A few words about physical theories as approximations of the real world.


    Foreword


    I decided to write a small article examining the current level of development of some physical theories (in my level of understanding) in the context of comparison with theories called classical non-relativistic physics.

    First of all, I want to point out that I call classical nonrelativistic physics a part of theoretical physics, which was created in the second half of the 18th - first half of the 19th centuries, Lagrange , Hamilton, and later expanded by other physicists during the 19th century (I don’t mention the names of these physicists who could help bring the theory and its mathematical apparatus to a modern form, including those born in the Russian Empire).

    Classical non-relativistic mechanics and the theory of gravity


    The foundations of classical mechanics were laid by I. Newton, who formulated his “3 laws” in his work “Mathematical Principles of Natural Philosophy” (year of publication - 1687), although the principle of relativity formulated by G. Galileo in 1632 (I also use the year of publication) should be mentioned.

    In the simplest case, we can say that Newton's mechanics (as well as Lagrange and Hamilton) can be formulated as:

    $ \ frac {dp} {dt} = F, $


    where p is the impulse, in the general case the so-called “generalized impulse”, and F is the force. In the absence of a magnetic field (which I don’t even mention here a weak or strong interaction), this force can be conservative. Such a force is called conservative, whose work on any trajectory does not depend on the shape of the trajectory and the speed of movement (this is including a reference to relativistic dynamics, in fact it turns out that the concept of "conservative force" does not exist in the SRT).

    For conservative forces, the above law can be rewritten as

    $ \ frac {dp} {dt} = - \ frac {\ partial U (x)} {\ partial x}, $


    where x is the generalized coordinate, and p is the corresponding generalized momentum.

    Such a formulation of "2 Newton's Law" is more general, because it is obtained by writing the Lagrange equation or the Hamilton equation. The Lagrange and Hamilton equations are derived from the principle of least action. The action is an integral that has the dimension J * s and is taken between 2 configurations of the system, that is, sets of coordinates and impulses (x, p). In general, it is expressed in different ways for different approaches to classical mechanics.

    If we talk about the classical theory of gravity, then it is formulated in the form of Newton's law of gravity (through force, or you can write it down through potential energy)

    $ F = G \ frac {mM} {r ^ 2}, $


    where the force acts in the direction of the attracting body (by this the force of gravity is different from the electric force, which creates repulsion for identical charges).

    The formulation of the law of gravity through the potential energy can be expressed by the simplest phrase:

    The sum of the kinetic energy T (v) and the potential energy U ( r ) remains constant the entire time the particle (system of particles) moves along their trajectory.
    From this law you can get the simplest equation:

    $ {\ frac {m} {2} \ left (\ frac {dr} {dt} \ right) ^ 2} + U (r) = E $


    In the event that we were able to reduce the task to a 1-dimensional coordinate r (the distance between the centers of mass of these 2 bodies), we can write the solution of the problem through the integral:

    $ {\ left (\ frac {dr} {dt} \ right) ^ 2} = {\ frac {m} {2}} (E - U (r)) $


    A further solution method is to take the root and then we get the simplest differential equation with separable variables. There are 2 problems:

    1. In the general case of an arbitrary potential U ( r ), we may not be able to take this integral at all.
    2. Instead of the usual solution of the problem r = r ( t ), we get the solution t = t ( r ).

    At the end of this section I want to add that before A. Einstein created his own form of the theory of relativity in the second half of the XIX century, J. Maxwell summarized the laws for the electric and magnetic fields (which began to be formulated 35 years before, but separately). Prior to that, such a theory was recorded. formulas, like the formula of Lorentz force.

    Heaviside's role in creating the very concept of '4 Maxwell's equations'
    Хевисайд упростил для использования учёными оригинальные результаты Максвелла. Эта новая формулировка дала четыре векторных уравнения, известных теперь как уравнения Максвелла. Хевисайд ввёл так называемую функцию Хевисайда, используемую для моделирования электрического тока в цепи. Хевисайд разработал понятие вектора и векторный анализ. Хевисайд создал операторный метод для линейных дифференциальных уравнений.

    The Lorentz force (divided by the electric charge of the particle) is interesting in that it is essentially an approximation for the concept of "electric field strength E in the reference frame of a particle moving at speed v " for speeds v much lower than the speed of light.

