Ah, Wulf,

you are a treasure. It's so good to have a list of Bohmians. It is not complete, of course, but is helpful. An acquaintance of mine said that for the Bohmians, Bohm's interpretation is a religion. If one presents an anti-Bohm proof, they argue ad infinitum.

I have friends among the Bohmians, many of them are nice people. Though, if one tries to publish an anti-Bohm proof in a journal, one should look who are the editors. I had a bad experience with a journal of which I wasn't aware who is the chief editor.

The questions about the Lagrangian of the scalar field have to do with classical, not quantum, physics.

The Lagrangian in question describes nothing more and nothing less than an infinite number of relativistic harmonic oscillators.

It suffices to take the Lagrangian of one harmonic oscillator and replace the kinetic term -φ(d^2/dt^2)φ, that, essentially, expresses just time translation invariance, with -φ(Box)φ, where Box= d'Alembertian, that expresses Lorentz invariance, along with invariance under spatial and time translations. The continuous, spatial, variable, labels each oscillator.

A Fourier transform then decouples them...

Dear Stam,

I thought you'd intervene - and thank you - I know you are competent in such things. But, please go slowly. (Please see, the article I mentioned, is supposed to give a Bohmian theory of the photon, i.e. to explain which-way measurements without resorting to the collapse - so I was told. I intend to understand exactly what the authors do, and if they succeed in their task.)

You say: "The Lagrangian in question describes nothing more and nothing less than an infinite number of relativistic harmonic oscillators."

Here I don't understand. Which infinity? Which variable in formula (1) sweeps that infinity of oscillators?

You say also: "replace the kinetic term -φ(d^2/dt^2)φ, that, essentially, expresses just time translation invariance, with -φ(Box)φ, where Box= d'Alembertian,"

Stam, it's not the Klein-Gordon equation that I don't understand, it's the equation (2). The authors take ∂ϕ/∂t = canonical momentum. It's no -ϕ ∂2ϕ/∂t2 in equation (2).

Note that for a plane wave ( exp[i(E - px)/ħ] ) one has iħ∂ϕ/∂t = Eϕ. E is energy, not momentum. The expression iħ∂/∂t has energy units, not linear momentum units!

So, if you understand the equation (2), please be kind and explain me too.

With thanks in advance!

Hi Sofia, I am going to propose a discussion that can help you in understanding this subject. Most of what I will discuss here are concepts applied to general field theories (not necessarily quantum field theories). Let's start by highlighting some of your interpretations that need to be rethought:

"The Lagrangian is defined as T - V, where T is the kinetic energy and V the potential energy."

Well, this is right when we are talking about classical particles. But the more general way to define the lagrangian is as the function that miniminises the action. We can still use the analogous "L=T-V" form to construct the theories, but we can't think about it as a definition. The Einstein–Hilbert lagrangian, for example, contains iformation about the geometry of space-time.

Another important thing is to know the difference betwenn lagrangian an lagrangian density. In classical mechanics, we define the action integrating a function L in time. L is said the Lagrangen function. In a field theory, we define the action integrating a function L' in time and space. That is why we call L' a "density of Lagrangean". We usually call both "lagrangean", for short, but is necessary to know the difference.

"If the kinetic energy is defined as ½(∂ϕ/∂t)2, then, what can be the other term, ½(∇ϕ)2 ? According to the definition of the Lagrangian, it should represent potential energy."

As i said, is still usefull to think of the lagrangean on a kind of "T-V" form, but we have to extend the concepts of the "kinetic" term and the "potential" term.

Let's think about a classical particle. Its movement is described by its position r in function of a parameter t (time), so r=r(t). The velocity of the particle is dr/dt, that is, the derivative of the function with respect to its only parameter, t. Kinetic energy is proportional to the square of this derivative.

Now, to the field theory: we're not talking about a particle that have a single defined position at a given time (r = r(t)); we're talking about a field, a function that assumes a value to each point in space, at each instant t. So, ϕ = ϕ (x,y,z,t). Not only the time, but position coordinates are also parameters of the field! This way, we can think about ½(∂ϕ/∂t) as the "temporal component" of the kinetic term, and ½(∇ϕ) as the "spatial component" of kinetic term (the minus sign of the gradient comes from the structure of Minkowski space). So, the gradient term has nothing necessarily to do with relativistic covariance; even a non-relativistic field theory, like Schrodinger field, can present similar structure.

Finally, about the canonical momentum: we define it as the time derivative of the lagrangean density (in further studies on field theory, you'll se how important this quantity is). But don't let the name trick you: it's not necessarily the linear momentum of the field (It happens that, in the classical mechanics, the canonical momentum often coincides with the linear momentum, but it's a special case). Think about relativity: the momentum 4-vector stores the information about the particle's momentum, but also its energy information. We have to abstract the meaning of the word "momentum".

I suggest the reading of the first sections of " Peskin and Schroeder - Introduction to quantum field theories", or the lectures of Prof. David Tong: http://www.damtp.cam.ac.uk/user/tong/qft.html

I hope this be helpful.

Sofia,

Why don't you try to understand first what a field is and how it behaves. Why does the squared modulus of the wavefunction describe a probability density distribution? Probability density of what? What is the relation of these distributions with fields? How do they interact? Do you know the mathematics that describes these interactions?

