Physics Diary

SiftS 2019

SiftS 2019 concluded on Friday. It was an enjoyable two weeks of study and discussion on topics in string theory and holography. Eran Palti and Kyriakos Papadodimas were for me the highlight of the event. This is not meant to take away from others, it is just that Palti and Papadodimas were one of the main reasons for my attending SiftS. I could sit and listen to Papadodimas talk physics for hours. And Palti’s lectures on the Swampland were outstanding, as expected.

If I had one minor personal grief about the summer school as a whole, it’s that there wasn’t enough pure string theory. But it is very likely that I would say this at a number of different engagements, with the exception perhaps of Strings 2019 and String-Math 2019, two of the main string conferences. So it is unfair to make any such complaint formal, and one must also be mindful that while string theory was the theme, the engagement wasn’t necessarily meant to serve pure stringy discussion.

All of this is to say that I am both thrilled and honoured to have had the privilege of attending SiftS 2019. To mark its conclusion, I want to take a moment to congratulate the SiftS organisers for putting together a terrific summer school. I also want to take a moment to thank everyone at the Universidad de Autonomous Madrid for their hospitality and support throughout my stay. My impression of the university before arriving was that it was one of the best in Europe, and I left the campus and the Instituto de Física Teórica UAM/CSIC with the same view. I can say with honesty that I very much look forward to my return at some point in the future.

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Now that SiftS is over for the year, and with the conclusion of my admittedly brief holiday during the weekend, I have returned to my research and studies at the University of Nottingham. There is a lot to discuss and catch up on with Prof. Padilla, with a number of possibly interesting ideas percolating. My return to Nottingham also means that I will start actively blogging again. In addition to covering some interesting topics from SiftS 2019, I am also working on a number of research projects which will be nice to write about in the coming days, weeks, and months. I will also be continuing my series of string notes, where the reader and I are on our way to covering the whole of textbook bosonic and superstring theory. We will start from where we left off, namely an introduction to conformal field theory. (In the background, I am going to continue working on my blog to fix the LaTeX of older posts as a result the move).

With regards to SiftS 2019 in particular. I will not write about all of the lecture series and topics covered. Instead, I will focus on sharing my notes and thoughts from the lectures by Palti on the Swampland and by Papadodimas on the Black Hole Interior. This will serve as a nice opportunity to also reflect on some of their respective papers, and to summarise key arguments.

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Physics Diary

Papers: Holography and the cosmological constant problem, plus a new publication from Zwiebach

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Non-perturbative de Sitter vacua via $alpha^{prime}$ corrections – Barton Zwiebach

Barton Zwiebach has a new paper out. It’s rather lovely.

The paper was written for, and submitted to, the Gravity Research Foundation 2019 Awards for Essays on Gravitation. Zwiebach’s contribution was awarded second place. However, given the content and general parameters of the first place paper, I believe Zwiebach’s contribution could have easily been given the top prize. (Granted, I have my stringy biases).

As for the paper itself, the premise is both subtle and interesting. The stage is set in the context of two-derivative supergravity theories, in which a large – indeed, infinite – number of higher-derivative corrections correlate with the parameter $alpha^{prime}$. One should note that $alpha^{prime}$ is indeed the dimensionful parameter in string theory.

One of the challenges since the 1980s has been to achieve a complete description of these higher-derivatives. It turns out a rather interesting approach may be taken by way of duality covariant ‘stringy’ field variables, as opposed to directly supergravity field variables.

From a quick readthrough, the idea presented in Zwiebach’s paper is to drop all dependence on spatial coordinates, such that only time dependence remains. The time dependent ansatz enables a general analysis on cosmological, purely time-dependent backgrounds on which all the duality invariant corrections relevant to these backgrounds may be classified. Leveraging the duality group $O(d,d, R)$, an $O(d,d, R)$ invariant action is constructed of the form

[ I_{0} equiv int dt e^{-Phi}frac{1}{n}(-dot{Phi}^{2} – frac{1}{8}tr(dot{S}^{2}) ]

Then, Zwiebach includes in the two-derivative action all of the $alpha^{prime}$ corrections. Around this point, the work of Meissner is referenced. Admittedly, I will have to review Meissner’s work moving future. Meanwhile, I rewrite the general duality invariant action from the paper as it is quite nice to look at,

[ I_{0} equiv int dt e^{-Phi}frac{1}{n}(-dot{Phi}^{2} + sum_{k=1}^{infty}(alpha^{prime})^{k+1}c_{k}tr(dot{S}^{2k}) + multi-traces ]

This action, it turns out, encodes the complete $alpha^{prime}$ corrections. Moreover, these corrections are for cosmological backgrounds. The aim is thus to classify the higher-derivative interactions, and then derive general cosmological equations to order of $alpha^{prime}$. Upon deriving these equations of motions, the key point to highlight is how one can then go on to construct non-perturbative de Sitter solutions.

