Transcript for:
Quantum Field Theory Introduction

foreign yes I'm Hong so uh you can just call me Hong um I I'm a theoretical physicist I welcome a high energy Theory including screen Siri quantum gravity line equivalent statistical physics Etc many different topics so now let me say a few words regarding the qft at the quantum field Theory itself so the goal of this Quantum field class is to develop concepts for momentum anti-clicks for Quantum Dynamics or fields say I'm sure all of you have studied Maxwell equations okay the Maxwell equations describe fully classical Dynamics say of electric and magnetic fields but we ID but we live in the quantum words and uh so we should also treat electric a nitrogen and magnetic fields using uh using Quantum language okay and they and then you find that once you do that and then and then the the concept of the photon uh once you treat the electric field and the magnetic field Quantum mechanically and then you get the concept of the photon uh under the photon now is the fundamental particle which mediates say electromagnetic interactions okay so you you actually get a completely different physical picture from what you get classically and yeah so the quantum uh uh so one goal of the class is for you to appreciate say uh to understand the quantum electrodynamics okay and and that's actually where we will end this class and this we will discuss uh the the quantum naked Dynamics and um Quantum field series is also very important for many other for other interactions in nature so among the four fundamental interactions in nature three of them are described by Quantum field Theory completely okay and you can also use quantum field Theory to describe gravity to describe quantum gravity but but at the moment the quantum field does not offer a complete description of a quantum gravity but still you can use the quantum field Theory techniques for for understanding certain questions of quantum gravity and also um Quantum philosophy over the years even though it's initially developed for particle physics but over the years has been also found many applications in many many other branches of physics in condensed matter statistical physics Etc and it's fair to say nowadays Quantum field 3 has become a universal language okay for uh for theoretical physics essentially many in all different fields and so if you are serious certainly you lead to master this language and even if you are only interested say in atomic physics or or statistical physics and the field Theory concept will be very important and for experimentalists to know basic concept of quantum field Theory and to have some basic understanding of it should also go a long way help you to appreciate say the most recent theoretical development and also for help you to communicate with series and yeah and the so so this class is the first of a series Master sequence and so so this semester we mostly developed the fundamental concept and the quantum field Theory two and three will be more about technical development and say if you're experimentalist and then you can view this uh Quantum fields that we want as a standalone class and which yeah just enough for you to to get the basic idea of quantum Fields Quantum serial fields which you don't yeah you may not depend on your leads and uh and whether you have two or three or not uh yeah may depend on your leads okay so under the main topics I plan to cover are listed in the outline this document of the outline which is already on the website and uh um so so I should emphasize that outline is a rough it's only a rough road map I may change it depends on the pace I may change things along the way and uh sometimes I change Minds say halves through the course somehow I feel uh uh um yeah anyway uh so so don't treat it too literally and any questions about this subject or Quantum field Series so far okay good and also let me say a few words that the quantum field three has a reputation of being a very difficult subject okay actually indeed myself have suffered a lot when I learned it myself okay but but with 20th 2020 hindsights and from also from interacting with 90 Lombard students through teaching various level of quantum field Theory classes along uh yeah in many years I can assure you now that actually quite a few Theory is actually not difficult at all okay if you landed the right way and of course the lending things yeah of course anything is not difficult if you lend it in the right way and so in the sense this is an empty words but uh but but keep it in mind uh whenever you find this you think it's too hot and there might be the reason might not be the reason might be the you have to change your perspective okay you have to change your perspective and uh so so quantity one thing people complain about is that the controversity often involve a lot of calculations and that's true uh that's just the fact of life you cannot avoid it but that's not what it makes it difficult okay a complicated calculations you can just go through them from one line to the left line to the left line if you're careful enough patience enough you can go through so so the difficulty I think for most people of quantum field Theory is more at the conceptual level it's it's because the this subject is not a very intuitive subject it's not something you can just understand just by thinking okay uh so that's where I emphasize earlier the exercise and the really working through it is very important it's a little bit like quantum mechanics in quantum mechanics your intuition was developed through examples by walking through many examples you slowly