Schema (Kant)

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In Kantian philosophy, a transcendental schema (plural: schemata; from Greek: σχῆμα, "form, shape, figure") is the procedural rule by which a category or pure, non-empirical concept is associated with a sense impression. A private, subjective intuition is thereby discursively thought to be a representation of an external object. Transcendental schemata are supposedly produced by the imagination in relation to time.

Role in Kant's architectonic system

Kant created an architectonic system in which there is a progression of phases from the most formal to the most empirical: "Kant develops his system of corporeal nature in the following way. He starts in the Critique with the most formal act of human cognition, called by him the transcendental unity of apperception, and its various aspects, called the logical functions of judgment. He then proceeds to the pure categories of the understanding, and then to the schematized categories, and finally to the transcendental principles of nature in general." It is within this system that the transcendental schemata are supposed to serve a crucial purpose. Many interpreters of Kant have emphasized the importance of the schematism.

Purpose of the schematism chapter

If pure concepts of the understanding (Kantian Categories) and sense perceptions are radically different from each other, what common quality allows them to relate? Kant wrote the chapter on Schematism in his Critique of Pure Reason to solve the problem of "...how we can ensure that categories have 'sense and significance.' "

A posteriori concepts have sense when they are derived from a mental image that is based on experienced sense impressions. Kant's a priori concepts, on the other hand, are alleged to have sense when they are derived from a non–experienced mental schema, trace, outline, sketch, monogram, or minimal image. This is similar to a Euclidean geometrical diagram.

Whenever two things are totally different from each other, yet must interact, there must be some common characteristic that they share in order to somehow relate to one another. Kantian Categories, or a priori concepts, have, according to Kant, a basic and necessary importance for human knowledge, even though they are totally different from sensations. However, they must be connected in some way with sensed experience because "… an a priori concept which cannot, as it were, establish any empirical connections is a fraud … the purpose of the Schematism chapter was to show that the categories at least do have satisfactory empirical connections." Kant was preoccupied "with bridging the otherwise heterogeneous poles of 'thought' and 'sensation' in the Schematism of the Pure Concepts of the Understanding (A 138/B 177)."

Explication of the Kantian account of schemata

Three types of concept and their schemata

There are three types of concept that require a schema in order to connect them to phenomenal sense perceptions so that they have sense [Sinn] and meaning [Bedeutung]. These three types are (1) empirical concepts, (2) pure (mathematical) sensuous concepts, and (3) pure concepts of the understanding, or [Kantian] Categories. The first two employ schemata. The third employs transcendental schemata.

Empirical concepts

An empirical concept is the abstract thought of that which is common to several perceptions. When an empirical concept is said to contain an object, whatever is thought in the concept must be intuited in the mental representation of the object. Examples of intuitive perceptions that are the content of empirical concepts are vague images that are imagined in order to connect a concept with the perceptions from which it was derived as their common feature. "Intuitions," Kant wrote, "are always required to verify or demonstrate the reality of our concepts." These examples ensure that "our abstract thinking has not strayed far from the safe ground of perception, and has possibly become somewhat high-flown or even a mere idle display of words." This is because "concepts are quite impossible, and are utterly without meaning or signification, unless an object is given for the concepts themselves, or at least for the elements of which they consist." For example, "The concept of a dog signifies a rule according to which my imagination can trace, delineate, or draw a general outline, figure, or shape of a four-footed animal without being restricted to any single and particular shape supplied by experience." In order to prevent the emptiness of "thoughts without contents," it is "necessary to make our concepts sensible, i.e., to add an object of intuition to them." In order to test whether a concept is sensible, we sometimes " … go back to perception only tentatively and for the moment, by calling up in imagination a perception corresponding to the concept that occupies us at the moment, a perception that can never be quite adequate to the (general) concept, but is a mere representative of it for the time being. … Kant calls a fleeting phantasm of this kind a schema."

Pure sensuous (mathematical) concepts

These are concepts that relate, prior to experience, to the external sense of space and the internal sense of time. As such, they are mathematical in that they refer to geometry and arithmetic. A pure, sensuous concept is the construction or mental drawing of what is common to several geometrical figures. These mathematical concepts are not based on objective visual images. They are based on schemata that exist only in thought. Any particular image could not be as general as the concept. The schemata are rules that allow the imagination to mentally construct or draw or trace a pure, general geometrical form that gives the pure, sensuous concept significance. "… [T]o possess the schema corresponding to the concept triangle is to be able to envisage the variety of things to which the word "triangle" applies." "[T]he schema of sensuous concepts (such as of figures in space) is a product and, as it were, a monogram of the pure imagination a priori. Images become possible only through the schema. But the images must always be connected with the concept only by means of the designated schema. Otherwise, the images can never be fully congruent to the general concept."

Pure concepts of the understanding (Categories)

A pure concept of the understanding, or category, is a characteristic, predicate, attribute, quality, or property of any possible object, that is, an object in general or as such. These concepts are not abstractions of what is common to several perceived, particular, individual objects, as are empirical concepts. "Since the categories are a priori and are therefore not abstractions from sense perceptions, they owe their origin to the very nature of the mind itself." They are not derived from perceptions of external objects, as are empirical concepts. Instead, they are the result of the way that the mind is constituted or formed. They come from within the mind, not from outside of the mind.

Kant claimed that the schemata of pure, non-empirical concepts, or categories, provide a reference to intuition in a way similar to the manner of empirical concepts. "If the concepts are empirical, the intuitions are called examples; if they are pure concepts of the understanding, the intuitions are called schemata." In the same way that examples provide signification for empirical concepts, schemata help to answer the question of "whether operating with the categories is anything other than playing with words."

Since the pure concepts of the understanding, or categories, are characteristics of all objects in general, they can never be associated with the image of any specific, particular, individual object. "Since they are pure, they cannot be pictures..." "Yet there must be some connection between the abstract idea and the experienced world to which the idea is expected to apply..." "In order for the pure categories to have objective validity (and not merely subjective validity) they must be related to sensibility."

