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field in physics

Lecture



A field in physics is a physical object that is classically described by mathematical scalar, vector, tensor, spinor field (or some combination of such mathematical fields), obeying dynamic equations (equations of motion, called in this case field equations or field equations, are usually partial differential equations in partial derivatives). In other words, a physical field is represented by a certain dynamic physical quantity [1] (called a field variable [2] ) defined at all [3] points of space (and accepting generally different values ​​at different points in space, also changing with time [4 ] ). [ source not specified 293 days ]

In quantum field theory, a field variable can be considered formally, just as the spatial coordinate is considered in ordinary quantum mechanics, and the quantum operator [5] of the corresponding name is associated with a field variable.

The field paradigm , which represents the whole physical reality at the fundamental level, reduced to a small number of interacting (quantized) fields, is not only one of the most important in modern physics, but, perhaps, undoubtedly dominant [6] .

  • It is easiest to visualize the field (when it comes to, for example, fundamental fields that do not have an obvious direct mechanical nature [7] ) as a perturbation (deviation from equilibrium, movement) of some (hypothetical or just imaginary) continuous medium that fills the whole space. For example, as the deformation of an elastic medium, the equations of motion of which coincide with or are close to the field equations of that more abstract field that we want to visualize. Historically, such an environment was called the ether, but later the term was almost completely out of use [8] , and its implied physically meaningful part merged with the concept of the field itself. Nevertheless, for a principled visual understanding of the concept of a physical field in general, such a representation is useful, given that in the framework of modern physics such an approach is usually accepted by and large only as an illustration [9] .

The physical field, therefore, can be characterized as a distributed dynamic system having an infinite number of degrees of freedom.

The role of a field variable for fundamental fields is often played by potential (scalar, vector, tensor), sometimes a quantity called field strength. (For quantized fields, in a certain sense, the corresponding operator is also a generalization of the classical concept of a field variable).

Also, a field in physics refers to a physical quantity, considered as a place-dependent: as a complete set, generally speaking, of different values ​​of this quantity for all points of some extended continuous body - a continuous medium, describing in its entirety the state or movement of this extended body [10] . Examples of such fields can be:

  • temperature (generally speaking different at different points, as well as at different points in time) in a certain medium (for example, in a crystal, liquid, or gas) is a (scalar) temperature field,
  • the speed of all elements of a certain volume of fluid is the velocity vector field,
  • displacement vector field and tensor stress field during deformation of an elastic body.

The dynamics of such fields are also described by partial differential equations, and historically the first, starting from the XVIII century, in physics were considered precisely such fields.

The modern concept of the physical field grew out of the idea of ​​an electromagnetic field, first realized in a physically specific and relatively close to the modern form by Faraday, mathematically consistently implemented by Maxwell - initially using a mechanical model of a hypothetical continuous medium - ether, but then going beyond the use of a mechanical model.

Fundamental Fields

Among the fields in physics emit the so-called fundamental. These are fields that, according to the field paradigm of modern physics, form the basis of the physical picture of the world, all other fields and interactions are derived from them. They include two main classes of fields interacting with each other:

  • fundamental fermion fields, primarily representing the physical basis of the description of a substance,
  • fundamental boson fields (including the gravitational field, which is a tensor gauge field), which are an extension and development of the concept of Maxwell electromagnetic and Newtonian gravitational fields; the theory of fundamental interactions is built on them.

There are theories (for example, string theory, various other theories of unification) in which several other, even more fundamental from the point of view of these theories, fields or objects play the role of fundamental fields (and the current fundamental fields appear or should appear in these theories to some extent as a “phenomenological” consequence). However, while such theories are not sufficiently confirmed or generally accepted.