develop intuition about the quantum mechanics and and if you learn all the lessons and then you uh you get good feeling about quantum mechanics and the quantum field series is the same thing I said you have to uh it's like some kind of intuition is has to be developed okay it's not something you can easily uh uh yeah just like mechanics which you can uh in some other subject maybe if you have very good intuition you can just you can just imagine it okay um yeah so um so so to help you develop good intuition about Quantum Fields you I can offer you three pieces of the devices okay so so first yeah so the first piece of a device is that the quantum field Theory is essentially quantum mechanics but dealing with infinite number of the good Freedom okay so so in your quantum mechanics class you always treat this final number because freedom but the quantum philosophy the difference for Quantum field is now you treat the infinite number to good feeder it turns out that this treating infinite numbers Freedom makes a difference okay so more is different and uh and actually sometimes make conceptual differences and so that's why sometimes uh uh uh the uh the uh the quantum Quantum field theory is unintuitive okay but that said I found for many people including myself when I learned it many difficulties you encounter in learning Quantum philosophy is not due to the difficulty in Quantum filter itself is actually due to your Gap in understanding of quantum mechanics so so whenever you encounter something you don't quite understand in quantum philosory try to step it back to say can I formulate this difficulty in terms of quantum mechanics with only finite numbers equals freedom and often you find actually your difficulty can already be formulated in quantum mechanics and then that way then you should be able to just settle it yourself okay because we are supposed already to be a master of quantum mechanics okay and certainly uh uh uh a Long Island Quantum philosophy I was stuck at a certain points for a long time and then later I realized just because I didn't understand certain Heisenberg fiction of quantum mechanics very well okay when you understood somehow I realized when I understood the Heisenberg picture of quantum Mechanicsville and those difficulties just went away and I have nothing to do with Quantum filter itself so that's why in your first preset you will get familiar with Heisenberg picture of quantum mechanics okay yeah that came from my own experience and also the second point is that Quantum philosophy deal with formalisms and sometimes this subject will look seems very formal okay you uh you have lots of formalisms okay but just keep in mind any formalism in physics no matter how abstract it is it was always designed to solve some concrete physical problems and physical questions very concrete physical questions and if you understand what kind of concrete physical questions Quantum filtering was designed to solve then that can give you very good perspective on those formalism and why people do this why people do that why people do districts where people do that trick and because they were invented to solve certain concrete problems okay and once you understand the questions understand the problems then the formalism become much easier to understand and the third thing is already already said is that the in Quantum field Theory as in quantum mechanics intuition was built through experiences okay through examples so so when you do your PSAT when you look at the examples in the class you should always ask yourself afterwards say say after you have done your PSAT problems always look back at that program say what did I learn from this problem okay and just think through it again think through uh think through what you learned from their problem again and that is a very good way to help you to learn from your experiences and to help you develop intuitions okay and uh so yeah so so also very important thing you should keep in mind he said in The Graduate course like this and most of the things should be learned outside the class so inside the class the purpose is to give you a guide okay is to emphasize the conceptual picture and the physical intuition Etc and so sometimes I will leave some details for you to finish in the preset and sometimes the PSAT will involve problems which I did not say fully discussed in lecture but I want you to work out yourself okay and uh so so P said this important part of the lending even new things okay not just to practice something but but but it's also a very important part for for learning new things [Music] um right good also finally I would like to make an apology so you will soon find that the rotations are used in the lecture are different from the rotations in the recommended reading books okay so I recommended reading passing and Weinberg and you will find that my notations are can be different from them also the order of presentations are also different from them okay I know this is very annoying but but they just know no proof no perfect textbooks and there's no perfect set of rotations everybody use and we all use the notations which we find the most convenient to use okay and uh so so even though I realized this problem but I don't have a good resolution okay so so just keep in mind the rotations in my lecture can be different from the rotations in those uh uh textbooks okay good so so do you have any other questions good okay so if you don't have any other questions so let's start so um so the chapter one will be about y we can see the quantum Field Strip okay so first we talk a little bit about the