Mordred

I always only buy textbooks that a full of the mathematical detail. Heuristic written books will always lead you down garden paths.

I agree they can, any methodology has its appropriate applications and range of applications. Hence why its best to understand a vast majority of methods. If your not adverse to buying a textbook pick up "Mathematical methods for Physicists" by Arfken. https://www.amazon.ca/Mathematical-Methods-Physicists-Comprehensive-Guide/dp/0123846544 You will find this aid incredible as it is not specific to any theory but provides the tools to understand any

agreed, anyways hope your back on track as to how to approach the topic. Its a huge subject that is far more involved and interconnected than many realize. One of the most useful aids I found was learning how to simplify the larger group theories under U(1), SU(2) and SU(3) any orthogonal group regardless of how many dimensions can be reduced to those. Ie the double cover of orthogonal groups

lol yeah it is a bit of a lengthy thread with a lot of misdirection.

Amazing enough even then. Though they may not read every post. You see that quite often in numerous replies where they quote a particular post but don't read the full thread.

You would be amazed how many readers we get that don't even register an account or post.

lol no prob I understand the sentiment In all honesty It took me decades of active study to reach the understanding I now have. I at one time tried to shortcut much like your tried. However after so many failed attempts I finally sat down and studied how every common formula was developed and derived via their individual proofs. After a while you will start to see some very common patterns. These patterns will apply to any application or theory. One of the most common patterns being kinematic motion under the action principle. The other is group theory itself and how to organize all this complexity. I would start with the U(1) guage then goto SU(2) then SU(3) followed by SO(1.3) take the time to fully understand the notation Dubbelosix used is an extremely handy set of mathematics. Unfortunately I haven't studied much on the extended groups as for the most part I never need them.

Umm this is a forum where we have readers of all age groups, so lets avoid the language. It is one of our forum rules, you might note Resident experts is part of the Moderator staff

Yes infinities arise quite often when dealing with any field treatments this is where one must apply renormalization and IR and UV cutoffs to avoid infinity issues. LQC does an interesting technique by applying Wicks rotation to avoid some of these issues. Under LQC renormalization is avoided via Wicks rotation. However that is just one of many numerous techniques. Another technique being killing vectors which will correspond to you IR and UV cutoffs

Understanding a spin 2 boson such as the graviton is also rather tricky as you are dealing with two individual polarity states which will correspond to creation and annihilation operators but would take me forever to latex how [math]h_+,h_x[/math] would be described in terms of path integrals. the above is one of the simpler examples being scalar spin 0. LOL you should see how bad the quaternion Higg's field looks under the above

Welcome to the complexity of physics, corrections applied to the above... There is no easy way to simply skip this stuff, the real physics behind how particles arise for the number of different particles and field treatments is a complex topic that takes years of study to understand. That is the reality of nature.

d Precisely because the details you need lie in how the creation and annihilation field operators apply to those equations. From those operators you can literally calculate the particle number density etc... Here is the process taken from Quantum Field theory Demystified. Though its not the greatest book as it has errors its written simple enough for the average person to get a handle on QFT. page 132. To find the energy of the vacuum we need to compute [math]\langle|\hat{H}|\rangle[/math] [math]\langle|\hat{H}|\rangle=\langle 0|\int d^3k\omega_k[\hat{N}\vec{k}+\frac{1}{2}]|0\rangle[/math] [math]\langle|\hat{H}|\rangle=\langle 0|\int d^3k\omega_k[\hat{a}^\dagger(k)\hat{a}(\vec{k}+\frac{1}{2}]|0\rangle[/math] [math]=\langle 0|\int d^3k\omega_k(\hat{a}^\dagger(\vec{k}\hat{a}(\vec{k}))|+\langle 0|\int d^3k\omega_k(\frac{1}{2}|0\rangle[/math] [math]=\frac{\omega_k}{2}\int\langle 0|0\rangle[/math] [math]=\frac{\omega_k}{2}\int d^3k[/math] which is reminiscent of the quantum harmonic oscilator [math]\frac{1}{2}\hbar\omega[/math] however this will lead to [math]\int d^3 k\longrightarrow\infty[/math] over all of momentum space so one must renormalize the Hamilton via subtraction of the term that leads to the infinity term and set the ground state at zero to arrive at the renormalized Hamilton. [math]\hat{H}_R=\hat{H}-\int d^3 k[/math] [math]=\int d^3k\omega\hat{N}(\vec{k})=\int d^3k\omega_k\hat{a}^\dagger(\vec{k})\hat{a}(\vec{k})[/math]

As long as you keep jumping to the end points and try to modify and apply the end point equations without understanding the proofs behind those equations you will never be able to calculate a single thing. You must take the time to properly understand how those equations got developed in the first place. For example try this question which I will later post how to solve. Find the energy of the vacuum for a real scalar field ?

yes the lecture is primarily on what the Higg's revealed to us However both equations clearly demonstrate a key aspect. The methodology to correctly understand any and all field theories is to understand how they affect kinematic motion under ACTION. The correct methodology to understand any lie group is to study the action path integrals for each Lie group. This includes relativity.... Once you understand how Action is involved in any and all lie groups the rest falls into place. ALL interactions can be described under action. This includes the Feyman diagrams themselves under S_matrix... The external lines of the Feyman diagrams requires quanta of action while the propagators (virtual particles) are the internal lines on the diagrams.