Please read

TheStructureOfPhysicalReality.pdf ; http://dx.doi.org/10.13140/RG.2.2.10664.26885

BehaviorOfBasicFields.pdf ; http://dx.doi.org/10.13140/RG.2.2.15517.20960

64 Shades of Space ; DOI: http://dx.doi.org/10.13140/RG.2.2.28012.46724

Mass.pdf DOI: http://dx.doi.org/10.13140/RG.2.2.10268.59528

So, obviously physical reality is ruled by stochastic processes and not by all kinds of strange forces and force carriers.

Marcos Souza,

Look for an alternative way of thinking at https://en.wikiversity.org/wiki/Hilbert_Book_Model_Project#Multi-mix_Path_Algorithm.

This starts with a stochastic hopping path and ends with the Lagrangian.

Hans van Leunen,

Thank you for your indication. I have a very poor knowledge of stochastic process, but I will save your page for future readings. In fact, I'm reading it right now, but I'll need time and math background to absorb the concepts.

Hans,

I don't know where from did you take that I have to be taught what is a field and what is a density of probability. Also, I had a look at some of your articles, they have nothing to do with my question.

"So, obviously physical reality is ruled by stochastic processes and not by all kinds of strange forces and force carriers."

My question does not discuss stochasticity, carriers, etc. I am testing a theory which is not the standard one. You did not bother to read the question.

If you want to discuss other issues than my question poses, open a thread of yourself. It's nonsensical to tell me to change my question.

Most physicists use the notion of field without a solid definition of that structure. In a quaternionic non-separable Hilbert space exist normal operators that provide fields as eigenspaces that are defined by quaternionic functions. Their behavior is defined by quaternionic differential equations that precisely specify the interaction between artifacts and field excitations. This is a purely mathematical description that is free from physical interpretations. It is described in BehaviorOfBasicFields.pdf ; http://dx.doi.org/10.13140/RG.2.2.15517.20960

Especially the solutions of the second order partial differential equations are interesting. Part of these solutions are ignored by physicists. Still, these solutions play an important role in physical reality.

Hans,

"Most physicists use the notion of field without a solid definition of that structure."

It's not my guilt, neither is it connected to my question!

Now, to be practical, please see the question:

https://www.researchgate.net/post/What_is_a_field

If the way I asked this question would displease you, I would modify the question in the way you would wish, or, if you would request, I would delete it.

As I mentioned there, I hope the discussion won't remain on mathematics, but would also contain phenomenology. Physics is firstly phenomenology.

Sofia,

I started a thread that treats the subject.

See: https://www.researchgate.net/post/What_part_of_a_field_carries_energy_The_field_itself_or_its_excitations

Yes, Hans, very well you did, I saw the thread.

Now, my problem is not what is a field in general, but what is a quantum field. I don't ask about equations, but rather phenomenologically. See my comment in my thread

https://www.researchgate.net/post/What_is_a_field

with the specific problem that bothers me.

Best regards

Sofia,

In my opinion, the field itself is not quantized. Instead field excitations can be quantized. Obviously, oscillations can be quantized. This is shown by the solutions of the Helmholtz equation. However, the set of solutions of the wave equation covers many other solutions and this includes solutions that cannot be represented as waves or wave packages. Two very particular solutions usually escape the attention of physicists. These are the pulse responses that only occur in odd numbers of participating dimensions. I call them shock fronts because they are not waves. During travel the front keeps its shape. The one-dimensional shock front also keeps its amplitude. This means that it can travel over huge distances without losing its integrity. The spherical shock front diminishes its amplitude as 1/r with distance r from the trigger location. It means that over time it integrates into the Green's function of the field. The (quaternionic) Green's function possesses some volume. Thus the spherical pulse response injects a bit of volume into the field. The dynamics of the spherical pulse response shows that this volume quickly spreads over the full field. Thus, the local deformation that is due to the volume injection quickly fades away. To generate a persistent deformation, a mechanism must recurrently regenerate the spherical pulse response. The effect depends on the regeneration rate and the density of the produced pulse location distribution. So the relation between deformation and the pulse response is rather complicated.

Instead photons can be represented by strings of equidistant one-dimensional shock fronts. In this case the Einstein-Planck relation E=h v defines a more direct relation between the energy of the photon and the energy of the shock fronts that constitute the photon. Planck's constant h relates the energy of the photon to the emission duration of the photon.

In any case, the shock fronts play the role of basic quanta. The one-dimensional shock fronts play the role of pure energy quanta. The spherical shock fronts play the role of a quantized mass-capability. The capability depends on the regeneration rate and on the density of the location distribution.

The mathematics of the discussed shock fronts are treated in detail in

TheStructureOfPhysicalReality.pdf ; http://dx.doi.org/10.13140/RG.2.2.10664.26885

and

BehaviorOfBasicFields.pdf ; http://dx.doi.org/10.13140/RG.2.2.15517.20960

I use quaternionic differential calculus to show that in principle all basic fields can produce the discussed field excitations. Each of these excitations require a correct actuator. For example, a spherical shock front requires an isotropic actuator. This is delivered by the embedding of a quaternion that breaks the symmetry or the arithmetic capabilities of the embedding field.

For me the fact that this embedding invokes a stochastic process that owns a characteristic function is still a mystery. Still, it is the reason of existence of the wavefunction of elementary particles.