Perhaps it remains to be said that de Sitter vacua are a hot topic in string theory. In the end, Zwiebach finds a non-perturbative de Sitter solution – that is, non-perturbative in $alpha^{prime}$, which is very cool.

After only a few minutes with this paper, I am eager to dig a bit deeper. Needless to say, it is intriguing to think more about the string landscape in the context of its study.

AdS/CFT and the cosmological constant problem – Kyriakos Papadodimas

This 2011 paper by Kyriakos Papadodimas is on the cosmological constant problem in the context of the dual conformal field theory. The preprint version linked is several years old, and so I should preface what follows by stating that I have not yet had a chance to review all of the papers that directly proceed it. Indeed, this paper ends with the allusion of a follow-up, and I look forward to reading it.

Another comment for the sake of historical context, this paper on the CC problem precedes by almost two-years the publication of two important papers by Papadodimas and Suvrat Raju, the focus of which is on the black hole interior and AdS/CFT.

Now, as for the paper linked above, one of the key questions entertained concerns whether holography theory can teach us anything new about the cosmological constant problem. Of course the CC problem – or what is sometimes dramatically referred to as the vacuum catastrophe – represents one of the big questions for 21st century physics. It is one which Prof. Padilla and I sometimes talk about.

In approaching Papadodimas’ paper, the following question may be offered: is it possible that Nature’s mechanism for apparent fine-tuning may reside in a fundamental theory of emergent fields?

In the first section or two, Papadodimas builds an analogue of c.c. fine-tuning in terms of 4-point functions. We come to understand that, in the CFT, 4-point functions can more easily be translated into correlation functions. Using Witt diagrams, which are like Feynman diagrams, one takes the external points to infinity – or, in this case, to the conformal boundary of AdS. (The basics of this approach will most certainly be studied in my series of blog posts on string theory, beginning with the collection of notes on Conformal Field Theory). The advantage, moreover, is that the 4-point functions become Witt diagrams, and, then, through the AdS/CFT correspondence, one can relate these to the correlators on the boundary.

So this is the plan, which, Papadodimas argues, will enable the study of fine-tuning in the bulk in terms of fine-tuning within the dual CFT.

In proceeding his study, Papadodimas gives quite a few comments as to how we might think of fine-tuning as it manifests in the dual CFT. The picture here is really quite interesting; because if, as noted above, we can express fine-tuning in terms of graviton correlation functions, what I have so far failed to mention is that these correlation functions of gravitons are duel correlation functions of the stress-energy tensor on the boundary.

The greater goal – or ideal outcome – would be to study fine-tuning on the boundary, particularly how it is resolved. What we come to learn is how the large N expansion, the concept of which one may be familiar from their study of QFT, plays an important role. Moreover, one of the interesting suggestions is how the $frac{1}{N}$ suppression of correlators, which relates to the large N gauge theories in the t’Hooft limit, gives us early hints of fine-tuning.

Now, one of the questions I had early on pertained to the comment that it would not be difficult to split up a correlator into a sum of terms, and then expose this broken correlator to some interpretation of fine-tuning. That is to say, nothing is stopping us from breaking up the correlator into a sum of terms and then pointing at this object and saying, ‘see we have fine-tuning!’. But Papadodimas is a smart guy, and he already knows this.

To be clear, and to summarise what I am talking about, part of the strategy explained by the author early on includes expanding “the correlators into sums of terms, each of which corresponds to the exchange of certain gauge invariant operators in intermediate channels i.e. into a sum of conformal blocks. While the sum of all these contributions is suppressed by the expected power of 1/N, individual terms in the sum can be parametrically larger, as long as they cancel among themselves” (p.2). These cancelations are what we’re looking for, because, the interpretation of this paper is that they are in a way an expression of bulk fine-tuning in the dual CFT.