Or study this equation from the Sean Carrol lecture

Oh my Here lets save you a ton of effort. You can scream at me later. [latex] \stackrel{Action}{\overbrace{\mathcal{L}}} \sim \stackrel{relativity}{\overbrace{\mathbb{R}}}- \stackrel{Maxwell}{\overbrace{1/4F_{\mu\nu}F^{\mu\nu}}}+\stackrel{Dirac}{\overbrace{i \overline{\psi}\gamma_\mu\psi}}+\stackrel{Higgs}{\overbrace{\mid D_\mu h\mid-V\mid h\mid}} +\stackrel{Yukawa-coupling}{\overbrace{h\overline{\psi}\psi}} [/latex] There ya go Dirac, Higgs, Yukawa coupling, Higgs and Maxwell action all under a single equation

Your missing more than that. Are you familiar with performing dinensional analysis via replacing each term with the SI unit to test that the lefthand side does equal the RHS of the equal sign? https://www.google.ca/url?sa=t&source=web&rct=j&url=http://web.mit.edu/2.25/www/pdf/DA_unified.pdf&ved=0ahUKEwitheutlNzXAhUB8mMKHZnJBjgQFggdMAA&usg=AOvVaw3hTYBubBOwEQX7596gEKFR Test the viability of your equation via the procedure in that link. Make sure your LHS equals the RHS

Redshift, muon decay rates, everyday particle accelerators, detection of GW waves, light deflection due to spacetime curvature, direct tests of time dilation, GPS satellites, weak and strong equivalency tests. The list goes on and on. Its one of the most strongly tested theories https://www.google.ca/url?sa=t&source=web&rct=j&url=https://arxiv.org/pdf/1705.04397&ved=0ahUKEwigo-PF2tvXAhUB3WMKHVD0B9IQFggmMAM&usg=AOvVaw054EguS3mqzq8k-KfOkm8o Here is an arxiv dissertation on the tests Considering all the attempts to prove SR/GR wrong they all prove SR/GR as being incredibly accurate. It was one of the more tested theories as the majority hated the implications. However despite all the competitive attempts its born out as being highly accurate. PS once you understand it properly its incredibly easy to understand how time dilation and length contraction is involved. Though it takes a suspension of disbelief and proper study

velocity: rate-of-change of position acceleration: rate of change of velocity jerk: rate of change of acceleration jounce (snap): rate of change of jerk crackle: rate of change of jounce pop: rate of change of crackle lock: rate of change of pop drop: rate of change of lock http://wordpress.mrreid.org/2013/12/11/jerk-jounce-snap-crackle-and-pop/ the mnemonic above came from another forum if I recall, but I can't remember which one. I found it so useful to remember I wrote it down. edit: ah found it Reference https://www.physicsforums.com/threads/what-is-jerk-and-jounce-conceptually.716152/ Its the only place I recall ever seeing the higher derivatives lol. anyways here is the 4th jounce https://en.wikipedia.org/wiki/Jounce#cite_note-PhysicsFAQ-1 I should note none of the higher ones were ever taken seriously in particular snap crackle pop and drop. So good luck finding anything on them got lucky found snap crackle pop https://infogalactic.com/info/Pop_(physics) if I recall the list went higher but I can't recall past there

Here i,j,k Vector Calculus by Michael Corral note distribution of this is under the GNU free documentation license which gives permission to copy paste portions one has to be careful on copyright notices lol Anyways here you go Schwartzchild with the QFT treatments... including the Einstein-Rosen bridge. https://arxiv.org/pdf/1708.00748.pdf should be enough that if you study it properly should take a month to understand just how each equation is derived. (to properly understand it) when your not that familiar with QFT or vector calculus...

[math]\mathbb{R}[/math] is the set of real numbers, a unit vector only ever has a single value of 1. its not a set. To answer the next possible question the reason we square the unit vectors in probability density functions is to take a negative number and make it positive as probabilities are always positive integers. unfortunately there is no agreed upon symbol for the set of imaginary numbers so many use [math]i\mathbb{R}[/math] or just [math]\mathbb{I}[/math]

http://www.hri.res.in/~debsadhukhan/HRI web/pdf/Units & Vectors . the above is part of it, here go through this then think back to Dubbelsix mentioning of dimensionality. which he was quite correct to do so Those three unit vectors only ever have 1 value (hint unity) https://www.khanacademy.org/math/precalculus/vectors-precalc/unit-vectors/v/intro-unit-vector-notation for those that don't like reading math lol

I know how those unit vectors work, no they don't work the way you think they do ie you need the Jacobian matrix to use them correctly ps I posted this a few days ago ie the Kronecher delta and Levi-Cevita connections section I posted earlier this thread.

I can already deconstruct it, I am asking you to start with the [math]\frac{8\pi G}{3}[/math] start with a scalar field then do a vector field

great I see several matrixes in the above can you identify which terms are actually matrixes in the above? or better yet can be simplified by using matrixes to organize?