Відео

With this kind of approach, we still have a semi-classical theory, because the gravitational field serves as a background for quantum fields, we have not quantized the gravitational field (in the sense that we have not computed quantum amplitudes by integrating over the metric tensor). In particular, if one dares to compute quantum amplitudes related to the gravitational field (by using the path integral for instance), it becomes impossible to remove the infinities that arise without introducing an infinite amount of couplings (this boils down to the non-renormalizability of gravity). The lecture series presented here "QUANTUM EFFECTS IN GRAVITY" aims to show that the gravitational action is affected by quantum corrections which become more and more important as we move towards the high-energy regime (high curvatures). We simply lose the predictive power of the theory with this approach.

2:00 Is the permutation equation ( del-Fij/del-xi) valid for any 4x4 tensor, or valid for any antisymmetric tensor, or valid because it was defined as curl of a 4-vector phi?

8:45 "sqrt(-g) is 1 in our frame of reference". Why? What is g? gij i guess is the metric tensor which is used to get F-upperIndex from the already defined F-lowerIndex.

@@rgudduu g represents the determinant of the metric tensor. This lecture is an excerpt taken from a section of one of my courses. That section is dedicated to Einstein's article: “The Foundation of the General Theory of Relativity”, so this video is not completely self-contained, because it uses concepts discussed in previous lectures. You might read the article or check out the course (I am not interested in selling anything, but all your questions could have "simple" answers if you dig deeper into tensors, whether you study by yourself or take whatever course you are interested in).

The "curl" should indeed be interpreted as an antisymmetric tensor, the two facts are not independent, one leads to the other.

@@rgudduu the "permutation equation" (related to Maxwell's equations) is valid for any anti-symmetric tensor written as a curl. Try to do the calculation yourself by substituting the expression of Fij written as a curl into the "permutation equation", and show that you obtain 0=0 (which means that the equation is satisfied). I have provided more details in a course related to the "covariant formulation of classical electrodynamics".

It depends on your level, with some effort it can be done, to the point that this can become "easy" to understand. It's all relative

What that equation is telling you is that the number of particles between 𝑥 and 𝑥+𝑑𝑥 at a later time 𝑡+𝜏 depends on the displacement of neighboring particles from nearby positions at the earlier time 𝑡. Intuitively, as the displacement Δ becomes larger, the probability 𝜑(Δ) decreases, meaning that particles far from 𝑥 contribute less. Thus, the main contributions to the particle count at 𝑥 come from particles that were close to 𝑥 at time 𝑡.

Is there any book which I can refer too?

convergence is a "tricky" concept when we think of path integrals. The "awesome" thing about the physics behind them is that, even if these expressions technically diverge, we can extract lots of useful information

yes, I will make a video about it soon.

1) I see what you mean. The way to interpret the action operator inside that inner product is this: the operator acts like a differential operator (on the right state, but also on the left state). Let me remind that this is a "heuristic" derivation, so the notation is not always rigorous here. Any books on QFT that deal with the Schwinger formalism should cover this more rigorously. 2) "delta" represents the variation of the state due to a small change of the field. In general, the state changes if we perturb the field, so I would not consider your interpretation correct. "delta" can act both on the "bra" and on the "ket". 3) yes, at 11:11 we have a simple exponentiation. The variable of integration on the left is the entire inner product between the "bra" and the "ket" you see in the expression. The result should be the logarithm as written in the video (the delta in the inner product does not disappear)

Another way to think about your first question. Consider the evolution of the change in the action between two states A and B. We can write the inner product: < B | δS | A >. Based upon the considerations in the video, δS should depend only on the final and initial states, so it could be written as: δS= S_B - S_A (where S_B is an operator that acts only on state B, and S_A acts only on state A). Let's assume C is another state (that you can think of as being "in between" A and B). You can write: δS = S_B - S_A = ( S_B - S_C ) + ( S_C-S_A ) Therefore: < B | δS | A > = < B | ( S_B - S_C ) | A > + < B | ( S_C - S_A ) | A > Now, those differences in the action are going to create some small perturbations. In particular, S_B - S_C will act on < B |, whereas ( S_C - S_A ) will act on | A >. This serves as another "justification" of the formula you were confused about.

In hydrodynamics you have a Eulerian picture and lagragian picture and the transport theorem is exactly the transformation between the operators between 2 pictures

More specifically if think f as density rho the theorem gives Lagrangian derivative of mass is in fact the conservation law of mass in eular picture which is the right hand side of the equation

Yes, Poisson's ratio can indeed be linked to the concept of mass conservation. Mass conservation can be used to evaluate it. Mathematically, if a material is stretched along the x-axis, causing a strain 𝜖_𝑥, the resulting strains in the perpendicular y and z directions are given by: 𝜖_𝑦=𝜖_𝑧=−𝜈𝜖_𝑥 ν is Poisson's ratio, which is typically a positive number for most materials. Mass conservation implies that during the deformation of a material, the mass of the material remains constant. For a given volume element of the material, if we assume the material is incompressible, then its density remains constant, which implies that any increase in one dimension must be exactly compensated by a decrease in other dimensions to maintain the same volume. Consider a small cubic element of the material with an initial volume 𝑉0=𝐿𝑥 𝐿𝑦 𝐿𝑧, where 𝐿𝑥, 𝐿𝑦, and 𝐿𝑧 are the initial lengths along the 𝑥, 𝑦, and 𝑧,axes, respectively. After deformation, the new dimensions become 𝐿𝑥′=𝐿𝑥(1+𝜖_𝑥), 𝐿𝑦′=𝐿𝑦(1+𝜖_𝑦), and 𝐿𝑧′=𝐿𝑧(1+𝜖_𝑧) The new volume 𝑉′ after deformation is: 𝑉′=𝐿𝑥′𝐿𝑦′𝐿𝑧′=𝐿𝑥(1+𝜖_𝑥)⋅𝐿𝑦(1+𝜖_𝑦)⋅𝐿𝑧(1+𝜖_𝑧) For small strains, the volume can be approximated as: 𝑉′≈𝑉0(1+𝜖_𝑥+𝜖_𝑦+𝜖_𝑧) For mass conservation in the case of an incompressible material, the volume must remain constant, so 𝑉′=𝑉0 This implies: 𝜖_𝑥+𝜖_𝑦+𝜖_𝑧=0 Substituting the relationship between longitudinal and transverse strains (using Poisson's ratio): 𝜖_𝑥−𝜈𝜖_𝑥−𝜈𝜖_𝑥=0 This implies that for incompressible materials, Poisson's ratio 𝜈=1/2

@@math.physics thank you for the comprehensive explanation

w(x,y) is the displacement of the generic point (x,y) of the plate in the direction z perpendicular to the plate. In the classical theory of thin plates, the derivatives of w are related to stresses, strains, bending moments, twisting moments, shear forces, etc.

There are several applications that demonstrate the versatility of the Black-Scholes model in the financial world, particularly in derivatives trading and risk management (I don't have much experience with that though)

@@plranisch9509 köszi, azt hiszem, egy év alatt szépen fejlődtem (mármint ehhez a videóhoz képest) :)