classical field series to stand the stage for Quantum field service okay and the first important concept is called the principle of locality so if you remember from your say high school days Newtonian mechanics so internet mechanics you have action as a distance okay for example if you look at gravity and gravity is exerted by the sound on the earth which but they are very far away okay and uh uh yeah and the same thing with the coolant uh interactions between the charged particles but then in the 19th century they came from this Prince of locality and that's formulated by variety Faraday around 1830 okay so so the principle of a locality said all points you actually don't have action at the distance is at all points in space participate in the physical process okay and the the effect s so if you have interactions okay so you facts propagate foreign Point okay okay so uh so in this principle of a locality you don't have action at the distance so acting at the distance the always conveyed the action is always conveyed from one point to another Point through the propagating in the space okay and the fields the complex the concept of the fields is the mathematical device okay or vehicle that the principle of locality is at work okay so so this is essentially the uh the the device we need to use to realize that this principle of locality and uh so um so the so the main idea of the field is that we associate each point with each point in space or dynamical variable okay or dynamical variables okay so for example so so for example if you look at say if you have electric field so your Nitric field is defined for all space Okay so so at each point x we can introduce a a electric field okay and then and then this uh electric field can also depend on time and uh uh uh uh um yeah so of course normally we write it this way e is x t and the reason I write it this way is to emphasize that in the definition of the electric field the space and time actually play very different role the space plays a role of a label okay so at each point we have electric field okay at each point of X we have electric field and and so X here is just a label okay and the T is used to divide to describe the evolution of the change of the electric field okay so the x and t plays a very different role and similarly you can do it for magnetic fields it's the same thing okay and here you should always View acts as labels okay so the spatial points which you develop this Vector X is the labels and we know that the uh the the the evolution of electrical in my left fields I describe by Maxwell equations write them down to remind you so we have dot b 0 dot e row so this is so-called differential form of the Maxwell equations so the reason I'm taking trouble to write them down is to emphasize in the following point so this set of equations exemplifies perfectly the principle of locality it's because you see those equations only involve the value of electric fields and the magnetic field at the single point okay so if here is at X point so here is also at X points you never say say level say x here and here is some other point why okay and so this is refracted the principle locality that the uh everything so so the effect is propagating from point to point okay you just have the derivative of the same points okay it's never uh you don't involve separated points okay so these are local equations so these are we called the local equations and they contains only e and B at the same point and also the charge density and the current density at the same point responding number derivatives okay so the derivatives are the ones to help you to propagate to propagate okay because it relates the uh the point to the labeling Point okay and so so that's how it propagates it through the derivatives okay so the derivatives are key okay the derivatives are key another example with examples locality which I will not go into here which some of you may know is the instant gravity so Einstein's general relativity so in ice and gravity the dynamical variables also called space-time metric so they are the object with two indices two space time indices okay and and then the Einstein equations are equations for this kind of object and again the equation is a local equations in the sense that they only depend on uh the G evaluated at the same point with five line number of derivatives okay so so United sense gravity you no longer have action at the distance okay so so the so the fact of the gravity is propagated okay through the space time through space-time so so in fact so here we are say these two examples exemplifies the principle of locality in fact the principle of locality played a very important role in formulating those equations okay because because as we will very soon see when you have principle of locality you can significantly constrain uh the CV you can write down okay so that's a very very powerful principle yes [Music] yeah you can have one local equations but the front but it's believed that the fundamental equations in nature they're all local yeah yeah and uh uh so far the equation is common or fundamental interactions or four different interactions in nature the equations are local other questions yes sorry x and t why did you pick it up particular time component um say it again I I don't quite understand the question yeah kind of treated yeah oh no no no no um here I'm just one emphasize again that the of course in Einstein's theory this these two are treated the same way but but if you think in terms of the fields and they actually have very different physical interpretation yeah yeah so that's