Story

Historically, among the fundamental fields, fields were first discovered (namely, as physical fields [11] ) that are responsible for electromagnetic (electric magnetic fields, then combined into an electromagnetic field), and gravitational interaction. These fields were discovered and studied in sufficient detail in classical physics. At first, these fields (in the framework of the Newtonian theory of incidence, electrostatics, and magnetostatics) looked to most physicists rather as formal mathematical objects, introduced for formal convenience, and not as a full-fledged physical reality, despite attempts at a deeper physical understanding, which remained however rather vague or not bearing too substantial fruits [12] . But starting with Faraday and Maxwell, the approach to the field (in this case, to the electromagnetic field) as a completely substantial physical reality began to be applied systematically and very fruitfully, including a significant breakthrough in the mathematical formulation of these ideas.

Fields corresponding to weak interactions and strong interactions (playing an important role in nuclear nuclear physics and particle physics; the latter — among other things, in explaining nuclear forces) are discovered much later, since practically they manifest themselves only in nuclear and particle physics and at such energies and distances, which in principle belong to the field of quantum theories.

Nevertheless, in principle (despite the fact that it is not easy to detect directly for all of them), all four mentioned fields manifest themselves as intermediaries in the interaction of charged (various types of charges) bodies (particles), transferring this interaction with a finite velocity (the speed of light), while the intensity (force) of the interaction is determined, in addition to the position and movement of bodies, their charges: mass (gravitational charge) for the gravitational field, electric charge for electromagnetic, etc.

Experimental confirmation of Maxwell’s theory in 1887 by Heinrich Hertz, who obtained direct experimental evidence of the existence of electromagnetic waves predicted by Maxwell (which, among other things, allowed us to finally connect optics, which had previously been an independent field of physics, to electromagnetic theory, and this was a very significant advance in the direction of increasing the internal connectivity of physics).

Gradually it turned out that the field possesses almost all the attributes of a full-fledged physical reality, including the ability to transfer energy and momentum, and even under certain conditions to have an effective mass [13] .

On the other hand, as quantum mechanics developed, it became increasingly clear that matter (particles) possesses properties that are theoretically inherent in fields.

State of the art

After the creation of quantum mechanics and the sufficiently deep development of quantum concepts, it became obvious that all matter, including matter, is described by quantized fields: individual fundamental fields (like an electron) or their collective excitations (like a proton composed of three quarks of an igluon field). Single quantum excitations of fundamental fields and are elementary particles. Photons, vector bosons, gluons, gravitons (not yet fixed as separate particles), leptons and quarks belong to such quantum excitations of fundamental fields of different types [14] . The field equations for free fields, their quantization, the interaction of various fields were discovered and studied in detail [15] .

Thus, it turned out that the physical picture of the world can be reduced in its foundation to quantized fields and their interaction.

To some extent, mainly within the framework of the formalism of integration along trajectories and Feynman diagrams, the opposite movement took place: the fields became possible to a large extent represented as almost classical particles (more precisely, as a superposition of an infinite number of almost classical particles moving according to imaginable trajectories) , and the interaction of fields with each other - as the birth and absorption of each other by particles (also with a superposition of all conceivable variants of such). And although this approach is very beautiful, convenient and allows you in many ways to psychologically return to the notion of a particle as a good old classic particle with a well-defined trajectory, it, nevertheless, cannot cancel the field view of things and is not even a completely symmetrical alternative. (and therefore, nevertheless, closer to a beautiful, psychologically and practically convenient, but still just a formal reception, than to a completely independent concept). The point here is in two key points:

  1. the superposition procedure cannot be “physically” explained in terms of truly classical particles, it is simply added to the almost classical “particle” picture, not being its organic element; at the same time, from a field point of view, this superposition has a clear and natural interpretation;
  2. The particle itself, moving along one separate trajectory in the formalism of the integral along trajectories, although very similar to the classical one, is still not completely classic: to the usual classical movement along a certain trajectory with a certain momentum and coordinate at any given moment even for one the only trajectory is that you have to add a completely alien to this approach in its pure form the concept of a phase (that is, some wave property), and this moment (although it is really minimized and about it is quite easy just not to think) also does not have some kind of organic inner interpretation; and in the framework of the usual field approach, such an interpretation is again, and it is again organic.