Applying pure concepts to sense impressions

The categories, or pure concepts of the understanding, are a priori logical innate forms that are conditions of the possibility of things in general, or of things as such. A thing can become a known object of thought when an a posteriori sense impression is comprehended through the forms of the categories. Categories and sense impressions are totally different from each other. Categories are utterly heterogeneous with the perceptions that are experienced through the sense organs. In order for specific phenomena of Nature to be thought from the combination of categories (pure concepts) and sense perceptions, there must be a third, mediating procedure that connects them. This mediator is a transcendental schema. Transcendental schemata meaningfully join the empty "thoughts without content" and the blind "intuitions without concepts."

Schemata that mediate between empirical (a posteriori) concepts or mathematical (pure sensuous) concepts and perceptions are similar to adapters. Just as adapters are devices for fitting together incompatible parts, schemata connect empirical concepts with the perceptions from which they were derived. Schemata are rules for the production of images. As rules, they are related to concepts. As image–producers, they are related to perceptions. "While the concept belongs to the understanding and its instance to perception, the schema has, so to speak, a foot in either domain. As rules for the production of images the schemata … are linked to the understanding; as rules for the production of images they are linked to perception." The "adapter" simile is even more apt in the case of transcendental schemata. This is because pure concepts of the understanding (Categories) are totally unrelated to perceptions. The pure concepts or Categories are original constituent components of the understanding and are not derived from empirical sense perceptions.

Transcendental schemata

Transcendental schema are not related to empirical concepts or to mathematical concepts. These schemata connect pure concepts of the understanding, or categories, to the phenomenal appearance of objects in general, that is, objects as such, or all objects.

Judgment

Transcendental Schematism results from the ability to make judgments. Judgment applies "the concepts of the understanding [the Categories] to phenomena." " … [T]he judgment … schematizes these concepts a priori and applies these schemata, without which no experiential judgment would be possible, to each empirical synthesis. … the transcendental schematism of judgment provides it [judgment] with a rule under which given empirical intuitions are to be subsumed." Kant defined the Greek word hypotyposis as a " … rendering perceptible to the senses, making sensual (Versinnlichung)." The usual definition is "example, pattern, outline, or sketch." If a hypotyposis is schematic, according to Kant, "...to a concept, which is comprehended by the understanding, the corresponding intuition is given a priori..." This is in opposition to a symbolic hypotyposis, like God, in which the concept can only be thought by Reason and to which no sensible intuition can be adequate. Schemata contain direct presentations of the concept. They make this presentation demonstratively, not by the use of analogies. Judgment, according to Kant, works mechanically with given appearances and brings them under concepts. It does this as a tool that is utilized and controlled by both the understanding and the senses. To avoid possible mistakes in judgments one must conduct the transcendental reflection.

Time

The schematism of the pure understanding is "the sensuous condition [time] under which alone pure concepts of the understanding [the Categories] can be used." Categories, or pure concepts of the understanding, are abstract representations of objects in general. However, they can result in thought about particular, specific internal or external objects if they are related to time. All concepts are derived from perceptions, therefore pure concepts [Categories] are based on pure perceptions. The purest perception, or schema, is time. Time has the purest relation to sensation that is possible. It is the mere form of sensation because sensations must be felt in succession. Therefore, time was designated by Kant as the purest possible schema of a pure concept.

All things are experienced in time, that is, in succession, one after another. This applies to our internal selves as well as to all external objects. Since categories are the forms through which every specific thing can be thought as being in time, categories are related to time. Thus, pure concepts, or categories, and phenomenal objects share time as a common feature. Therefore, time is the means by which an intuited phenomenon is subsumed under a pure concept. Schemata are transcendental time determinations. "Hence it will be possible for the category to be applied to appearances by means of the transcendental time determination, which, as the schema of the concepts of the understanding, mediates the subsumption of appearances under the category."

Schemata are procedural rules, not images

Because schemata are determinations of objects in general, not specific, individual objects, they are not particular images. Kant asserted that "… a schema must be distinguished from an image." A schema is a procedural rule. The rule prescribes the way to relate a pure concept to an object in general. Schemata are ways of applying pure concepts (categories) to sense impressions. They are prescriptions for graphically illustrating a pure concept. A schema is a method for representing a non-empirical concept in any image as such or any image in general. " … [F]or Kant a schema is not an image, but a capacity to form images or (perhaps) to construct models." "The schema of a pure concept of the understanding is something which can never be made into an image..."

Lewis White Beck wrote, "The pure concepts of the understanding…are applied to the pure form of intuition (time) to give rise to the 'transcendental schemata' or rules of the application of pure concepts to whatever sense–impressions we receive." To exemplify this, he continued, "To take the most important example, we have the hypothetical (if–then) judgment, which is the mode of judgment under the category of cause. This is applied to phenomena in time by the schema of causation, namely, the rule that the cause of a phenomenon is another phenomenon that invariably precedes it in time…." In this way, Beck showed that a transcendental schema is the rule that leads to the understanding of successive [in time] sensations according to various pure concepts [Kant's "categories"].

Examples of different categories and their schemata

Each category has one schema. Some schemas are shared by other categories in their class.

