its own or several so for now i'm just going to give you a quick sketch of the different ingredients and how they fit together in our relativistic notation we can write

the klein gordon lagrangian much more compactly like this where ada is another bit of notation we defined last time it's a four by four diagonal matrix with minus one in the top left corner and ones for the rest it just serves to take care of the relative minus signs for us between the time and space terms because remember it's implied that we're summing over both mu and new here when we expand out the sum we get the same lagrangian as before and that'd be good practice for you to check for yourself to get more comfortable

with this abstract notation i'm also going to work in units where c equals 1 for the rest of the video since that's the standard convention in field theory and it makes the formulas look a lot simpler as we've seen l is invariant under the u1 rotational symmetry where i've inserted a factor of q here in the exponent since that's going to become the electric charge in a second this is a global symmetry meaning that the angle alpha is a constant we're performing the same rotation of the field at every point in space-time but electromagnetism and

the standard model as a whole are examples of gauge theories in which the defining symmetries are promoted to local transformations in space-time in other words the u1 gauge symmetry of electromagnetism is obtained by demanding that the lagrangian is invariant under this rotation for any choice of alpha of x including one that depends on what point you're at in space and time our original lagrangian is not invariant under rotations when alpha is a function of x the second term is still okay because the two rotations coming from phi and phi bar cancel each other out but

the terms with the derivatives are no longer invariant when alpha was a constant we were able to pull those rotations right outside the derivatives and then they canceled against each other and left the lagrangian unchanged but when alpha of x becomes a function of x we'll get additional terms from the product rule when the derivatives hit the alphas and these do not cancel out the way to resolve this problem is to replace these ordinary derivatives with what's called a covariant derivative it's defined by the regular derivative of phi plus i q a mu phi where

a mu is a new field the electromagnetic potential and you've almost certainly met at least part of it before it's another four vector with a regular old electric potential in its first slot in other words the voltage and in the spatial slots is the magnetic vector potential which is needed in addition to v once magnetic fields are thrown into the mix they're related to the electric and magnetic fields by taking a few derivatives i'll hopefully go through those details in a later video what this covariant derivative does for us is restore the nice transformation property

that leaves the lagrangian invariant even when alpha is a function of space-time provided that when our transformation rotates phi and phi bar it simultaneously sends a to a plus d alpha this is called a gauge transformation and it reflects the fact that the electromagnetic potential isn't uniquely defined different choices for a mu related by gauge transformations will describe the same electric and magnetic fields i'll leave it as a little exercise for you to check this transformation property the upshot is that if we replace the ordinary derivatives with covariant derivatives in our original lagrangian we get

a theory that's invariant even when alpha is a function of space-time because the rotations again come outside the derivatives and cancel each other out the result of this procedure which just looks like a trick at first glance but has deep theoretical underpinnings is that the field phi now carries electric charge q and phi bar carries charge minus q another current is now a source for the electromagnetic field just like any electric charges and currents would produce electromagnetic fields according to maxwell's equations but speaking of maxwell's equations there's still one thing missing from our lagrangian maxwell's

equations determine how electromagnetic fields are produced by charges and currents and how they evolve with time they're the equations of motion for the electromagnetic potential a mu just like the klein gordon equation was the original field equation we had for phi and there ought to be terms for a mu in the lagrangian that give us maxwell's equations when we apply the principle of least action again i'm just going to quickly tell you the answer right now and in the future i hope to tell you more about where it comes from and what it means from

the electromagnetic potential a we define the electromagnetic field strength by taking a couple of derivatives f mu nu equals d mu a new minus d nu a mu it's a four by four matrix again that packages up the electric and magnetic fields in a particular way like this but we don't really need this whole matrix right now using f munium the lagrangian for pure electromagnetism is actually very simple it's basically just f squared where this combination with the upper and lower indices is another shorthand sorry there is so much notation in this subject it's what

we get by matrix multiplying with eta to raise the indices it looks like something fancy but again the point is just to take care of those pesky relative minus signs between time and space terms that always come up in special relativity altogether then here's the lagrangian for the electromagnetic potential a and our electrically charged field phi if all this is new to you then this is a ton of information to take in so quickly so just take this as a teaser for future lessons where we can dive into more of the details speaking of which

let me finish with one last teaser what we've been talking about in this lesson by putting the klein gordon theory together with electromagnetism is called scalar electrodynamics and it's a good example for starting to learn the ideas of symmetries and gauge theory without too many extra technical complications and it is physically relevant because the piece of the standard model lagrangian that describes the higgs boson is a generalization of this theory but the most fundamental theory of electromagnetism is the theory of the electron field and the electromagnetic potential that's called quantum electrodynamics or qed and it

was the first piece of the standard model of particle physics to be understood the electron is not described by a scalar field like phi it's called a spinner field usually denoted as psi here's what its lagrangian looks like spinners make things a little bit more mathematically complicated but the ideas we've learned here go over directly to qed it starts with the theory of a free electron which has a global u1 rotation symmetry we gauge the symmetry by replacing the ordinary derivatives with covariant ones which means that the field becomes electrically charged and finally we add

on the maxwell lagrangian for the electromagnetic field itself the rest of the standard model is a generalization of all this with more fields and a larger gauge symmetry these symmetries and in particular gauge symmetries are fundamental to the construction of everything we know about particle physics we went through a huge number of concepts in this video so pull up the notes from the link in the description to take your time processing all the information also remember to check out the link to blinkist where you can start exploring your curiosity with bite-size summaries of some fantastic

books thanks again to blinkist for sponsoring this video and thank you for watching and i'll see you next time