Papadodimas quickly extinguished my one concern, however; because we are not arbitrarily breaking up the correlator. He recognised that one may perform such an act artificially, which wouldn’t have much meaning. Instead, he searches for and performs an canonical procedure. I found this very interesting to follow. What we end up with is an expansion in conformal blocks that looks like this,

[ langle tilde{T}(x_{1})tilde{T}(x_{2})tilde{T}(x_{3})tilde{T}(x_{4})rangle_{con} = sum_{A} mid C_{TT}^{A}mid^{2}G_{A}(x_{1}, x_{2}, x_{3}, x_{4}) ]

Then, following a brief review of holography, some comments on hierarchy vs. fine-tuning, and also a study of important cutoffs, the idea of a sort of dual picture emerges, in which the physical description may appear natural in one picture – such as in the $frac{1}{c}$ expansions discussed in the paper – and then finely tuned in the second picture.

There is too much to summarise in such a small space. So I will just pick up on how, in the dual CFT, I found it intriguing that, should the arguments hold, the double operator product expansion of the correlator written above displays partial sums over the conformal blocks that cancel. As alluded early, in the context of the particular CFT, the interpretation is that the conformal blocks appear fine-tuned. But the mechanism behind this exhibited fine-tuning is not fine-tuning as I read it, such that it becomes clear that in large N gauge theories this expansion of the conformal block is natural. The naturalness of the expansion is consistent with what we know of the large $frac{1}{N}$ expansion in the t’Hooft limit.

Papadodimas closes with the explanation that in the context of a CFT with a large central charge, there is a notable c.c. problem in the holographic dual. As I focused on, it is found in this CFT a semblance of fine-tuning in the correlator blocks of the $frac{1}{N}$ expansion. However this may be interpreted, the interesting idea is that: “this fine-tuning may be visible when the correlators are expressed in terms of the exchange of conformal primaries, it may disappear if the correlators are written in terms of the fundamental fields of the underlying QFT” (p.32).

If this is in any way true, it raises some very intriguing questions and raises some very interesting potential pathways for future study, the likes of which Papadodimas spells out in explicit terms. The suggestion of the use of a toy theory, as opposed to a non-perturbative study, makes me think of the SYK model. It will be curious to follow-up on the papers that directly follow.

One last closing comment: I am excited to say that Papadodimas will be one of the lecturers at a special engagement I will be attending this summer at the University of Madrid. The engagement, SIFTS 2019, comprises of two weeks of discussion, review, study and brainstorming on issues in string theory (and fundamental physics broadly). Papadodimas will be lecturing on the black hole interior and AdS/CFT. It shall be brilliant.

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Physics Diary

Some papers I’ve recently read, including a new one from Susskind (12/05/19)

I thought I would experiment with a new type of weekly post. The premise is simple: I collect and describe some of the papers that I have read that are my favourite or that standout for whatever reason. The papers could be from the last calender week or fortnight (we’re working with loosely defined parameters). Or they could simply be papers I read some time ago that have been on my mind as of late.

I imagine these posts will be primarily research based. There will be heavy focus in string theory, and certainly on new research. But I am also one for obscure papers, and for reading across other areas, which means that one should expect an occasional mixture. Within this mixture, also expect some pedagogical literature to be flagged. The papers listed will be old and new. I am also still trying to figure out the balance between specialist and pedagogical language as a blogging principle, so I imagine these posts will be a product of trial and error during the fledgling stage production.

Without further ado, and in no particular order:

Complexity and Newton’s Laws – Leonard Susskind

Last week Leonard Susskind uploaded a new paper to the archive. It is was originally forwarded to me by my Professor, as it relates to some interesting questions which may form the basis of a future research project in holography theory.

In this paper, Susskind follows recent efforts to explore the holographic origin of gravitational attraction with a study of the size-momentum correspondence. We’re working in the SYK model here, which simplifies things rather nicely. Susskind argues that Newton’s laws are a consequence of, or can be retrieved from, this improved version of the size-momentum correspondence.

A theory of gravity on the boundary is something I’ve started thinking about in recent time. So this paper was enjoyable to read. I take inspiration from the Susskind’s efforts here, and also from the surrounding literature, insofar that I have more or less being intuiting my way to the same domain of enquiry. That is always nice. I will have to dig a bit deeper into some of the background literature (like on the CV correspondence, etc.). The notion and treatment of complexity in this paper is also intriguing (I had a time where I was obsessed with complex systems, generally, and I maintain interest in the study of their evolution).