    Special theory of relativity


    The special theory of relativity (STR) was created in the years 1892-1905 by the works of H. Lorentz, A. Poincaré and A. Einstein. Describes inertial reference systems (ISO), strictly speaking its postulates are violated as soon as the reference system ceases to be inertial (the nature of the system’s motion ceases to be uniform and rectilinear). In quantum field theory (in my humble understanding), such a “law” works that, after finding CO in a state of non-inertial motion, the first of the postulates mentioned below ceases to be fulfilled altogether, even for the time of the future uniform and straight-line motion.
    Probably everyone remembers the postulates of SRT, from which the Lorentz transformations are derived, but I will formulate them as follows:

    1. The formulation of all laws of physics does not depend on whether the system is at rest or moves uniformly and rectilinearly .
    2. The invariance of the phase of an electromagnetic wave relative to the transition to another ISO, also known as the conservation of the square of the interval between two events.

    Of the necessary formulas for further consideration, I will mention the following:

    $ E ^ 2 = (pc) ^ 2 + (mc ^ 2) ^ 2 \: \: \: (1) $


    It describes the relationship between particle energy, momentum, and rest mass .

    One of the consequences of SRT is that a particle with a rest mass above 0 cannot reach the speed of light, although energy can still grow above the “classical” limit.

    $ E = {\ frac {mc ^ 2} {2}} $


    This statement is consistent with the fact that the elementary particle can have a kinetic energy that is substantially greater than this value.

    And of course we should mention the Lorenz metric, also known as the Minkowski metric:

    $ g = diag (1, -1, -1, -1) $


    Through this metric, you can introduce the concept of "length of 4-vector", 4-vectors include:

    $ 4-coordinate \: (t, r), \: 4-speed \: ({\ Gamma}, v {\ Gamma}), \: 4-pulse \ :( E, p) $


    In this case, I applied the notation, in which time is measured in meters , and the speed of light is equal to one . That is, a “good” 4-vector record requires that it consist of 4 values ​​of the same dimension.

    An important property of any 4-vector is that its value, when moving to another frame of reference, is transformed in the same way as the corresponding components of the 4-coordinate.

    In electrodynamics, there is such a value as 4-dimensional current density. The 4-current vector can be written as:

    $ J ^ \ mu = (c \ rho, j) $


    $ J_ \ mu = (c \ rho, -j) $


    It should also be mentioned that there are covariant (as the first 4-current record) and contravariant (as the second record) vectors. The transition between these vectors is carried out according to the formula:

    $ J_ \ mu = g _ {\ mu \ nu} J ^ \ nu, $


    here the Einstein agreement is applied, which means that this record implies a summation over a pair of identical indices located at the top and bottom.

    And since the article on approximations, I will certainly mention how one can show the STO approximation to Newton's mechanics and how one can use it. From the formula (1) can be expressed energy through momentum:

    $ E = ((mc ^ 2) ^ 2 + (pc) ^ 2) ^ {\ frac {1} {2}} = mc ^ 2 * \ left (1 + \ left (\ frac {p} {mc} \ right) ^ 2 \ right) ^ {\ frac {1} {2}} \ approx mc ^ 2 \ left (1+ \ frac {1} {2} \ left (\ frac {p} {mc} \ right ) ^ 2 - \ frac {3} {8} \ left (\ frac {p} {mc} \ right) ^ 4 \ right) $


    Kinetic energy can be expressed as the difference between the total energy E and the energy of rest:

    $ T = E - mc^2 \approx mc^2 * \left(\frac{1}{2} \left(\frac{p}{mc}\right)^2 - \frac{3}{8} \left(\frac{p}{mc}\right)^4\right) \:\:\: (2) $


    And in the approximation p << mc we get one function for recording the kinetic energy through a pulse:

    $ T = \frac{p^2}{2m} $


    Without taking into account any fields (electric, magnetic, gravitational, etc.) that create potential energy, this formula can be written as a special case of the Hamilton function (see above the mention of Lagrange mechanics and Hamilton mechanics):

    $ H = \frac{p^2}{2m}, $


    in the more general case

    $ H = \frac{p^2}{2m} + U(r) $


    Do not do in the theory of relativity without the energy-momentum tensor (the tensor can be written in the form of a matrix of 4 by 4 dimensions). I will write down the definition of this tensor:
    The energy-momentum tensor is a symmetric second-rank tensor describing the density and flow of energy and momentum of matter fields.