• The categories of quantity all share the schema of number. Quantity is related to a numerical time series. William H.S. Monck claimed that Kant can be understood as saying "In apprehending an object I always successively add part to part, and thus generate a series of determinate magnitude." Unity is one moment in time. Plurality is several moments. Totality is expressed as all moments. But Monck noted that "Kant does not trace out specially the Schemata of the Categories of Unity, Plurality, and Totality." Monck supposed: "In the Category of Unity I presume we should stop at the first term of the Time–series: for the Category of Plurality we should represent the addition of unit to unit without laying down any determinate limit; and for the Category of Totality we should limit the number of units and complete the addition up to this number."
• The categories of quality all have degrees of reality as their schema. Quality is related to the content of real being in time. Kant metaphorically explained the schema of quality as the filling up of time with a sensation. Monck paraphrased Kant as saying, "We may speak of time as being more completely filled when the sensation is more vivid, and less completely filled when the sensation is weaker. If the sensation is sufficiently vivid the present moment is so filled with it that it seems impossible to attend to, or even be conscious of any other sensation simultaneously; but when it becomes weaker we have leisure to perceive other sensations also. This occurs by degrees. "We can represent this filling as taking place by a gradual increase from zero (empty time) to any given degree of vividness, or by a similar descent from the sensation of the moment to zero." Reality is the experience of sensation in time. Negation is the absence of sensation in time. Limitation is the range of degrees, between the transition from full to empty, by which things are sensed in time. But Monck stated that "Again Kant does not here give in detail the Schemata of the Categories of Reality, Negation, and Limitation." He qualifiedly suggested that "In the first case [Reality], we should, perhaps, represent the sensation as occupying the present moment to the exclusion of everything else; in the second [Negation] the sensation as entirely absent from the present moment; in the third [Limitation] the sensation as occupying the present moment along with others.
• "The Schema of the Category of Relation is the Order of Time." In the class of relation, each category has its own schema. Substance's schema is the permanence of the unchanging substance (subject) to which accidents (predicates) belong, or the permanence of the object in time. Causality's schema is the necessary succession of a consequent to an antecedent. That is, "… the Schema of the Category of Cause is its regular Antecedence in Time (i.e. Succession in Time determined by fixed law)." The category of community has the schema of the necessary coexistence of the accidents of one substance with the accidents of another substance. This communal interaction consists of the changing accidents of one substance having their cause in the changing accidents of another substance, and vice versa. This can be understood as "… the Simultaneity of objects in Time."
• "The Schema of the Category of Modality according to Kant is Time itself as related to the Existence of the object." In the class of modality, the category of possibility has the schema of possibility at any time. The schema of actual existence at a certain time belongs to the category of existence. Finally, the category of necessity has the schema of being an object at all times.

Even though Kant provided these illustrations and examples of schemata, author John Mahaffy claimed that the topic remained obscure. He wrote, "I may add, that these illustrations of the various schemata are developed and explained by the succeeding chapters on the Principles which embody them, and that it is impossible to make them clear to the reader until he has studied the theory of the Principles."

Schematized and unschematized categories

The schemata give the categories a "cash value", as though the category is like paper money and sense experience is analogous to precious metal. A schema is the agreement or harmony of a category with sensual phenomena. For example, "Number is the quantity of the phenomenon; sensation is the reality of the phenomenon; the permanence and endurance of things is the substance of the phenomenon, eternity is the necessity of the phenomenon, etc." In this way, the schemata restrict the categories to conditions of sensibility. "Schematism and the schemas thus have the property of 'realizing' the categories at the same time as restricting their scope to appearances." Categories cannot be realized in objects that are not detectable by the senses, that is, are not phenomenal objects (objects that appear to an observer).

"The schemata of the pure concepts of the understanding are, therefore, the true and sole conditions for providing these concepts with a reference to objects and hence with signification. And therefore the categories have, in the end, no other use than a possible empirical one." In order for categories to refer to perceived, experienced objects, they must be schematized. If a category is not schematized, then it has no reference to perception. An unschematized category can be thought, but can not be known. If something can never be perceived, it can never be known. Schemata represent things in general as they appear, not as they might otherwise exist. "Categories, therefore, without schemata are only functions of the understanding necessary for concepts, but do not themselves represent any object." This act results in the formation of one abstract concept from various perceptions or other concepts. With the transcendental determination of time as the transcendental schema, " … use of the categories is clearly restricted to the range of things that fall within time — meaning, for Kant, restricted to phenomena." Metaphysical entities that are not related to time, such as spontaneous or uncaused movements, immortal souls, and eternal gods, are products of unschematized categories. They can be thought, but not known.

Unschematized concepts

Two kinds of concepts do not use schemata in order to exhibit their empirical bases: rational concepts (ideas of reason), and Platonic Ideas. With these concepts or ideas, there is no intermediate nexus between abstract entity and concrete sense perception. These unschematized concepts do not contain or subsume a representation of an object that is the basis of a concept.

Kant listed three rational concepts or ideas of reason: God, freedom, and immortality. They are not found in experience. “…[T]he difficulty about ideas of reason is that ‘absolutely no intuition commensurate with these can be given’ (Critique of Judgment, § 59).”  “To make sense of such ideas we must accordingly have recourse to an alternative procedure, in which use is made of ‘symbols’ as opposed to schemata proper. What happens here is…that we find some empirically intuitable situation which can serve as a model by reference to which the idea can be made comprehensible.” For example, Kant tried to show how sense can be made of the idea of an unseen God “by making a [loving or angry] father’s relationship to his children the [metaphorical, figurative] symbol of God’s relationship to the world.” The rational idea of a non-deterministic occurrence can be spoken of in reference to the analogies of the universe as a mechanism [such as a clock] or organism. Thus, while pure concepts of understanding, empirical, and pure sensible concepts are directly exhibited and made sensible through schemata, ideas of reason are indirectly exhibited through relations by the use of symbolic analogy.

Professor W. H. Walsh attempted to explain how schemata are used to make sense of, or exhibit, the twelve Kantian categories (pure concepts of the understanding) while symbolic analogous relationships are used for the three rational concepts (ideas of reason). "It may be useful in this connection to compare what Kant says about the schematization, or quasi-schematization, of ideas of reason with what he says about the schematization of categories. The problem in the two cases is in essentials identical: how to make a concrete use of concepts which are by nature remote from sense. To show that such a use is possible we need, in Kant’s technical terminology, to find intuitions corresponding to them. In the case of pure concepts of the understanding this can be done, since we can point to the appropriate schemata; the difficulty about ideas of reason is just that 'absolutely no intuition commensurate with them can be given' (Critique of Judgment, §59). To make sense of such ideas [of reason] we must accordingly have recourse to an alternative procedure, in which use is made of ‘symbols’ as opposed to schemata proper. What happens here is, roughly, that we find some empirically intuitable situation which can serve as a model by reference to which the idea [of reason] can be made comprehensible. The idea of God, for example, is incapable of being schematized, but we can nevertheless make partial sense of it for certain purposes by making a father’s relationship to his children the symbol of God’s relationship to the world....What is noteworthy in this discussion…is…the sharp contrast drawn between schema and symbol, and hence between the meaningfulness of categories and that of ideas [of reason]….every concept capable of schematization…seems to allow that the relationship between idea [of reason] and symbol is altogether less intimate: symbolizing is a relatively arbitrary process, and hence each idea [of reason] can be symbolized in a variety of ways." A category has one transcendental schema; an idea of reason can have multiple symbols for its analogy.