One last thing before moving on: the notion that entropy may behave like observables over a code space is super intriguing.

Modular invariance and orbifolds – Stefan Huber

This paper offers a survey of the some of key tools and ideas pertaining to
modular invariance in string theory. It uses Di Francesco et al., “Conformal Field Theory” (1997), as its main resource. The contents of discussion are also those covered by Polchinski in Volume 1. In any case, the paper offers a useful review of modular transformations on the torus, focusing particularly on the constraints of modular invariance in the context of CFTs defined on the torus.

Lectures on Two-Loop Superstrings – Eric D’Hoker and D.H. Phong

Lecture notes by D’Hoker and Phong from 2002. Though some time has passed, I’ve found these notes helpful. The main attraction is their review, and treatment of, multiloop superstring perturbation theory. The emphasis to start is a first principles construction of a two-loop superstring measure on moduli space. Much of the discussion, and certainly also the techniques on display, are useful to review. The section on the vanishing of the cosmological constant (CC) is interesting, as is the chapter on compactifications and the CC. The subtlties of chiral splitting is something I need to look into more thoroughly. In fact, this paper is filled with facts and assumptions that I need to still need think about.

Graviton Dominance in Ultra-High-Energy Scattering – G. ‘t Hooft

I’ve been thinking a lot about graviton scattering and more generally about the uniqueness of solutions in string theory. There is actually a lengthy story to be told here, including a motivating discussion with my professor, which relates to what is below.

In this paper from 1987, ‘t Hooft studies high-energy scattering of two particles in which the energy is so great that the gravitational field of the particle comes into direct focus. Here, ‘t Hooft describes how this field consists of a “shock wave”. The physics and the calculations are interesting, and I recommend going through it. But the main reason I was thinking about this paper again in recent days can be found on p.62. ‘t Hooft notes that there is a rather striking similarity between the scattering amplitude computed and the well-known Veneziano amplitudes. Anyone familiar with string theory will know about the Veneziano amplitudes or will be on course to become familiar with them. The similarity is most curious, indeed! It is interesting to think about from a number of perspectives.

Anything by ‘t Hooft is brilliant. He’s one of my favourite physicists, and I’ve said before that I hope I will get to meet him one day.

Magic: The Gathering is Turing Complete – Alex Churchill, Stella Biderman, and Austin Herrick

To end with something off track, and also quite fun, a new paper was uploaded to the archive which seeks to argue that optimal play in Magic: The Gathering (MTG) is at least as hard as the Halting Problem. As an avid player, one thing that stands out about Magic is that the gameplay has incredibly high variance. This high variance almost renders the notion of optimal play to be a sort of platonic and not quite attainable concept which one nevertheless continuously strives to achieve. It is generally what makes MTG thrilling. One strives for optimal gameplay through a mixture of logical and well-reasoned decision making. Each choice, or play, tends to matter. Pattern recognition is essential. However, even the best of players, who, at any given time, may be playing one of the more optimally constructed decks given the format and the current meta, will inevitably suffer a series of loses. One reason, as the linked paper argues, is that deterministic outcomes in MTG are essentially non-computable. In some sense, as one necessarily strives for as optimal of gameplay as possible, there is some definite limit to which one can effectively configure a logical and well-reasoned structure for decision making; because, by design, the game is configured to produce a complex and even mildly chaotic system of variables and inconsistencies.

As I have yet to dissect the paper, including the methodology, I will reserve further comment on the author’s study. Given some free time, it will be intriguing to go through it systemically.

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Physics Diary

Notes from this week’s particle cosmology conference

As I mentioned in a past post, this week I attended a particle cosmology conference. The talk that I had circled, and which was the main reason for my attendance, was Eran Palti’s presentation on the string landscape and swampland conjectures. It did not disappoint.

The talk was very much a repeat of Palti’s lecture at the CERN Theory Colloquialism. A link to his notes can be found here.

It was interesting to listen to him speak on a number of matters, including the primary conjectures on the swampland. Moreover, attempts to derive the weak-gravity conjecture, distance conjecture, etc. from the idea of emergent fields is something I find to be intriguing. That the swampland conjectures may form a coherent framework is an idea I have been questioning in my own notes, as I think it is an important development, so it was nice to listen to Palti talk about such an interlinked coherent framework. I am excited to work through the derivations myself and also reconstruct these links in my own notes.

