The Biggest Ideas in the Universe | 9. Fields

If a field is a mathematical sum of modes, how is the wave function of the field constructed from the wave functions of the separate modes?

How can the three types of energy of a field, defined here as the square of the time derivative, square of space derivative and the field squared, all have dimension of energy?

I have learned things from these videos that I missed entirely after previously watching more formal, mathematically rigorous presentations. Great job!

He’s trying to isolate. Let him be.

What's field in classical mechanics and what's spacetime in quantum mechanics

bigbang,
how much faster than light in the beginning did the world expand, now that we have been waiting for the light that came after us for 13.8 billion years?
Isn't it possible that this light has gone before us?

Just watching these videos now, but when Sean talks about phi_k(h) should it really be a function of h? Since it is a field mode, isn't it a function of R^3? so maybe it should be written as phi_k,h(R^3)?

This lecture broke me. I was mostly keeping up before it.

the field is decomposed into Fourier modes, and then, the wave function is evaluated mode by mode. Does that mean, the wave function is linear?

Professor Sean Carroll debated William Lane Craig and not only did he destroy every single 'fine tuning' argument Craig presented then he said "but god can do what he wants, he's god, - He can do what he wants.and it just let the air out of Craig's self-righteous soap box lol

Re: wave function... do you literally mean a function like math 'F of X' or its functionality is wave like?

In classical mechanics a field exerts a force on an object. If this object keeps moving in an uniform straight line in curved spacetime even next to a large mass, there is no force acting on it. Would be GR gravity a field in classical therms? If gravity is not a "force-field" could we consider GR as non-classical?

Thank you Sean for your work. The voice of reason in this trying times 😊

Very nice introduction of field. I had one doubt the electron produces electromagnetic field. But electron is again particle of Dirac field. What produces Dirac field

It is not logical that every point in space can have several particles excited there -- particles occupy space and most are definitely not points -- identical fermions in particular can have at most two symmetric solutions. Thus, it is unclear how particle fields might contribute to the cosmological constant.

I think the sum of the zero-state energies has to be cut off starting at some high values of the k vector. Probably, the lowest wavelengths possible in a field are about the size of the Plank length - having a photon with a higher frequency would create a black hole. A finite cosmological constant is hinting that the space is discrete and not continuous.

fields are not fundamental. this video does not seem to keep up with the developments in the field. just peddling old ways of picturing things.

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correction: at 33.00 he states that all energies (kinetic, gradient, potential) per mode are proportional to the square of the height h. This is true for the graient and potential energy. The kinetic energy is proportional to the square of the time derivative of the height h. This is exactly what is required for a simple harmonic oscillator.

Does a field “look” like anything?
👁👄👁

Watching these now a year after they were published and they're exactly what I was looking for: a down-to-the-metal description of how all this stuff works without all the pop-sci fluff that surrounds these topics. Thank you so much for taking the time to make this series. It's a monumental effort.

An interesting note: you use the word "quantize" to (I think) refer to taking a classical phenomenon and bringing it into the paradigm of quantum mechanics. This terminology is disorienting to an electrical engineer: "quantize" to me invariably means to take a sampled continuous amplitude signal and place each sample in an amplitude "bin" that can be labelled with a fixed-width binary number. We talk about quantization noise and various ways to mitigate its effects. That seems emphatically to not be what you're talking about here. You're talking about taking something with a single classical configuration and describing it as a superposition of configurations characterized by a probability function, all of which exhibit fully continuous behavior in space, time, and amplitude.

Sooooo....................No free beer then?

Question: Here "probability" is really probability density. How is the probability density for a field configuration defined? For probability density rho(x) the probability for x in [x, x+dx] is rho(x)*dx. What is the analog for a field configuration? But I guess with Fourier analysis I don't have to worry because for fixed \vec{k} we are back to rho of the height, h. The classical energy of phi(x, t) is the space integral of what you wrote down (the 3 terms) and for a plane wave that integral is infinite.

The subject of this video is exactly what I have been wondering about, namely, what is a quantum field and how does one get particles from fields.

May I say that though I understand little of the talk, I am enchanted by the utter humility and spirit of generosity of this beautiful person.
Makes me think of Socrates.
Thank you and all the best!

@Sean Carroll, your video's have the feel of delivering individualized attention. I watch them with the appreciation as if you're focused on helping me understand these awesome lessons. Thank you so very much. For generously sharing your time, passion, lifes work with all your years of learned experience.

Higher dimensions are smaller and curled up? I call bullshit. A 2D creature only experiencing a cross-section of our 3D world would also say our dimension was smaller and curled up. In reality the 10th dimension is infinitely more massive than our our own because God needs the space to create any universe he wants

Thank you, Dr. Sean 🙏

I'll not compare you to my teacher nor to someone else, because you are absolute 👌

You talk about the Fourier transform as if the collection of sine waves that sum to a function is unique. Is that really true? naively, I think there is a requirement that you seek the smallest such collection: because you can add two of half values of any one wave to replace it. So I feel you might be making things simple for us, which is appreciated. :) if that is so, what does the real world basis for enforcing this uniqueness criteria? Particles cannot be located at the same spot?

Clearly the very best explanation I've ever seen for describing QFT principles, many thanks

Dr Carrol, thank you very much for this video series. I have a question regarding fields, and hope you could address it.
If what we call a particle is a perturbation on a field, then the speed of the particle would be determined by the field, just like the speed of a wave is determined by the media in which it travels.
In your lecture, you reconstruct the particle-like behavior by adding an infinite number of planar waves. Wouldn't the speed of those waves, and thus the particle they form, be determined by the media?. In this case the media is the quantum field. Could you give me some pointers of where can I find the answer to this.
Best Regards.

Omg we live on a hollodeck! Computer, end programme! Computer, exit!

Thank you for all you have done to help improve our understanding of current perspectives in physics.
Regarding the zero vector potential: is not the zero of the potential arbitrary like other potentials?

Hi, I hope I'm not too late to ask a question about this video. Around 20 mins you start discussing a particular classical field and how to quantize it. What I didn't get from the discussion following is: how do you choose a classical field which, when quantized, will lead to a specific particle type? If e.g. the electromagnetic field leads to photons, which classical fields would lead to electrons, neutrinos etc? Do you e.g. have to "design" a classical field which has to have some set of required properties?

Physicists are not allowed to make up words without peer review. Cease the use of "mathy" immediately.

Or to put it differently, the height of each mode in the superposition of the quantum field tracks how many particles of the momentum corresponding to that mode's k-value are present?

Soooo we can think of each mode as the field corresponding to a particle, and when that mode's height/energy goes up, that amounts to adding a particle which has the momentum associated to the wavelength k of the mode?

fields are useful, but unessential, mathematical book-keeping devices to study interacting quantum particles. They are just part of a formalism and can be removed from the formalism. How was this not mentioned in this video?

Are we gonna be tested on this?

Is it accurate to say antiparticles that pop into existence are simply a down wave in a field, or inversely charged quanta at that location?

Thanks for taking the time to do these videos

In three dim, all functions can be written as a sum of volume wave not plane wave I think.

I

I'm another one like Chris Moon. I have had it in mind for decades to understand this stuff and although I'm just making a start on listening to you, I can tell that you are the man that's going to do it for me.