"Every electron in the universe is a ripple moving about in this electron field...these fields are always there."
"Reality then is a projection. When you study matter, it's not matter that's emitting a field, it's actually those complex patterns that are creating matter."
"So, particles themselves are essentially vibrations of a field."
"The field theory concept will be very important, even if you're only interested in atomic physics or statistical physics."
"Action principle for classical fields is significantly constrained due to the principle of locality."
"That energy could be the energy stored in this field and so some people would say that's what happened."
"So we now know that electric charge is a sort of shadow of the ancient fields from the birth of the universe."
"Anything that's producing its own field, it's going to be a toroidal field."
"When the Higgs field gets a non-zero value in the vacuum, what used to be an interaction now becomes a potential energy."
"...Q adjoin root2 is the set of all A + B < tk2 where A and B are rational numbers."
"...the smallest extension field F of Q the rationals that contains i i.e. the square otk of1."
"All particles are described as fields, quantum excitations of the field."
"The degree of the extension \( [E:F] \) is the dimension of \( E \) as a vector space over \( F \)." - This relates to extension fields as vector spaces.
"Field automorphisms are central to Galois theory."
"A field is a variable, an observable variable, a measurable quantity which depends on space and time."
"The ground state is any one of these possible positions for the field but always sort of locked together."
"A quantum field theory gives us a possibility to evaluate the result with high precision."
"The energy flowing through a specific point is proportional to the field strengths at that point."
"Particles are better understood as excited states of quantum fields."
"Particles are best understood as emergent mathematical constructs from underlying fields."
"It's more or less obvious that if you have a pure magnetic field in one frame, then in other frames you will also have an electric field."
"The only unusual thing about \( F \) is that it's anti-symmetric."
"Fields are things which vary in space-time, so they depend in general on all four components of position."
"That's what flux is: the amount of field flowing through an area."
"The flux does not depend on the shape or the size of the closed surface."
"The gravitational field only has a relative existence in a manner similar to the electric field generated by electromagnetic induction."
"A field is something which has a value at all points in space."
"There is a field that underlies all existence."
"When we get to quantum field theory, the relationship between particles and fields becomes more manifest."
"Any kind of invariance under a field transformation is considered to be a kind of symmetry."
"A gauge theory is a type of field theory in which the Lagrangian doesn't change or is invariant for a continuous group of local transformations of the field."
"If K is a field, is K[X] a field? The quick answer to this is no."
"It's F mu nu F mu nu, it's proportional to sums of squares or differences of squares."
"All fields are principal ideal domains because the only two ideals that they have are principal ideals."
"The Triumph of field theory is like a Victory Parade that made the spectator gasp with awe and laugh with joy."
"All particles are excitations of the field."
"The field really becomes interesting when the two points of view are combined."
"Classical field theory is the classical mechanics with an infinite number of degrees of freedom."
"Now we want to quantize this infinite number of freedoms, and now this becomes quantum field theory."
"A field F is a set of elements with 2 operations addition and multiplication satisfying certain properties."
"Magnetic fields run perpendicular to electric fields and are part of the same force."
"Maxwell's equations describe electric and magnetic fields that vary in time and that time-varying electric and magnetic fields that don't satisfy these equations can't exist."
"Not every field has an infinite number of elements the way that the rational numbers do."
"The cyclic group of prime order actually is, with the usual operations of modular addition and multiplication, a field."
"Every field is an integral domain, but not every integral domain is a field."
"Welcome to this next video in the playlist on field theory."
"Let's begin then with the definition of a splitting field."
"Any field at all, its characteristic, if it's not zero, has got to be a prime number."
"The standard model is an effective field theory."
"Quantum mechanics is the one category case, quantum field theory is the n category case, and then the hope would be that quantum gravity is some infinity category case."
"Field essentially embodies this cause-and-effect relationship."
"...everything I do with the brains, I can mimic with the behavior of this field when the brains hit, that means the field goes to zero, and the brains move apart, that means the field grows."