Plato’s Ideas (also known as notions, forms, paradigms, or archetypes) are concepts that function as patterns or models. They are related to objects in the experienced world Ideas are related to natural objects or their perceptual representations by the processes of participating, partaking, and copying. “The particular objects which we perceive are imperfect copies or reflections of the eternal patterns.”

Link to the Unconscious

In von Hartmann’s Philosophy of the Unconscious he declared that Kantian Transcendental Schemata connect unconscious Categories to conscious knowledge. Kant’s Pure Concepts of the Understanding, or Categories, are unconscious representations or ideas that lie beyond knowledge. According to von Hartmann, these unconscious Categories produce conscious knowledge through the mediation of the Schemata of the Pure Understanding.

Alternative schemata

Kant said that the schema of a concept is the representation of a general procedure of the imagination by which an image can be supplied for a concept. Kant claimed that time is the only proper and appropriate transcendental schema because it shares the a priori category's generality and purity as well as any a posteriori phenomenon's manner of appearance. However, it may be true that time is not the only possible schema.

Space

"Even more remarkable, however, is the fact that in order to understand the possibility of things as consequent upon the categories, and hence in order to establish the categories' objective reality, we need not merely intuitions but indeed always outer intuitions." Since space is the form of all appearances of the outer senses, it may seem that space could serve as a schema. Indeed, any phenomenon that requires space, as well as time, as a form would also need a spatial schema. "This suggests that he may have thought at one point of recasting the Schematism argument in a fundamental way, by substituting space for time; but if he had this idea, he did not carry it out." In the editor's introduction to his translation of the Critique, Paul Guyer asserted that "…although the content of the transcendental schemata for the categories may be explicated in purely temporal terms, the use of these schemata in turn depends upon judgments about the spatial properties and relations of at least some objects of empirical judgment." Guyer claimed that this declaration was clarified in Kant's "The System of All Principles" section. In this way, the use of schemata is supposed to involve both space and time, instead of merely time.

Norman Kemp Smith claimed that there is apparently no good reason why Kant did not consider space to also be a transcendental schema for his Categories. Kemp Smith argued that consciousness of space is as fundamental as that of time. He concluded that "…Kant’s concentration on the temporal aspect of experience is exceedingly arbitrary…." and therefore without reason. Experience, Kant had claimed, is always spatio-temporal. Inner sense, however, may be exclusively temporal. Kemp Smith then guessed, in opposition to his earlier statement, why Kant ignored space as a transcendental schema. "Possibly Kant’s very natural preoccupation with his new revolutionary doctrine of inner sense and productive imagination has something to do with the matter [i.e., is the reason for his elimination of space]." Thus, Kant's doctrine of inner sense and its emphasis on time resulted in his exclusion of space as a transcendental schema.

A. C. Ewing claimed that the reason that Kant did not use space as a transcendental schema was because space is not needed in order to understand the relational concepts of substance or causality. According to Professor Ewing, Kant did intend to use space as a transcendental schematic mediator between a category and a sensible intuition. Kant, according to Professor Ewing, for one category did use space as a transcendental schema to unify a pure concept with a sensible intuition. Even though space is not needed in order to understand the relations of substance and causality, it is required in order to understand the conceptual relation of community or reciprocity.

Werner Pluhar explained why time, rather than space, is used as a connection to include (subsume) sensory intuition in the Kantian categories or pure concepts of the understanding. “…[A]ll [transcendental] schemata connect the categories with time; the reason for this is that time is the only form of intuition that applies to any intuition whatsoever, even to the inner intuition we have of ourselves, whereas space applies merely to all outer intuitions.” (Immanuel Kant, Critique of Judgment, translated by Werner S. Pluhar, Translator’s Introduction, p. xxxvi, Indianapolis: Hackett Publishing Co., 1987)

Organism

In order to show how time may not be the only schema, Professor Walsh suggested that there is "… the possibility of making sense of the categories in organic as opposed to mechanical terms." He hypothesized that "Elements in an organic complex would here take the place of elements in a temporal situation. Substance might be interpreted in terms of growth and form as opposed to what underlies mechanical change, and causality be thought of in terms of purpose and function." However, Professor Walsh concluded that Kant's choice of time as schema was more precise than any alternative choices. In spite of the general difficulty in understanding Schematism, he asserted that "… Kant's doctrine of schematism, if not altogether satisfactory at the theoretical level, will continue to stand on the strong empirical ground that the schemata offered do enable us to give real meaning to the categories and find for them a genuine use."

Schemata of systematic unity

In his discussion of the Architectonic of Pure Reason, Kant utilized the concept of schema in a way that was similar to his discussion of the schemata of the Categories. A science's whole systematic organization consists of parts. The parts are various cognitions or units of knowledge. The parts are united under one idea which determines the relation of the parts to each other and also the purpose of the whole system. A schema is needed to execute, carry out, or realize this unifying idea and put it into effect. This schema is a sketch or outline of the way that the parts of knowledge are organized into a whole system of science. A schema which is sketched, designed, or drafted in accordance with accidental, empirical purposes results in mere technical unity. But a schema that is drawn up from an a priori rational idea is the foundational outline of architectonic unity. Science must have architectonic unity. "For the schema of what we call science must contain the whole's outline (monogramma) and the whole's division into parts in conformity with the idea — i.e., it must contain these a priori — and must distinguish this whole from all others with certainty and according to principles." This use of the concept of schema is similar to Kant's previous use. It is a minimal outline, monogram, or diagram that realizes or executes an abstract, general concept or idea (Idee) as actual, perceptual experience.