why I read this way yeah yeah it's in the same uh same way I write it like here yeah yeah same reason just here to emphasize for this uh uh uh uh uh to emphasize the different role played by x and t yeah but then but I think you're asking a very good question so I'm going to mention later uh uh uh in the writing Mystic series then of course then these two become the same yeah become a play the very equal drop yeah does that answer your question yeah okay good good also yeah let me also very quickly mention the different types of fields okay so you can have What's called the scalar field scalar so these are quantities which are given point given point there's only one value okay say for example the temperature so it's just something just it's just a single quality single quantity defined as the uh at the point for example the temperature okay and exactly maybe other quantities and then you also have a vector field and E and B the examples of a vector field because at each point you have a vector okay so here I'm using the three-dimensional notations but in the rativistic series you can also use the four Vector rotations say for example the vector potential we have the following form say a mu will be a four factor and then the space time I combine them together into a four Vector okay and so so aside each point and then you have a four Vector okay and so um yeah and also you have tensor field so this Matrix is an example of a tensor field here you have two components you have two indices okay so so you have many many different components now depending on the uh so again this is relativistic notation because I can write it as a right basic notation okay so at each point and now you have a now you have some object with two indices okay and you can also have later we will see you you also have something called spinner fields Alpha and this Alpha is some other indices which we will Define later okay okay so so you can also have so-called Spindler and Alpha some other indices okay and that's our convention is that mu is always from 0 to 3. okay then when I sometimes I just write X when I write X just means a four Vector means X mu you can do c t and X so X always denotes X Vector always below the spatial vector okay and um yeah and then this is the same I see t x i okay so the spatial in this is always denoted by I and yeah and uh so um so partial mu will be the same as 1 over C partial T and then the derivative on the gradient on the on spatial directions and say if you have a four vector then again you we have the convention that a0 then the a vector is the same also as a0 and a I I use the I to get out the spatial components and and this is the last time I will write speed of light so the C will always take to be 1 and H bar will take into the Y okay that's for rotational convenience you have questions okay so now so this this preparation then we can talk about action principle for classical fields okay so first we call in in your classical mechanics so we introduce the action which is the integral of lagrangian and lagranging is a function of x is your variable and x dot and the time okay and so this is a one-dimensional yeah this is 801 one dimensional particle motion and you can also introduce a momentum canonical momentum which is defined by partial L partial X Dot and then you can also Define a hamiltonian and Tommy Tony is related to the lagrangian by p x dots minus L okay from a legendary transform and the equation motion is obtained by extremise okay so s is considered to be a functional of your trajectory so whatever check you have and you optimize this s and then you get the equation motion okay so this is a so now we can generalize to field Theory okay so now from principle of locality so for Fields TV from Principal locality so the form of the lagrangian so the form of the lagranging L okay so so again we're here for field we can again Define s as a Time integral over lagranging and the L and the formal L is significantly a constraint okay to have the phone in form so you must so you must have the form l equal to a spatial integral so this is integration of all spatial directions okay so d3x and the sum scene so an image I will first write down the notation and then I will explain the rotation okay yeah maybe just write partial eye okay so um so let me not just explain notation a little bit well so so here I use a shorthand notation to download the fields so so file a so it would be a function of spatial Direction on the time is General fields and a label different fields okay so this index a labels different fields okay so for example a can label different scalar field if you have multiple scalar Fields then can they can enable them and they can also refers to indices a space-time indices like a mu and also yeah uh Etc okay so a just label whatever field you have okay and then the second point he said this l the script l is a function the emphasize here is a function okay of Phi a and uh and and these derivatives okay and these derivatives so in other words the L so this is a key Point okay so that is L only depends on the value of Phi a and its derivatives at a single point okay say say x and then you integrate over Ox okay so so AO is called the lagrangian density sorry say it again yeah yeah yeah yes yeah yeah I will mention that yeah I will mention that so um so here just here I just uh explained the rotation okay so now let me just make some remarks on why the actual the LaGrange must have this form so first the principle of locality implies here must only involve a single integral okay because the locality that's not allow something