To that end, one paper that I certainly need to re-visit is a recent publication by Ooguri, Palti, Shiu, and Vafa titled “Distance and de Sitter Conjectures on the Swampland”. For anyone interested, you can find it on the archive.

Eran Palti, Higgs Potential and Weak Gravity Conjecture. Source: http://wwwth.mpp.mpg.de/members/palti/

The swampland constraints are deeply quantum gravitational in nature. As Palti emphasised, the proposal is thus that the swampland conjectures are “consequences of the emergent nature of dynamic fields in quantum gravity”. The conjectural properties of the swampland are not only shown to be related, but it is also outlined how they form an emerging coherent picture that raises interesting questions about the microscopic physics at work.

I have some posts planned for my particle physics blog that will examine all of this in more detail and will discuss some of the more important papers on the string landscape and swampland. Meanwhile, one last note of interest was Palti’s reference to Witten’s 1979 paper on an emergent gauge field toy model CP^N. This is one of several papers that I am eager to read. It was also nice to listen to Palti touch on a variety of other topics, including cosmological implications of the swampland, not least how inflation is in exponential tension with the conjectures.

Lots of things to think about.

It was by far the best talk of the day.

 

 

 

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Physics Diary

Teaching myself Quantum Field Theory and String Theory

I’ve been a reader in String Theory (ST) for some time. And since my formal admission to university, I have been revising what I had learned and also expanding deeper into ST, whilst satisfying other formal university course requirements. From my understanding, the university is currently considering ways in which my degree may be fast tracked, so that I can continue unabated with my post-grad/post-doc. studies. This would enable me to more wholly focus on research. Meanwhile, in teaching myself ST, I have picked up both volumes of Polchinski’s “String Theory”, a great and much celebrated work. Although I haven’t been able to dedicate much focused time, with many other things pulling at my attention, in a couple short late night sittings I have powered through the first two chapters and enjoyed every moment. I can also honestly say that string theory is the most difficult and challenging thing I have taken up so far in life. It is in no way easy, but it is immensely enjoyable and I find stringy ideas motivating. (David Tong’s lecture notes are also superb, and I highly recommend that any interested reader also engage with these).

As alluded, I have grown frustrated with more of my time being consumed and taken, especially as I must continue to undergo the formal university process, and I am trying my best to adapt. If I had things my way I would sit and devour Polchinski in a week and then move on to some other books on my list, with topics in scattering amplitudes and elsewhere requiring urgent attention. But such is the nature of the current circumstance.

In that I have also taught myself and am actively extending my knowledge in Quantum Field Theory, I am currently reviewing a few QFT texts alongside my Polchinski readings. Whilst perhaps unorthodox (?), I have found it fruitful to go through both simultaneously. These activities have happily coexisted with my active thinking on matters to do with the string landscape and the swampland conjectures, compactifying all of this with other studies in particle physics, cosmology and mathematics and whatever else that I currently consider worthwhile. (For anyone interested in QFT, there are a number of books that I recommend, some that I am also eager to read through. These include the likes of Schwartz’s “Quantum Field Theory and the Standard Model”, Steven Weinberg’s “The Quantum Theory of Fields”, and Klauber’s “Quantum Field Theory”. Of course there are also some classics that have gone unstated, which anyone interested should read).

Now that the winter break is here, I would like to cross several of these activities off my list in the coming weeks. This is my primary aim, and then I can continue to advance toward some other things. Unfortunately, the winter break also means a break from my sessions with Prof. Padilla, to whom I am grateful for spending time with me and talking with me about all sorts of subjects. We discuss stringy things and also lots of cool cosmology-related theories and concepts. He has also been guiding my self-studies in ST and QFT and in other areas. I am also continuing to sketch some potential PhD theses, encircling different concerns and potentially fruitful research projects. Nima Arkani-Hamed’s works, among others, has been a lasting source of inspiration.

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On another note, I have been encouraged to write more personal blogs and so I thought I would write my first personal post about some of the studies I am actively pursuing. In time, these sorts of posts will become more refined as I understand more how to write them and generally why one might communicate in this way. With my Asperger’s, such engagements and communication don’t come naturally.