Criticism

Obscurity of the concept "schema"

Kant introduced the concept of the transcendental schema in his chapter entitled "Of the Schematism of the Pure Concepts of the Understanding." It is considered to be one of Kant's more difficult chapters. Even though he knew that he was not writing for a popular readership, Kant twice tried to apologize for this chapter by calling it "very dry" and "dry and tedious." Kant entered into his 1797 notebook: "In general, the schematism is one of the most difficult points. – Even Herr Beck cannot find his way about in it." Professor W.H. Walsh, of the University of Edinburgh, wrote: "The chapter on Schematism probably presents more difficulties to the uncommitted but sympathetic reader than any other part of the Critique of Pure Reason. Not only are the details of the argument highly obscure (that, after all, is a common enough experience in reading Kant, though one is not often so baffled as one is here): it is hard to say in plain terms what general point or points Kant is seeking to establish." Arthur Schopenhauer referred to it as "…the strange 'Chapter on the Schematism of the Pure Concepts of the Understanding,' which is well known for its great obscurity, since no one has ever been able to make anything out of it." Schopenhauer's notebooks contained entries that described Kant's chapter on schemata as "an audacious piece of nonsense" and the schema as "an absurdity whose non–existence is plain." Schopenhauer also remarked on "...the futility of such intermediate things between intuitive perception and concepts." In Schopenhauer's criticism of Kant's schemata, he attempted to clear up the obscurity by attributing Kant's concept of schemata simply to a psychological need for architectonic symmetry in his writings. Empirical concepts are based on empirical perceptions. Kant, however, tried to claim that, analogously, pure concepts (Categories) also have a basis. But this contradicts his previous assertion that pure concepts simply exist in the human mind and are not based on pure, schematic perceptions. Schopenhauer also alleged that schemata were introduced merely to give plausibility to Kant's description of the categories or pure concepts of the understanding. The article on Kant in the Encyclopedia of Philosophy calls Kant's schematism a "baffling doctrine " with "cryptic sentences." Josiah Royce referred to "the perplexing doctrine of the Schema." The Scottish philosopher Robert Adamson wrote: "Kant's manner of explaining the functions of schematism is extremely apt to be misunderstood, and to mislead." Kant's early critics (1782 –1789) did not discuss schematism because they couldn't follow Kant's explanation. Heidegger wrote of "the dryness and tediousness of this analysis…." After more than two centuries, Kant's explanation of schema still seems to be unclear to many readers. In their book on parallel distributed processing, the PDP Research Group discussed Kant's schemata when they appropriated that word to designate their concept of image schemata. "The schema," they wrote, "throughout history, has been a concept shrouded in mystery. Kant's … use of the term has been provocative but difficult to understand." After this sentence, no further attempt was made to discuss Kant's term and the concept that it designates. H. H. Price, in Thinking and Experience, page 292, referred to Kant's Schema and wrote, "…I must confess I do not fully understand it." In 2004, Professor Georges Dicker of SUNY Brockport stated: "I find the Schematism especially opaque…." Kant scholar Norman Kemp Smith judged  the schematism chapter to be a "highly artificial argument." Kant's explanation seems, to Kemp Smith, to be contrived and to have no natural progression. Hermann Weyl described his reaction to Kant: "While I had no trouble making this part of Kant’s teaching [regarding a priori space and a priori synthetic judgments] my own, I still had much trouble with the Schematism of Pure Mental Concepts…." In his English lectures, published in 1796, Kant's pupil Friedrich August Nitsch warned his listeners and readers against the difficulty in comprehending Kant's concept of the schema. He wrote: "It will require great efforts of abstraction in the reader to conceive the Schematism of the intellect, in a perfectly clear manner. In his essay "Kant’s Conception of the Categories," T. K. Seung called "…the 'Schematism,' perhaps the most oracular chapter of the Critique….". Professor Eva Schaper wrote that “Schematism…has posed problems for interpreters, and many have wondered whether Kant’s thought had fully matured at the time he wrote it.”

According to Kant, a transcendental schema is a mediating nexus, a third thing (tertium quid; ein Drittes), between a pure concept and a phenomenon. This mediation was never satisfactorily explained by Kant, and Charles Sanders Peirce declared that it is a major part of Kant's system. Kant's "doctrine of the schemata can only have been an afterthought…," Peirce wrote. The theory of mediating schemata was "an addition to his system after it was substantially complete." The enormous importance of the concept of the transcendental schema was emphasized by Peirce when he wrote that "if the schemata had been considered early enough, they would have overgrown his whole work."

Discrepancies

According to Professor W. H. Walsh, there is an apparent discrepancy in Kant's central arguments about schematism. Kant, according to Professor Walsh, first claimed that empirical concepts do not require schemata. Only pure concepts need schemata in order to be realized. This is because pure concepts are totally different from intuitions, whereas, empirical concepts are abstracted from intuitions and are therefore homogeneous with them. But in another part of his chapter, Kant states that mathematical concepts have schemata. "In fact," he wrote, "it is schemata, not images of objects, that lie at the basis of our pure sensible (i.e., geometrical) concepts." In discussing schematism as the method of representing in one image a certain mathematical quantity according to a certain concept, he wrote: "This representation of a general procedure of the imagination by which a concept receives its image, I call the schema of such concept." With regard to pure concepts, Kant then declares, "The schema of a pure concept of the understanding, on the contrary, is something which can never be made into an image … ."

Kant, according to Professor Walsh, has two distinct ways of describing schemata. "Sometimes, as at the beginning of his discussion, he speaks as if a schema were a feature of things which could be pointed to … ". In another place, Kant " … speaks as if schematism were a procedure … ."

Reconciling Irreconcilables

Problematic mediation

The perennial mind-body problem examines the relation between cognitions that are inside of a knower’s brain and objects that appear to be outside of that brain. This dualism is reflected in many philosophical dichotomies. These dichotomies consist of two opposing parts that are totally heterogeneous to each other. Kant was concerned with the problem of uniting these opposites for much of his middle and later life. Descartes incorrectly claimed that the dichotomy of soul and body is unified when the brain’s pineal gland is psychokinetically moved by the brain’s thoughts. Salomon Maimon likened Kant’s Transcendental Schemata to Descartes’ pineal gland. Kant’s concept of Transcendental Schema is understood as being a variation of the enduring mind-body problem because his Transcendental Schema subsumes a sensuous intuition under a pure concept, combining conceptual understanding [mind] with perceptual sensation [body].