Thus, it can be concluded that the path integration approach is, although very psychologically convenient (after all, a point particle with three degrees of freedom, for example, is much simpler than an infinite-dimensional field that describes it) and has proven practical productivity, but it’s just some reformulation albeit a rather radical, field concept, and not its alternative.

And although in words in this language everything looks very “corpuscular” (for example: “the interaction of charged particles is explained by the exchange of another particle — the carrier of interaction” or “the mutual repulsion of two electrons is due to the exchange between them by a virtual photon”), but behind this are typically typical field reality, like the propagation of waves, albeit fairly well hidden for the sake of creating an efficient computation scheme, and in many ways providing additional opportunities for qualitative understanding.

Currently (2012), the fundamental boson (gauge) fields are considered to be several fields associated with electroweak, strong game gravitational interactions. The spinor fields of several “generations” of leptons and quarks belong to the fundamental fermion fields.

List of fundamental fields

In the framework of the Standard Model, the following fields appear as fundamental

Fundamental fermion fields

Each fundamental fermion (each type of quarks and each type of leptons) in the Standard Model has its own field, mathematically represented by a spinor field.

Fundamental boson fields (fields - carriers of fundamental interactions)

These fields in the framework of the standard model are calibration fields. Their types are known:

  • Electroweak
    • Electromagnetic field (see also Foton)
      • Electric field
      • A magnetic field
    • The field is a carrier of weak interaction (see also W and Z bosons)
  • Gluon field (see also Gluon)
    • nuclear field
  • Gravitational field

Hypothetical fields

Any theoretical objects (for example, fields) that are described by theories that do not contain internal contradictions can be considered hypothetical in a broad sense, which clearly do not contradict the observations and can at the same time give observable consequences that allow making a choice in favor of these theories compared to those which are accepted now. In practice (in order to cut off a vast amount of potentially possible, but useless theories), the principle of falsifiability is also applied. Below, we will speak (and this generally corresponds to the usual understanding of the term) mainly about hypotheticality in this narrower and more strict sense, implying the validity and falsification of the hypothesis that we call a hypothesis.

In theoretical physics, many different hypothetical fields are considered, each of which belongs to a very specific definite theory (in their type and mathematical properties, these fields may be quite or almost the same as known non-hypothetical fields, and may differ more or less strongly; in that and in another case, their hypothetical meaning is that they have not yet been observed in reality, have not been detected experimentally, and in relation to some of the hypothetical fields there may be a question Oguta whether they occur in principle, and even if they exist at all - for example, if the theory in which they are present, suddenly appears self-contradictory).

The question of what should be considered a criterion that allows you to transfer some specific field from the hypothetical category to the real one is rather subtle, since the confirmations of a particular theory and reality of certain objects contained in it are often more or less indirect. In this case, it usually boils down to some kind of rational agreement of the scientific community (whose members are more or less detailed in what degree of evidence they are actually talking about).

Even in theories that are considered to be fairly well confirmed, there is a place for hypothetical fields (here we are talking about the fact that different parts of the theory have been tested with varying degrees of thoroughness, and some fields that play an important role in them have not yet clearly appeared in the experiment that is, it looks like a hypothesis, invented for various theoretical purposes, while other fields that appear in the same theory have already been studied well enough to speak of them as reality).

An example of such a hypothetical field is the Higgs field, which is important in the Standard Model, the remaining fields of which are by no means hypothetical, and the model itself, albeit with unavoidable reservations, is considered to describe reality (at least to the extent that reality is known).