There are many other things I would like to talk about, such as Jackson’s book on electrodynamics. It is a book I have yet to read, although I have already studied the contents. So I am also looking forward to spending time with that book over the break. I would also like to write about it when time permits. It may sound odd, but I first learned Special Relativity through its connection to electrodynamics. Other books I am eager to engage with are Hooft’s Yang-Mills and Nima Arkani-Hamed et al. title on the positive Grassmannian and scattering amplitudes.

Finally, I’ve been encouraged to think about making more maths and physics videos. Making maths and physics lectures or tutorials is a way for me to practice communicating, and I generally find it to be quite fun. At first I wanted my youtube channel to be a complete tour, capturing all of mathematics and physics, so naturally I decided to start from the basics and build. My plan was to also incorporate history – such as the history maths and important mathematical concepts. But that quickly became boring and I have trouble with keeping focused on things not directly related to my active research and study interests, so I sort of lost motivation. But in that I’ve been encouraged to continue making lecture videos and perhaps instead reformat my channel, the idea is that I start with a series on QFT or ST (I can talk about how I’ve taught myself these subjects, how I continue to approach them, and also offer lectures for anyone else interested in learning). I would also like to write a series of blog posts, when time permits, where we could even just begin with introducing notation and history and then deepen from there.

However, I am still not certain what approach I would prefer to take for such a series of lectures and posts, whether it is best for wider audience to start with the RQM entry or the path integral formalism.

In any case, I certainly have lots of notes that I will aim to post on here in the future.

Bye for now.

 

 

 

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Physics Diary

Particle cosmology conference (17 December 2018)

This week, I will be at a particle cosmology conference. The conference is at my home school, University of Nottingham, the perfect venue for such an engagement with our Particle Cosmology Group hosting some fantastic researchers.

I will be attending, especially focused on my ongoing research and studies in string theory, particularly the string landscape and swampland conjectures. I haven’t written much about my ongoing research activity on the swampland conjectures, though I have made a note to do so. There is certainly a lot to be said about these matters, and certainly a lot of work to be done. I also have plans to write a wider series of posts on string theory as part of my particle physics blog.

It remains to be said that for these reasons I am super excited for Eran Palti’s talk on the “cosmological aspects of the string theory swampland”. Palti recently published a paper with Cumrun Vafa, Gary Shiu, Hirosi Ooguri titled “Distance and de Sitter conjectures on the Swampland“, which I am looking forward to reading.

Additionally, Susha Parameswaran’s engagement on “runaway quintessence and the swampland” also sounds promising.

Perhaps if I have time I will write a follow-up post reflecting on the talks. Meanwhile, for anyone interested, visit the above link and also watch for my post next post on the string landscape and swampland conjectures.

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Klein-Gordon equation
Physics Diary

Klein–Gordon equation

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The other day I was thinking about the Klein-Gordon equation, otherwise known as the Klein–Fock–Gordon equation. I had to use it for something, and afterwards I found myself thinking of its derivation. So, for fun, let’s derive it!

Klein Gordon equation traveling wave plot5

[For the reader not familiar with this equation, it is a relativistic wave equation related to the Schrödinger equation. You can find a very general entry into its use and importance here. I also found some lecture slides that go over its derivation in more detail than my own treatment here].

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In that we are dealing here with relativistic quantum mechanics, in the following derivation of the Klein–Fock–Gordon equation we’re going to employ the Einstein energy-momentum relation. Also, to simplify things let’s take the invoke the standard convention for the units in which $hbar = c =1$. This will allow us to not have to focus on $hbar$ and $c$ terms that appear throughout. It’s not so much lazy, just practical.

We start with the Schrödinger equation in natural units,

[ i frac{partial{d}psi}{partial{t}t} + frac{1}{2m} nabla^2 psi + Vpsi = 0 ]

As we continue to set the stage, let’s now also remind ourselves of the equation for the nonrelativistic energy of a free particle (this will be important for reasons that will become clear in a moment),

[ E= frac{vec{p}}{2m} ]

Now that we have our equation written, we can proceed. The key here is that we’re going to want to quantise this nonrelativistic equation. The result we get is $hat{H} psi = E psi$, where $hat{H} = T + V =  frac{p^2}{2m} + V$.

(Recall: the inner product of momentum (lorentz invariant) gives, $p^{mu}p_{mu} = m^2c^2 = frac{epsilon^2}{c^2}=mid{p}^smid$).