like this say for example does not allow a term like this okay that's not the dial term like this which involving the uh the file at a different points okay why it's because if you have terms like this in your LaGrange okay so so later you will oh sorry l air script so if you look around you have this kind of terms and then so as we will describe the equation motion later I you will see from the equation motion then your equation notion will not be local okay so equation notion will involve the behavior of the uh your field at one point and then influence by point at some point far away okay and then and then you will not have local so the locality significantly constrained because if you could give away locality and in principle your logonian can be arbitrarily complicated it has many integrals as you want okay but because of locality you only allowed to have such a simple integral a y integral of a function okay so this is the key okay so so that's not allow so here is key point so so the second point is that we only allow first derivative in time okay we only allow first derivative in time we don't allow the second derivative in time so the reason is that again as you will see equation motion if you involve the second derivative in time in your in your action or in your lagrangian and then when you get the equation motion you will get equation motions involving more than two derivatives in time okay so this will lead to so this implies the equation motion only contains two derivatives two times the reviews okay so this constraints come from so you can stream come from our experiences it's the same reason here we only include uh the first derivative in time okay it's because the uh in in real life all the experiment is determined by the initial condition the initial condition you only needs to specify the location and velocity okay you don't need to specify more if you increase in motion involving more than two derivatives then you need to specify a more General initial conditions and so yeah so here is the same thing uh we only allow the first Duty with time but you can in principle now or or arbitrary Rumble derivative in spatial Direction okay and but for Simplicity for the most of the time as we will see we will restrict to Quantum field Theory in special relativity okay means that there will be a relativistic invariant maybe lorenzing warrant and you know renting wire in the series space and time they can transform to each other play equal role so if you only have single derivative in time you only have single time in in spatial derivatives so the example we will see are they will all have only single derivative in spatial directions okay but certainly non-relativistic systems you can have a higher a number of spatial derivatives okay good any questions on this okay good so now so as in classical mechanics allowing we can introduce the canonical momentum chromatonin Etc okay so so here we can so so here we can introduce so-called the canonical momentum density the reason we call it densities will be clear so so we can so remember for this Phi so X is just a label okay so you can just view this Theory essentially has infinite number of the such kind yeah it's an unfortunate rotation here we use x at the dynamical variable okay but here the x is only a label okay here x is a label so here you can just imagine you have infinite number of degrees Freedom just each one is labeled by X okay so now just imagine here you have many many X just uh you have some labels for it and so so for each such one we can deduce this momentum Okay so so each Phi a we can introduce is canonical momentum okay if I a DOT derivative okay oh yeah yeah maybe let me just so so defined as the derivative of the Argonian density divided by time derivative of Phi a okay so so this is just the direct generation of here so remember so this is evaluated at each point okay in spatial Direction and uh each yeah so this thing will also depend on x and t okay oh this is just the capital pi yeah Capital pi okay and this five a DOT just a Time derivative of I okay and remember X are the labels okay so X does not do anything here so the reason we call this lagonian density is because the momentum density is because L is an LaGrange density okay and uh um yeah and then the we can also Define the hamiltonian density the same way freedom so we have to find the script h which is defined by Phi a DOT Pi a again this is all defined on the same spatial point minus l okay so this is the quantity density and then the then the hamiltonian it just uh you just integrate over all space essentially sum over all the good freedom foreign okay just give him yes Udacity yeah yeah you just like you sound so if you treat the uh uh uh yeah so yeah yeah yeah yeah so so the a is summed so I will always yeah I forgot to mention good this is a great question so so I always assuming this Einstein convention in the sense that all the repeated indices are assume to be summed so so no matter how many components are there you just sum over a okay so a is sound here yeah good other questions good so now we can talk about equation motion so we just it's the same thing as classical mechanics so classical filter is just like classical mechanics you generalize to infinite language freedom so now with each point in space your you associate with some degree freedom okay so so now let's look at equation motion okay so again you just do the vibration of the action you extremise of the action to be zero okay so the action is again just defined by the DT over lagrangian uh the lagrangian okay so um so the s is the dtl