    There are formulas for the components of this tensor of various substances and fields, for example, a fluid at rest or an electromagnetic field (that is, SRT operates with an electromagnetic field as a field with energy density, energy and momentum flow). In the latter case, the energy-momentum tensor can be written through the electromagnetic field tensor F :

    $T^{\mu \nu} = -\frac{1}{\mu _0}(F^{\mu \alpha}F_{\alpha }^{\nu} + \frac{1}{4}{\eta}^{\mu \nu}F_ {\alpha \beta}F^{\alpha \beta} )$


    As the end of this section, I will mention the concept of Lorentz invariance, more precisely, the case of application to physical quantities. This property is defined as follows:
    Lorentz invariance refers to the property of some value to be preserved under Lorentz transformations (usually a scalar value is meant, however, this term is also applied to 4-vectors or tensors, meaning not their specific representation, but “ geometric objects themselves ").

    Values ​​possessing the above property are called invariants . The set of invariants of special relativity is mentioned here , some invariant mass is of some interest among them .

    General theory of relativity


    Immediately I warn you that I am not an expert in this part of physics, so I’ll write about what I remember a little from the course of the Physical and Technical Institute and from various sources, like Wikipedia.

    First of all, the principle of general covariance should be mentioned . It is a modification of the first of the postulates of SRT mentioned by me and can be formulated as follows:

    Mathematical equations describing the laws of nature should not change their form and be valid when converting to any coordinate systems, that is, to be covariant with respect to any coordinate transformations.

    I would like to start the distinction of general relativity from special relativity with the fact that the metric tensor in general relativity receives a difference from the type of the Minkowski tensor, while retaining at least one of its properties:

    $g_{ij} = g_{ji}^*$


    where the symbol * I applied here in the sense of complex conjugation. Of course, by definition, it is not very good to introduce a metric with complex elements of the tensor, but physics does not always operate with real values, so I leave the expression in this form. In general, you can try to substitute any kind of metric (that is, not valid) into the equations in general, but you can then get a complex energy-momentum tensor. All components of the metric tensor can depend on coordinates, but at the same time these dependences should remain fairly smooth, since the tensor is a solution of a differential equation.

    The concept of space-time curvature is introduced into general relativity through concepts such as Christoffel symbols.and the covariant derivative (in the sense I need, the covariant derivative is written here ).

    The curvature tensor was first introduced by the German mathematician Bernhard Riemann in “Ueber die Hypothesen, welche der Geometrie zu Grunde liegen” ([1]), first published after the death of Riemann. With the help of the symbols mentioned above, this fourth-rank tensor can be written in the following form:

    $R^\iota_{\sigma \mu \nu} = \partial_\mu \Gamma^{\iota}_{\nu \sigma} - \partial_\nu \Gamma^{\iota}_{\mu \sigma} + \Gamma^{\iota}_{\mu \lambda} \Gamma^{\lambda}_{\nu \sigma} - \Gamma^{\iota}_{\nu \lambda} \Gamma^{\lambda}_{\mu \sigma}$


    And a sufficient condition for all components of the curvature tensor to be zero is the equality to zero of all Christoffel symbols:

    $ \Gamma^{\lambda}_{\nu \sigma} = 0$


    A trivial condition for this is the diagonal matrix g and the condition for any permutation of indices

    $ \frac{\partial g_{\nu \sigma}}{\partial x_\lambda} = 0 $



    I now turn to how to get space-time with a zero curvature tensor, or rather, the Ricci tensor. The Ricci tensor is called the convolution of the curvature tensor by the first and last index:

    $ R_{\sigma \mu} = R^\nu_{\sigma \mu \nu} $


    Looking ahead, I will say that, according to Einstein's equation, the Ricci zero tensor can only be in empty space (when all components of the energy-momentum tensor are zero). In such a space, we will not get gravity according to Newton's theory. Those interested can try to find a metric that is different from the Minkowski metric, but retains the Ricci zero tensor. It is possible that you will open gravitational waves .

    By convolving the Ricci tensor over the remaining 2 indices, we obtain the scalar curvature:

    $ R = R^\nu_{\nu} $


    I now turn to the Einstein equation itself, also known as the Einstein-Hilbert equation.