There are many theories containing fields that have (so far) never been observed, and sometimes these theories themselves give such estimates that their hypothetical fields seem to be (due to the weakness of their manifestation, following from the theory itself) and cannot be detected in the foreseeable future (for example, a torsion field). Such theories (if they do not contain, besides practically unverifiable, also a sufficient number of easier verifiable consequences) are not considered of practical interest, unless some nontrivial new way of their verification emerges, allowing to circumvent the obvious limitations. Sometimes (as, for example, in many alternative theories of gravity — for example, the Dicke field), such hypothetical fields are introduced, about the force of manifestation of which the theory itself cannot say anything at all (for example, the coupling constant of this field with others is unknown and can be quite large , and arbitrarily small); с проверкой таких теорий обычно также не торопятся (поскольку таких теорий много, а своей полезности каждая из них ничем не доказала, и даже формальнонефальсифицируема), за исключением случаев, когда какая-то из них не начинает по каким-то причинам казаться перспективной для разрешения каких-то текущих затруднений (впрочем, от отсеивания теорий на основании нефальсифицируемости — особенно из-за неопределенных констант — тут иногда отказываются, т.к. серьезная добротная теория иногда может быть проверена в надежде, что ее эффект обнаружит ся, хотя гарантий этого и нет; особенно это верно, когда теорий-кандидатов вообще немного или некоторые из них выглядят особенно фундаментально интересными; также — в случаях, когда можно проверять теории широкого класса все сразу по известным параметрам, не тратя специальных усилий на проверку каждой в отдельности).

Следует также заметить, что принято называть гипотетическими лишь такие поля, которые совсем не имеют наблюдаемых проявлений (или имеют их недостаточно, как в случае с полем Хиггса). Если же существование физического поля твердо установлено по его наблюдаемым проявлениям, и речь идет лишь об улучшении его теоретического описания (например, о замене ньютоновского гравитационного поля на поле метрического тензора в ОТО), то говорить о том или другом как о гипотетических обычно не принято (хотя для ранней ситуации в ОТО можно было говорить о гипотетическом характере тензорной природы гравитационного поля).

В заключение упомянем о таких полях, сам тип которых достаточно необычен, т.е. теоретически вполне мыслим, но никакие поля подобных типов никогда не наблюдался на практике (а в некоторых случаях на ранних этапах развития их теории могли возникали и сомнения в ее непротиворечивости). К таким, прежде всего, следует отнеститахионные поля. Собственно, тахионные поля можно назвать скорее лишь потенциально гипотетическими (то есть не достигающими статуса обоснованного предположения ), т.к. известные конкретные теории, в которых они играют более или менее существенную роль, например, теория струн,

Еще более экзотические (например, лоренц-неинвариантные — нарушающие принцип относительности) поля (при том, что абстрактно-теоретически вполне мыслимы) в современной физике можно отнести к стоящим уже достаточно далеко за рамками аргументированного предположения, то есть, строго говоря, их не рассматривают даже в качестве гипотетических [16] .

Традиционные варианты употребления термина поле

see also

  • Фундаментальные взаимодействия
  • Квантовая теория поля
  • Maxwell's equations