and then can be written as the four integral DT and DX of the density okay so now let me just write let me now just write it in the more writer basic notation just assuming there's one derivative once time and one spatial derivatives okay so the mule run now yeah combine the space and the time derivative together and you can straightforwardly generalize to say moving more spatial derivatives good so so here so uh so let's just do the variation so the so we want zero equal to Delta s so you want the variation of this s to be zero so now let's just worry s so you have d4x now just remember this nagonian density is just ordinary function because of the locality just ordinary function of Phi a and its derivatives okay and we can just do the um we can just do the variation in the straightforward way you can just write a partial l partial file a in Delta Phi a plus partial l partial partial mu Phi a and Delta function mu Phi a okay and in this case yeah when you so the Delta is a variation and the partial mu so these two operations are independent on each other so you can exchange them okay so you can put extra Delta inside so this is the same as partial Mu Delta Phi okay so now you can integration by part of this second term okay now now you can do because here is a little bit awkward because you move in the partial mu data Phi a so again here the the repeated indices they're all summed okay so you should sum them so now we can do integration by part here so so then you get the 4X you get partial l 525 A minus mu plus l I should passion mu by a okay and then then the whole scene data Phi a and then pass boundary terms so the boundary terms come from U2 integration by part here okay you're doing integration by parts here so so the boundary return will be proportional to to Delta 5A okay yes thus I just started to evaluation yeah just arbitrary variation other questions okay so so the boundary so we always assume we have the boundary conditions okay and so that the boundary return vanishes okay so so here we are not going to detail so we always choose 35a so that boundary terms vanishes okay the term come from integration by parts okay and the second uh just remember the heated indices or something okay and now so this has and then then we don't have to worry about boundary terms so the boundary term vanishes then now we just have this has to be zero but this has to be zero for any variation so so this Delta x view is can be any function of x okay can be any function of s the only way this can happen is that this pre-factor must vanish okay so this implies that the E Delta X partial Phi a minus partial mu pressure l foreign okay so this is the general equation motion for for classical Fields okay good any questions on this okay so it's so from now on we will make two restrictions okay just for Simplicity also they are the most uh uh uh will make two restriction just for Simplicity uh um yeah most of our discussion can be generalized uh Beyond those situations uh um yeah so the first you said we restrict to Phil C which are translationally in warrant means that there's no special point there's no special means there's no special or or space-time point in this series okay if you do experiment here in in Boston it's the same as you do experiment in Washington DC Etc okay and if your theory it's not translating warrant and then when you do experiment in Boston then we'll be different from you do it in say in New York and the second is that we assume it's low renting warrant so in some condensed meta applications which you don't have fluorine Symmetry and uh then you don't have uh uh Lawrence symmetry but uh um yeah but but similar techniques we are going to talk about can be uh can be applied yeah and we will also elaborate a little bit more on both of these aspects okay so now let me give you some simple examples of the classical field series okay which satisfy these two conditions um examples so the first one is the Maxwell Theory so for markzware series the dynamical variable and these four Vector potential a mu okay so now I'm using the four vax rotation so that will often suppress this uh the the index on X that means this is a four Vector okay and then from the AMU you can Define the field strings and the E and B then can be obtained from this F me mu okay yeah and we can be a painful and then they uh then the action for the year and M then can be obtained by okay that's these okay so this is the uh just the action for the for the back for the uh e and the M we not charged matter okay so if you do the variation here okay to get the equation motion then you get the maximum equations with that without sources okay you get the vacuum maximum equations equations and so so you see that this action have the form we mentioned here is corresponding to a local Theory okay a single integral of space time okay and the second example is Einstein gravity there's a lot of classical field Theory okay so so let me just write down the action so this is written as G Newton so G is the metric yeah anyway if you don't know the Einstein series okay okay so R is a rich scalar and uh if you don't know the Einstein series okay and just write it down just to show that this is a local Theory involving only a single space-time integral of some quantities and at the same place Quantum field Theory called the scalar field series because I think I erased it because this involving the vector field okay so this is a vector field people that each space time Point there's a vector here here the Einstein Gravity the dynamic variable is a tensor so it's even something more complicated