    Briefly about the role of Hilbert in creating the Einstein equation
    Цитата из Википедии:

    Летом 1915 года Эйнштейн приехал в Гёттингенский университет, где прочитал ведущим математикам того времени, в числе которых был и Гильберт, лекции о важности построения физической теории гравитации и имевшихся к тому времени у него наиболее перспективных подходах к решению проблемы и её трудностях. Между Эйнштейном и Гильбертом завязалась переписка с обсуждением данной темы, которая значительно ускорила завершение работы по выводу окончательных уравнений поля. До недавнего времени считалось, что Гильберт получил эти уравнения на 5 дней раньше, но опубликовал позже: Эйнштейн представил в Берлинскую академию свою работу, содержащую правильный вариант уравнений, 25 ноября, а заметка Гильберта «Основания физики» была озвучена 20 ноября 1915 года на докладе в Гёттингенском математическом обществе и передана Королевскому научному обществу в Гёттингене, за 5 дней до Эйнштейна (опубликована в 1916 году). Однако в 1997 году была обнаружена корректура статьи Гильберта от 6 декабря, из которой видно, что Гильберт выписал уравнения поля в классическом виде не на 5 дней раньше, а на 4 месяца позже Эйнштейна. В ходе завершающей правки Гильберт вставил в свою статью ссылки на параллельную декабрьскую работу Эйнштейна, добавил замечание о том, что уравнения поля можно представить и в ином виде (далее он выписал классическую формулу Эйнштейна, но без доказательства)...

    When deriving the gravitational field equation, scientists used 2 principles:

    • general covariance principle
    • the assumption that in the approximation of a weak gravitational potential the equations of mechanics should be reduced to the mechanics of special relativity with Newtonian gravity

    Taking this into account, it was found that the effect of the gravitational field can be a function of only 2 values ​​— the scalar curvature R (in the absence of gravitating masses and other energies, the curvature should be zero) and the determinant of the metric tensor g (for the Minkowski metric g = -1).

    These statements I consider proven scientists. Other scientists could introduce a modification of Einstein's action, the most famous example being the Brans-Dicke theory . Sufficient evidence of these theories in the observations has not yet been received. Those who wish to study the theory itself can read for example here .
    Taking into account the above notation, the Einstein equation can be written as follows:

    $ R^{\mu \nu} - \frac{1}{2}g^{\mu \nu} R + 8 \pi G T^{\mu \nu} = 0, $


    where G is the gravitational constant. The short meaning of the equation can be formulated as follows:

    • The source of the curvature of space-time is the energy-momentum tensor of all matter and energy in this space.

    In this case, I do not mention the dark energy (cosmological constant), although I consider its presence on a global scale to be the result of astronomical observations.

    Quantum mechanics


    Quantum mechanics was created by physicists to describe microscopic systems. One of the first achievements of quantum theory, confirmed in the observed data, was the semiclassical model of the atom N. Bohr, created in 1913. I will use such liberty to write the equations of quantum mechanics - I denote the reduced Planck constant by the letter h (instead of the symbol " h with a line"). The postulate of the Bohr theory, which has a minimal relation to real quantum mechanics, is a postulate about quantization of the angular momentum of an electron of mass m in orbits in an atom:

    $ mvr = nh, $



    where n is a positive integer (in real quantum mechanics, the moment of impulses can be 0, but this number n , called the “principal quantum number”, is natural).

    A further stage in the development of quantum mechanics was the formulation by E. Schrödinger of an equation, later named after him. This equation is written through a special operator called the “Hamiltonian”. The operator is obtained from the Hamilton function by replacing the classical momentum with the momentum operator :

    $ p_x = ih \frac{\partial }{\partial x} , $



    where x is the generalized coordinate corresponding to the classical generalized momentum p x .

    In the general case, the Schrödinger equation is written for the wave function (denoted by the Greek letter "psi") as nonstationary:

    $ ih \frac{\partial \Psi}{\partial t} = \left(- \frac{h^2}{2m} \nabla ^2 + U(x,t)\right) \Psi, $



    here a special case is applied when the generalized impulse in the Hamilton function of a classical system has the form of an ordinary classical impulse. And for the case of conservative systems, the Schrödinger equation can be written in stationary form, which can be considered as an equation for finding the eigenfunctions and eigenvalues ​​of the Hamiltonian operator:

    $ \left(- \frac{h^2}{2m} \nabla ^2 + U(x,t)\right) \Psi = {E \Psi}, $



    where E is the corresponding operator eigenvalue.

