5 equivalents in physics

We can express the same thing in many ways, and different people may choose different words or speak different languages or dialects.

In the world of science, the same theory of physics may be expressed in many ways. For years, many scientists have been pondering the question of what exactly we are saying when we say that two theories are "equivalent."

In a review paper, James Owen Weatherall, philosopher of science, reviews in detail the equivalence of physical theories in numerous philosophical papers in the past.

Experience is equivalent

Empirical Equivalence

Empirical equivalence is a relatively weak form of theoretical equivalence. The understanding of the relationship between empirical and theoretical equivalence can also be said to vary from person to person.

We can start with a counterexample. Both Newton's theory of gravity and general relativity can describe the orbits of the planets in the solar system. They agreed on highly agreed orbital projections for planets far from the Sun, including Earth, but the predictions for Mercury's orbit were not exactly the same.

That is, the two theories diverged in terms of empirical predictions, and they could be distinguished from each other by observation or experiment. In other words, these theories are empirically unequal, and they cannot be said to provide an equivalent description of the world in any sense. Empirical equivalence is therefore often considered the minimum necessary condition for theoretical equivalence.

There are also philosophers of science who argue that empirical equivalence is a sufficient condition for theoretical equivalence. The fact that two theories are equivalent means precisely that they can be applied in the same case and always bring the same prediction.

But the problem is that determining that two distinct theories are empirically equivalent or unequal is often a very delicate one. Moreover, the fact that merely two theories give very different theoretical descriptions of a situation does not in itself mean that they are empirically unequal, since there may be some systematic transition between different descriptions.

A well-known example is that de Broglie-Bohm guided wave theory (also known as Bohm mechanics) is widely considered to be empirically equivalent to the standard von Neumann-Dirac formulation of quantum theory, but they actually differ so much in various characteristics that few would argue that the two are equivalence theories.

In Bohm mechanics, for example, particles always have precise and definite positions, and every measurement one makes in theory is ultimately understood as a measurement of position, whereas in the von Neumann-Dirac formulation, particles never have a definite position, and in general, one can measure any number of different quantities.

Define equivalence

Definitional Equivalence

The weak equivalence of experience has led many philosophers of science to realize that we need a more nuanced concept of equivalence.

In the 1970s, philosophers of science such as Clark Glymour proposed a stronger concept of equivalence, which is defining equivalence. They argue that if theories are interchangeable, then they should be considered equivalent. Or it can be simply understood as translating a book, although it is expressed in different languages, but the content is essentially the same.

Glymore had proposed a concept of "freedom" for defining equivalence, and he did not define equivalence precisely, but thought that it should imply the fact that for two theories, if any model of one theory could be uniquely transformed into a model of the other, the two theories would be equivalent.

He gave an example to illustrate this idea. Newtonian gravity can be standardly expressed as a theory: an object accelerates in the presence of a gravitational field, which in turn depends on the distribution of matter.

But there is another formulation of theory, which philosophers are interested in because of some qualitative similarities with general relativity, in which gravity is "geometrically" derived. In this theory, there is no gravitational field, and objects do not accelerate due to gravitational force, but instead follow the trajectory of geodesics in curved space-time, whose curvature is proportional to the mass distribution.

In Grimmall's definition, the two theories are not equivalent because the transitions between them are not unique. In this example, the definition of equivalence is a necessary condition for theoretical equivalence, but it still cannot provide sufficient conditions for theoretical equivalence for the time being.

Categories are equivalent

Categorical Equivalence

The disadvantages of defining equivalence are obvious, it is either inadequate or has limited usefulness in practice. Therefore, in the 21st century, the academic community has proposed a new equivalence standard between physical theories, which adopts the method from category theory, so it is called category equivalence.

Category theory usually deals with mathematical concepts in an abstract way, turning them into groups of objects and state projections. If we have two theories, each theory is represented as a category whose object is the theoretical model, the state emission retains the structure of the model in some appropriate sense. If there is an equivalence relationship between these categories, we can say that the two theories are categorically equivalent.

The key to this concept is that many of our physical theories can actually be described as sets of certain mathematical structures. Because of this inextricable property between mathematics and physics, it contains a chain of logic: if category equivalence is a valid concept of equivalence between mathematical theories, then it can indicate that the mathematical structures used in those theories are equivalent. In addition, if the theory of physics is empirically equivalent in the above sense, then it can be obtained that the theory of the equivalence of the two categories actually uses equivalent mathematics to capture the same empirical laws.

Category equivalence can be easily applied to real cases and provides intuitive, credible judgments, for example, clarifying the equivalence of Hamiltonian and Lagrange mechanics.

Explain the equivalence

Interpretational Equivalence

As the philosophy of science developed, many people began to realize that the formal equivalence standards mentioned earlier might not work.

Whether two theories are equivalent, or whether they are different presentations of a fundamental theory, is more importantly how we interpret them. This is not a question of theoretical formulation, but of our intentions and interpretations.

Interpretation equivalence can be simply understood as the idea that two theories are equivalent as long as they have the same explanation.

Some philosophers argue that whether the two theories are equivalent depends on whether they can simulate the same "target system" and whether the provided model guarantees the same claims and inferences. When two theories lead to the same claims and inferences in the same case, they should be equivalent.

Duality

Duality

In addition to the much-talked about concept of "equivalence," there has been another parallel thread of discussion over the past few decades about the meaning and meaning of those "different but (to some extent) equivalent" theories, known as duality.

That is to say, one may have two theories, each of which can be expressed separately, but there is some correlation between the two.

At present, in the context of the popularity of string theory, which hopes to unify the Standard Model of gravity and particle physics, the description of the duality of the same physical situation has become particularly important. For example, the most important achievement in string theory, the AdS/CFT duality, is that it links two very different theories. It tells us that there is duality between gravity theory in five dimensions and quantum field theory in four dimensions.

#创作团队:

Compile: M ka

Typography: Wenwen

#参考来源:

https://arxiv.org/pdf/1810.08192.pdf

#图片来源:

Cover source: Dee Ashley via Flickr under CC BY-NC

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