Kant’s Three Mediators

In each of his three critiques, Kant described the inclusion of a particular, sensible, concrete intuition in a universal, super-sensible, abstract concept. This inclusion or subsumption occurs, according to Kant, by means of a mediator.

"In the first Critique, Kant introduces the [transcendental] schema by arguing that it is needed to mediate between the pure concepts of the understanding and imagination (intuition). In the second Critique Kant similarly introduces the typus as needed to mediate between reason’s moral law and understanding….In the Critique of Judgment,… Kant justifies his treatment of judgment as…a cognitive power in its own right partly by showing how it mediates between the other two higher cognitive powers, understanding and reason…."

The three mediators (transcendental schema, typus, and judgment) have nearly equivalent functions. Kant presented the typus mediator (in the second critique) and judgment (as mediator in the third critique) as having nearly the same properties as the transcendental schema mediator (in the first critique).

"The model of the mediating item developed in the Schematism has five features….The mediator must be 1) a third thing, 2) homogeneous with both [heterogeneous] parties, but 3) independent of both as well. It must also be 4) analogical, but, in the case of non-schematic mediation, 5) it must remain at the level of analogy--it must not mediate substantively or materially between the two parties, unlike schematic mediation…."

Schopenhauer criticized Kant’s use of the first Critique’s organization in Kant’s construction of the second and third Critiques."The [first critique’s] table of categories is now supposed to be the guiding line along which every metaphysical, and in fact every scientific, speculation is to be conducted (Prolegomena, § 39). In fact, it is not only the foundation of the whole Kantian philosophy, and the type according to which its symmetry is carried through everywhere…but it has also really become the Procrustean bed on to which Kant forces every possible consideration…."

The second Critique’s typus matches the first Critique’s transcendental schema in obscurity. "[E]ven a reader as acute and sophisticated as Jacob Sigismund Beck wrote to Kant in [3 July] 1792 to express his puzzlement over the Typic."

Judgment as Mediator

Judgment, according to Professor Pluhar, mediates between understanding and reason in the following way: "…in a syllogism the power of judgment subsumes the particular under some universal (i.e., under some principle) supplied by understanding and thereby enables reason to make an inference from that universal to the particular." Kant hypothesized that judgment is "…the ability [capacity] to subsume the particular under the universal…." In this way, the two discrete, heterogeneous cognitions, viz., abstract universal and concrete particular, are linked together by means of the mediating judgment. Kant called this mediate linkage "subsumption" [Subsumtion], a verb that denotes the inclusion of a part into a whole.

A Third Thing (ein Drittes)

Kant taught that a Transcendental Schema is a third thing that exists between a perceived phenomenon or appearance and a conceived Category (pure concept of the Understanding). Through the mediation of time, as a third, shared thing, a pure Category that is merely thought can be applied to a phenomenon that is experienced as a sense perception, or, in other words, a phenomenon can be subsumed under a Category.

Previous philosophers had analyzed similar applications or subsumptions while they were investigating heterogeneous relations. They had claimed that an attempt to relate disparate things results in infinite degrees of relation. For example, Plato wrote about an endless series or infinite regress of Forms (i.e., Ideals, Ideas, Paradigms); Aristotle deliberated upon the Third Man Argument.

In Plato’s "Parmenides" dialogue he had Socrates discuss the relationship between a Form and a thing that has a share in the Form. In order for them to relate to each other they must be like each other in some way; they must share some common property. But, in order to relate to the common property, they must share a further common property, and so on to infinity.

Aristotle commented on the relation between Plato’s Universal Form of Man and the phenomenal appearance of an individual physical man.

Aristotle’s claim is that, if [the Platonic Form of] Man Itself is separated from the individual men, then, since the individuals and [the Platonic Form] of Man Itself are both men, a "third" man will be predicable (and thus separate from them) of both -- a "fourth" of those three, a "fifth" of the resulting four, and so on.

The problem of the infinite regress of mediated relations is by-passed if Kant’s use of the "third thing" analogy is understood as being inappropriate. According to Norman Kemp Smith, the Transcendental Schema is not a third thing between a sensed appearance and a thought Category. "[The] relation holding between categories and the material of sense is that of form and matter…," not "…between a class concept and the particulars which can be subsumed under it." "…[T]he true Critical teaching is that category and intuition, that is to say, form and content mutually condition one another, and that the so-called schema is simply a name for the latter [intuition or content] as apprehended in terms of the former [category or form].

Adamson's interpretation

Scottish philosopher Robert Adamson wrote from a Hegelian standpoint. He believed that Kant's analysis of knowledge into the separate topics of intuition, schema, and concept was mechanical and artificial. Adamson claimed that "Thought and Intuition are organically united in the schema." "We are not to suppose that the subsumption [of the intuition under the pure notion] is mechanical; that the particular is something distinct from the universal. The union is organic; the particular is only the universal under a special form. The same function of synthesis, which in pure abstraction we call category, is, in realization, the schema, and the intuition is not apart from the schema." Kant's abstract analysis of perceptual knowledge was, according to Adamson, the misleading separation of an organic unity into individual components. He asserted that "… we must on no account regard Notion, Schema, and Intuition, as three parts of perception which would exist in isolation." This amalgamation is typical of the Hegelian "dialectical" formula in which two apparent opposites are always subsumed or reconciled by some third entity.