Notes

↑ Показывать компактно

  1. Скалярного, векторного, тензорного или спинорного характера; в любом случае эта величина как правило может быть сведена к представлению числом или некоторым набором чисел (принимающих вообще говоря различные значения в разных точках пространства).
  2. В зависимости от математического вида этой величины различают скалярные, векторные, тензорные и спинорные поля.
  3. Поле определено во всем пространстве, если это фундаментальное поле. Такие поля, как поле скорости течения жидкости или поле деформации кристалла, определены на области пространства, заполненной соответствующей средой.
  4. В современном изложении это обычно выглядит как поле на (в) пространстве-времени, таким образом зависимость полевой переменной от времени рассматривается почти равноправно с зависимостью от пространственных координат.
  5. В принципе аналогично тому, как координата частицы как физическая наблюдаемая в обычной квантовой механике представлена оператором координаты, который позволяет вычислить среднее значение координаты итд (аналогично полевой оператор позволяет вычислить среднее значение поля итд).
  6. Несмотря на наличие более или менее удаленных от ее стандартного варианта альтернативных концепций или переинтерпретаций, которые однако не могут пока ни получить решительного перед ней преимущества или даже равенства с ней (не выходя, как правило, за пределы достаточно маргинальных явлений переднего края теорфизики), ни, как правило, слишком далеко от нее удалиться, оставляя ей в целом всё же (пока) центральное место.
  7. В отличие от упомянутого несколько ниже класса физических полей из физики сплошных сред, имеющих достаточно нагляжную природу сами по себе, упоминаемых в статье дальше.
  8. По разным историческим причинам, не последней из которых была та, что концепция эфира психологически подразумевала достаточно конкретную реализацию, которая могла бы дать экспериментально проверяемые следствия, однако в реальности физически наблюдаемых нетривиальных следствий некоторых из подобных моделей не было обнаружено, следствия же из других прямо противоречили эксперименту, поэтому концепция физически реального эфира постепенно была признана излишней, а вместе с ней вышел из употребления в физике и сам термин.
  9. То есть за ним не признается обычно какого-то большого самостоятельного теоретического значения на современном этапе. Это означает, что о подобной гипотетической среде ничего конкретного и достоверно проявляющегося в эксперименте или наблюдении не известно, кроме собственно полевых уравнений, почему стандартно последние и принято рассматривать абстрактно, без привязки к конкретной механической итп модели (разве что в каких-то сугубо вспомогательных целях, список которых едва ли не исчерпывается целями наглядности). Это усугуб***яется тем, что для одних и тех же полевых уравнений может существовать много разных механических итп моделей (из которых не представляется возможным сделать обоснованный выбор), и наоборот, для некоторых физических полей трудно придумать хотя бы обну адекватную механическую модель (к чему, впрочем, обычно и не стремятся).
  10. Под состоянием и движением может иметься в виду макроскопическое положение и механическое движение элементарных объемов тела, а также это могут быть зависимости от пространственных координат и изменения со временем величин такого характера, как электрический ток, температура, концентрация того или иного вещества итд.
  11. Вещество было, конечно, известно даже раньше, но долгое время было совершенно не очевидно, что концепция поля может иметь отношение к описанию вещества (которое описывалось преимущественно "корпускулярно"). Таким образом, сама концепция физического поля и соответствующий математический аппарат был исторически развит сначала применительно к электромагнитному полю и гравитации.
  12. За исключением случаев, когда и самые туманные соображения приводили к серьезным открытиям, т.к. служили стимулом к экспериментальным исследованиям, приводившим к фундаментальным открытиям, как при открытии Эрстедом порождения магнитного поля электрическим током.
  13. Peter Galison Einstein's clocks, Poincaré's maps: empires of time. — 2004. — P. 389.
    См. статью Пуанкаре “Динамика электрона”, раздел VIII (А. Пуанкаре. Избранные труды, т. 3. М., Наука, 1974.), доклад М. Планка (М. Планк. Избранные труды. М., Наука, 1975.) и статью Эйнштейна и Лаубе “О пондемоторных силах”, § 3 “Равенство действия и противодействия” (А. Эйнштейн. Собрание научных трудов, т. 1. М., Наука, 1965.) (все за 1908 год).
  14. Тем не менее, имеющим очень много общего.
  15. Часть свойств полевых уравнений удалось прояснить исходя из достаточно общих принципов, таких как лоренц-инвариантность и принцип причинности. Так принцип причинности и принцип конечности скорости распространения взаимодействий требуют, чтобы дифференциальные уравнения, описывающие фундаментальные поля, принадлежали кгиперболическому типу.
  16. Это описание того положения, которое существует на настоящий момент. Конечно же, они не означает принципиальной невозможности появления вполне достаточно мотивированных теорий, включающих такого рода экзотические поля в будущем (впрочем, вряд ли следует считать такую возможность и слишком вероятной).

Literature


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Basic Physics

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