so the simplest case is a scalar which at each space time point you just have a single value quantity okay and so so let's just consider the simplest case just consider a real value scalar field Phi okay so so this is just take a real value you can also can see the complex value okay so so but the simply is just a real value so each space time point you have a single value okay and so that's the dynamical variable and so in real life there are many examples of scalar fields so for example the Hicks okay the the Higgs field which is uh celebrated Higgs field which was discovered uh a number of years ago was a is a real scalar field and also pions okay the pyram particle can be considered as excitations of a scalar field anyway so so simplest family of a scalar field action is a following form okay so so here I just write this action down but essentially you can argue this is the most General action you can write down which are compatible with locality okay with locality and with translation symmetry and the Lorent symmetry okay so so here you see the Lorent indices are contracted in the Lowrance indices in the derivative or contracted so that guarantees this series laurentine warrant and V just some function of Phi okay B here is just some some functional side so so this is the uh essentially the theory uh uh yeah the simply see you can write down based on localities okay so say we can take V5 in general we can take it to be polynomial yeah just some function of five okay so when you have a translation symmetry means that in this series if you do experiment it's the same everywhere that implies that V Phi so here there's nothing there's no parameter here V5 can must yeah just yeah sorry all parameters in V5 must be constant okay they cannot depend on space time okay must be constant so if you have some parameter depending on the space time then then of course a different space and a different space time point will behave differently and it will not be translation environment okay so so let's work out let's try to work out a work out this area a little bit in more detail so let's find this like one in this let's try to find its uh momentum and hamiltonian so it's momentum density here there's only single Fields so so there's no in the index so pi x would be take derivative of this so this is l okay so this thing inside the bracket is l l partial five dots okay time derivative of Phi and so if you look at here here the attack at the time derivative just Five Dot Square okay using these four Vector notation and if you expand it and then and then the time the momentum density Justified dot X okay because the uh because partial mu by partial view file is equal to Minus 5 dot square plus okay and then and then you can find the hamiltonian density dot minus l and then you find that this is given by being expressed in terms of momentum and equation motion if you apply here plus uh if you apply here then you find given by partial squared equal to partial V partial Phi you okay and then again here I'm using a shorthand notation partial Square is defined to be partial mu partial mu okay so this is the same as minus Pi t squared okay so here let me also give you some simple examples of the V in the simplest case so in the simplest case is you take V just of course to be a constant but the content does not do anything because the uh yeah so the simplest case is just V5 equal to linear term and sometimes translation symmetry this F has to be a constant okay it cannot be dependent on space time if it is depend on space time and then you will violate the space translation symmetry okay so so when you have this then your equation motion given by pi Square equal to 5 to F okay f is a constant so in this case so essentially you have some kind of scene which some kind of constant which source this file so in this case we say this system have external Force okay we call this F external Force because because the this if f is non-zero then Phi cannot be zero okay if it has to belong zero and so so in the sense the five will be always be excited uh it will always be uh be generate a long triple five we always be generated if s is non-zero okay so here we we call it external Force and so so normally we don't normally we like to consider five and only we will consider yeah normally in the situation uh of your increase in the situation there's no external Force okay the system develops on its own so in that case so if we forget about external Force then the simplest situation will be V5 equal to quadratic okay some polynomial a a five square okay so that's this is the next simplest function of Phi and M square has to be a constant okay again from the translation symmetry so in this case then you have equation motion partial Square Phi equal to M Square Phi squared or M Square Phi equal to zero so this is a very famous equation so this is called the client golden equation and the next time we will uh yeah you will uh see very quickly why why this is a famous equation okay and um um yeah yeah so this would correspond to the simplest field Theory we can consider and then you can consider more communicative field series with V5 to be some higher polynomial or more complicated functions but but keep in mind that this square is special we go in square this is a linear equation but whenever you have a something which is not linear or quadratic and you will get done in the equation and then the story becomes complicated okay not in the equations are always complicated and anyway so uh so uh so uh so this will be the simple series we will consider okay and uh okay so let's stop here