Pluhar's interpretation

In the translator's introduction to his version of Kant's Critique of Judgment, Werner Pluhar tried to explain schemata. He noted that perceptual intuitions and Kant's conceptual categories are very different, yet they relate to each other. This exposition by Professor Pluhar paraphrases Kant's doctrine that perceptions are based on concepts. Kant's position can be contrasted with Schopenhauer's opposite teaching that concepts are derived or abstracted from perceptions, thereby giving content to the concepts and allowing them to make sense. This is the very reason why pure concepts, or categories, require schemata. "Something is needed," Pluhar wrote, sharing Kant's viewpoint, "to mediate between intuition in general and the categories, viz., a rule or 'schema' that stipulates what conditions the intuition must meet so that it can match a category." Professor Pluhar then gave a specific example of how time is utilized to accomplish the matching or mediation. His explanation does not resort to presenting schemata through the use of visual analogies such as "sketches" or "outlines." Pluhar's schemata are rules. "In the case of causal relation, the schema is the rule that the effect must follow the cause in time." After providing this particular instance, he declared generally that "…all schemata connect the categories with time…." Professor Pluhar then asserted the reason for this schematic connection: "…time is the only form of intuition that applies to any intuition whatsoever, even to the inner intuition that we have of ourselves, whereas space applies merely to all outer intuitions." Oddly enough, schemata do not have to be added as mediators to the categories of causality and substance. These are already temporalized. Time is intrinsic to the relation between cause and effect. Substance, by its very nature, is a thing that continually endures.

Watson on time and schematism

Canadian professor John Watson, in his discussion of Kantian philosophy, wrote about supposed supersensible, atemporal beings such as God or the soul. Such things are said to have a timeless existence. As such, though, they cannot be known or experienced. Watson asserted that "…whatever cannot be ‘schematized’...cannot be known...."  He considered Kantian schematization as "...conforming to the process by which the definite or concrete becomes a possible object in time...." Schematizing an object is representing an object in time. Accordingly, timeless "...supersensible realities...are not capable of being 'schematized,' do not admit of the application to them of the [Kantian] categories and can never become objects of actual sensible experience." "In the Critique of Pure Reason it has been maintained that no knowledge of supersensible realities can be obtained, since such knowledge always implies a process of determining [or schematizing] objects in time, whilst the supersensible is necessarily free from the limits of time." If supersensible objects cannot be schematized because they are not in time, then "…to the supersensible world…the schematized categories have no application….". If a supersensible thing, like God or the soul, is not in time, then it can't be schematized, can't be applied to Kantian categories, and therefore can't be a known object. Such supersensible entities would have to be schematized through the form of time if they were to be known as having sequentially countable magnitude, gradations of intensive reality, permanent substantiality, or successive causality.

Elaborations of Kant's notion of schema in cognitive science

The philosopher Mark Johnson discusses Kant's conception of a schema with respect to developing a theory of the imagination within cognitive science. Johnson's theory makes use of Kant's insights that analogy is the cognitive mechanism which links sensible percepts to their conceptual categories, and that creative analogy—or what Johnson calls conceptual metaphor—is the cognitive mechanism by which we come to have our understanding of those abstract concepts and categories of which we have less direct sensible experience. He proposes that we use imaginative schemata to structure abstract concepts largely in terms a set of spatial analogies he calls image schemata. In Johnson's view, we acquire image schemata primarily from recurrent patterns of experiences in infancy and early childhood, and then reuse these image schemata in a metaphoric fashion both to reason abstractly and as we speak our language.

In an increase of ambiguity and confusion, some cognitive scientists today have appropriated the often–misused technical term "schema" to mean Kantian Category. In his book Cognitive-Behavioral Therapy: Basic Principles and Applications (Jason Aronson Publishers,1996), Robert L. Leahy of the American Institute for Cognitive Therapy in New York City and the Weill Cornell Medical College of Cornell University exemplifies this misuse. In Chapter 2, "Historical Context of Cognitive Therapy," he wrote of how, for Kant, "reality is never directly knowable, but rather is 'known' through 'categories of thinking.'" Leahy then stated, "According to Kant, all knowledge was based on the 'categories' (which today we would call schemas). Consequently, reality was never directly knowable--we only knew the schemas." In this way, Kant's concept of "category," or "pure concept of the understanding," is no longer defined as being a predicate, property, quality, or characteristic of any and all objects in general. A Kantian Category is now vaguely considered by cognitive scientists to be a "schema," which was a term that Kant had already used to designate the subsumption of an empirical intuition, through time, under a category or pure concept.

Axion

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Axion

Axion

Interactions	Gravity, electromagnetic
Status	Hypothetical
Symbol	A0, a, θ
Theorized	1977, Peccei and Quinn
Mass	10−5 to 10−3 eV/c2
Electric charge	0
Spin	0

An axion (/ˈæksiɒn/) is a hypothetical elementary particle originally postulated by the Peccei–Quinn theory in 1977 to resolve the strong CP problem in quantum chromodynamics (QCD). If axions exist and have low mass within a specific range, they are of interest as a possible component of cold dark matter.

History

Strong CP problem

As shown by Gerard 't Hooft, strong interactions of the standard model, QCD, possess a non-trivial vacuum structure that in principle permits violation of the combined symmetries of charge conjugation and parity, collectively known as CP. Together with effects generated by weak interactions, the effective periodic strong CP-violating term, Θ, appears as a Standard Model input – its value is not predicted by the theory, but must be measured. However, large CP-violating interactions originating from QCD would induce a large electric dipole moment (EDM) for the neutron. Experimental constraints on the currently unobserved EDM implies CP violation from QCD must be extremely tiny and thus Θ must itself be extremely small. Since Θ could have any value between 0 and 2π, this presents a "naturalness" problem for the standard model. Why should this parameter find itself so close to zero? (Or, why should QCD find itself CP-preserving?) This question constitutes what is known as the strong CP problem.

Prediction

In 1977, Roberto Peccei and Helen Quinn postulated a more elegant solution to the strong CP problem, the Peccei–Quinn mechanism. The idea is to effectively promote Θ to a field. This is accomplished by adding a new global symmetry (called a Peccei–Quinn (PQ) symmetry) that becomes spontaneously broken. This results in a new particle, as shown independently by Frank Wilczek and Steven Weinberg, that fills the role of Θ, naturally relaxing the CP-violation parameter to zero. Wilczek named this new hypothesized particle the "axion" after a brand of laundry detergent because it "cleaned up" a problem, while Weinberg called it "the higglet." Weinberg later agreed to adopt Wilczek's name for the particle. Because it has a non-zero mass, the axion is a pseudo-Nambu–Goldstone boson.