    To consider the transition from quantum mechanics to classical, consider the replacement of the wave function in the Schrödinger equation by the following variable:

    $ \Psi = A exp \left(\frac{i}{h} S(x,t)\right) $



    The Schrödinger equation can be solved by expanding the function S (having the dimension of the action) in powers of the Planck constant:

    $ S = S_1 +h S_2+ h^2 S_3 + ... $



    After substituting the function S into the equation, it takes the following form:

    $ \frac{\partial S}{\partial t} + \frac{1}{2m} \left( \frac{\partial S }{\partial x}\right)^2 + U(x) - \frac{i h}{2m} \nabla ^2 S = 0, $



    where the constant A was shortened.

    To obtain the equation of classical mechanics (known as the Hamilton-Jacobi equation), we should point out that the magnitude of the action S on any classical trajectory has a value much larger than the Planck constant. After that, the last member of the equation can be folded.

    If necessary, a more accurate solution of the equation applies the aforementioned expansion of the action in powers of h . The function S 1 is found as a solution of the Hamilton-Jacobi equation, after which it is substituted into the system of equations obtained by expanding the equation in powers of h(that is, that the left and right parts must coincide or, if they are shifted to one side, the coefficients of the conditional polynomial should become zero).

    The ideology of the approximate solution of the Schrödinger equation (more precisely, finding corrections to energy levels) can be formulated as follows:
    Using several wavelengths of the unperturbed Hamiltonian H 0 and the value of the perturbation H 1 (equal to H - H 0 ), we can find new energy levels E.

    Hamiltonian physical system is represented as:

    $ H = H_1 + H_2 + ..., $



    where ... imply that in different cases we need to take into account a different number of amendments, which, as a rule, have a different order of smallness. These corrections to the Hamiltonian are called perturbations, and the wave functions of the Hamiltonian H 1 must be exactly known. The corresponding equation theory is called perturbation theory .
    If we know the wave functions of the Hamiltonian H 1 , then they form the basis of linear space (EMNIP). This means that, in general, any wave function can be represented as a linear combination of the wave functions of the unperturbed Hamiltonian. With this in mind, it can be shown that the first order of perturbation theory leads to a change in the energy of the level numbered n by value

    $ dE_n = < \Psi_n |H_2| \Psi_n> $



    This expression is called the matrix element of the H 2 operator with respect to the wave functions corresponding to the states with the numbers n and n .

    The very first (by the time of discovery) and (EMNIP) the largest in magnitude deviation of the energy levels of a hydrogen atom from the prediction of nonrelativistic quantum mechanics can be obtained provided that the system of the kinetic energy operator is substituted in the form of a perturbation of the Hamiltonian in the form of formula (2):

    $ dE_n = < \Psi_n |mc^2 * \left( - \frac{3}{8} \left(\frac{p}{mc}\right)^4\right)| \Psi_n> $



    You could see that this value is negative. There are 2 comments. Firstly, the impulse operator here corresponds to a relativistic impulse that can exceed mc - it means that in the relativistic case the first term in the decomposition of kinetic energy also grows. Secondly, by the time formula 2 begins to fall as the impulse grows, you know exactly what you should have taken into account:

    • next term expansion;
    • the following order of perturbation theory;
    • many corrections to the physical model (the size and shape of the nucleus, the magnetic moment of the electron and the nucleus, the reduced mass of the electron).

    According to my very conditional estimates, this method of complicating the model can work for calculating the energy of the 1s energy level on a number of chemical elements from hydrogen to lanthanum (inclusive), and for higher energy levels - further (taking into account the correction for the second example the order of the perturbation theory uses the value of this level itself, that is, there is already an error). For these atoms, it is already necessary to take into account the Dirac equation , and for the most accurate (at the present level of development) real-world mapping, quantum theory of the (electromagnetic) field must be taken into account.

    Instead of an afterword


    This concludes my review, as it is close to the boundaries of my area of ​​expertise. But science does not stand still. For 100 years after the formulation of GR, gravitational waves were discovered, and for 100 years after the formulation of Bohr's postulates, a whole set of elementary particles was discovered and, in fact, 3 new fundamental interactions. SRT and quantum mechanics have already found application in practical devices (we are talking not only about experimental scientific installations, but also about a lot of optical devices).

    List of sources cited:
    1. Ueber die Hypothesen, welche der Geometrie zu Grunde liegen // Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen, vol. 13, 1867
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