Taking the quantum mechanical operators, we get:

[ E rightarrow i hbar frac{partial}{partial t} ]

[ P rightarrow -i hbar vec{nabla} ]

With these operators defined, the purpose for doing this is because a natural attempt is to want to use the Einstein energy-momentum relation. We can now turn our attention to this relation, which one may already know takes the form:

[ hat{E}^2 = p^2 +m^2 ]

Substituting for our quantum operators, we get something immediately looking like this:

[ (i hbar frac{partial}{partial t})^2 = (-i hbar vec{nabla})^2 + m^2 ]

Or, one will sometimes see this written as $(frac{E}{C})^2 = p^2 + m^2c^2$, then substituting for $E$ and $vec{p}$ you will find $(frac{hbar}{c} frac{partial}{partial t})^2 phi = (hbar nabla – m^2 C^2) /phi$. But because we’ve set $hbar = c =1$ our approach is slightly different. In our case, when we simply expand the brackets and simplify,

[ – frac{partial^2 phi}{partial t^2} = m^2 phi – nabla^2 phi  (*) ]

This is essentially the Klein-Gordon equation. However, we have some conflicts in notation. Really, we want our equation above to be in four-vector notation. So, what do we do? I’ve already given some hints in the graphic I posted above (a sketch from my notebook).

In four-vectors, one will already likely be familiar with the idea of having one time component and three space components. We can write this as $X^{mu} = (X^o, bar{X})$, where $bar{X}$ is just short for our x, y, and z components.

Now, with that sorted, we need to think about our four-gradient, which we can write as $partial_{mu} = (frac{partial}{partial t}, vec{nabla})$. Taking the dot product, or, in other words, taking the Einstein summation convention into consideration, $partial_{mu}partial^{mu} = partial_{mu} g^{mu nu} partial_{nu}$, where one might recognise $g^{mu nu}$ as our metric.  Understanding the metric convention here has the signature (+ – – -), we come to the following:

[ partial_{mu}partial^{mu} = partial_{mu} g^{mu nu} partial_{nu} implies (frac{partial}{partial t} nabla) begin{pmatrix}
1 & 0 \
0 & bar{-1}
end{pmatrix}
begin{pmatrix}
frac{partial}{partial t} \
bar{nabla}
end{pmatrix} ]

which, after performing standard matrix multiplication, comes out to

[ frac{partial}{partial t} nabla begin{pmatrix}
frac{partial}{partial t} \
bar{nabla}
end{pmatrix} = frac{partial^2}{partial t^2} – bar{nabla}^2]

So, we have (invoking the d’Alembertian at the end)

[ partial^{mu} partial_{nu} = frac{partial^2}{partial t^2} – bar{nabla}^2 implies Box = frac{partial^2}{partial t^2} – bar{nabla}^2 ]

As we approach our final result, recognise that what we have now looks very much like (*). That’s because we have arrived at a representation for our d’Alembertian, and we need to perform a substitution. All that is then left is some algebra and we’re done! Here is what I mean: let us now return to our previous equation, (*), rearrange it so that we can substitute directly for our d’Alembertian:

[ – frac{partial^2 phi}{partial t^2} = m^2 phi – nabla^2 phi ]

[ – frac{partial^2 phi}{partial t^2} + nabla^2 phi – m^2 phi = 0 ]

[ frac{partial^2 phi}{partial t^2} – nabla^2 phi + m^2 phi = 0 ]

Sub for the d’Alembert operator,

[ Box^2 phi + m^2 phi = 0 ]

[ implies (Box^2 + m^2) phi = 0 ]

And here is the version of the Klein-Gordon equation you will see in many texts, except it does not include $hbar$ and $c$. In that case one will often see it written as, $(Box + frac{m^2c^2}{hbar})phi = 0$ where, again, $Box = (frac{1}{x} partial t)^2 – nabla $.

Concluding remarks

What is really cool about this equation is that you can find plane wave solutions to it relatively easily. The caveat being that the plane wave is a solution to the Klein-Gordon equation so long that energy and momentum follows Einstein’s relation. This last comment provides a hint for further study, should the inquisitive reader immediately think of connections with GR and RQM.

I’ll also save some discussion on some of the problems, or limitations, pertaining to the KG equation for another time. For now it is just nice to appreciate the result – an attempt at relativistic quantum mechanics!

In a following post, I will show alternative way (there are a few) to derive the KG equation which is much more terse or abrupt. I think it is useful to know.

Standard