Axion dark matter

QCD effects produce an effective periodic potential in which the axion field moves. The oscillations of the axion field about the minimum of the effective potential, the so-called misalignment mechanism, generate a cosmological population of cold axions with an abundance depending on the mass of the axion. With a mass above 5 µeV/c² (10⁻¹¹ times the electron mass) axions could account for dark matter, and thus be both a dark-matter candidate and a solution to the strong CP problem. If inflation occurs at a low scale and lasts sufficiently long, the axion mass can be as low as 1 peV/c².

There are two distinct scenarios in which the axion field begins its evolution, depending on the following two conditions:

(a)	The PQ symmetry is spontaneously broken during inflation. This condition is realized whenever the axion energy scale is larger than the Hubble rate at the end of inflation
(b)	The PQ symmetry is never restored after its spontaneous breaking occurs. This condition is realized whenever the axion energy scale is larger than the maximum temperature reached in the post-inflationary Universe.

Broadly speaking, one of the two possible scenarios outlined in the two following subsections occurs:

Pre-inflationary scenario

If both (a) and (b) are satisfied, cosmic inflation selects one patch of the Universe within which the spontaneous breaking of the PQ symmetry leads to a homogeneous value of the initial value of the axion field. In this "pre-inflationary" scenario, topological defects are inflated away and do not contribute to the axion energy density. However, other bounds that come from isocurvature modes severely constrain this scenario, which require a relatively low-energy scale of inflation to be viable.

Post-inflationary scenario

If at least one of the conditions (a) or (b) is violated, the axion field takes different values within patches that are initially out of causal contact, but that today populate the volume enclosed by our Hubble horizon. In this scenario, isocurvature fluctuations in the PQ field randomise the axion field, with no preferred value in the power spectrum.

The proper treatment in this scenario is to solve numerically the equation of motion of the PQ field in an expanding Universe, in order to capture all features coming from the misalignment mechanism, including the contribution from topological defects like "axionic" strings and domain walls. An axion mass estimate between 0.05–1.50 meV was reported by Borsanyi et al. (2016). The result was calculated by simulating the formation of axions during the post-inflation period on a supercomputer.

Recent progresses in determining the present abundance of a KSVZ-type axion using numerical simulations lead to values between 0.02 and 0.1 meV, although these results have been challenged by the details on the power spectrum of emitted axions from strings.

Phenomenology of the axion field

Searches

Axion models carefully choose coupling strengths that are too weak to have been detected in prior experiments. It had been thought that these "invisible axions" solved the strong CP problem while still being too small to have been observed before. Current literature discusses "invisible axion" mechanisms in two forms, called KSVZ (Kim–Shifman–Vainshtein–Zakharov) and DFSZ (Dine–Fischler–Srednicki–Zhitnitsky).

The very weakly coupled axion is also very light, because axion couplings and mass are proportional. Satisfaction with "invisible axions" changed when it was shown that any very light axion would have been overproduced in the early universe and therefore must be excluded.

Maxwell's equations with axion modifications

Pierre Sikivie computed how Maxwell's equations are modified in the presence of an axion in 1983. He showed that these axions could be detected on Earth by converting them to photons, using a strong magnetic field, motivating a number of experiments. For example, the Axion Dark Matter Experiment converts axion dark matter to microwave photons, the CERN Axion Solar Telescope converts axions produced in the Sun's core to X-rays, and other experiments search for axions produced in laser light. Currently, there are dozens of proposed or ongoing experiments searching for axion dark matter.

The equations of axion electrodynamics are typically written in "natural units", where the reduced Planck's constant ${\displaystyle \hslash }$, speed of light ${\displaystyle c}$, and permittivity of free space ${\displaystyle {\epsilon }_0}$ are all set to unity. In this unit system, they are:

Name	Equations
Gauss's law	${\displaystyle \mathrm{\nabla }\cdot \mathbf{E}=\rho -g_{a\gamma \gamma }\mathbf{B}\cdot \mathrm{\nabla }a}$
Gauss's law for magnetism	${\displaystyle \mathrm{\nabla }\cdot \mathbf{B}=0}$
Faraday's law	${\displaystyle \mathrm{\nabla }\times \mathbf{E}=-\dot{\mathbf{B}}}$
Ampère–Maxwell law	${\displaystyle \mathrm{\nabla }\times \mathbf{B}=\dot{\mathbf{E}}+\mathbf{J}+g_{a\gamma \gamma }(\dot{a}\mathbf{B}-\mathbf{E}\times \mathrm{\nabla }a)}$
Axion field's equation of motion	${\displaystyle {\ddot{a}}^2-{\mathrm{\nabla }}^{2}a+m_a^{2}a=-g_{a\gamma \gamma }\mathbf{E}\cdot \mathbf{B}}$

Above, a dot denotes a time derivative and the axion-photon coupling is ${\displaystyle g_{a\gamma \gamma }}$.

Alternative forms of these equations have been proposed, which imply completely different physical signatures. For example, Visinelli wrote a set of equations that imposed duality symmetry, assuming the existence of magnetic monopoles. However, these alternative formulations are less theoretically motivated, and in many cases cannot even be derived from an action.

Analogous effect for topological insulators

A term analogous to the one that would be added to Maxwell's equations to account for axions also appears in recent (2008) theoretical models for topological insulators giving an effective axion description of the electrodynamics of these materials.

This term leads to several interesting predicted properties including a quantized magnetoelectric effect. Evidence for this effect has been given in THz spectroscopy experiments performed at the Johns Hopkins University on quantum regime thin film topological insulators developed at Rutgers University.

In 2019, a team at the Max Planck Institute for Chemical Physics of Solids published their detection of axion insulators within a Weyl semimetal. An axion insulator is a quasiparticle – an excitation of electrons that behave together as an axion – and its discovery is consistent with the existence of the axion as an elementary particle.

Experiments

Despite not yet having been found, axion models have been well studied for over 40 years, giving time for physicists to develop insight into axion effects that might be detected. Several experimental searches for axions are presently underway; most exploit axions' expected slight interaction with photons in strong magnetic fields. Axions are also one of the few remaining plausible candidates for dark matter particles, and might be discovered in some dark matter experiments.

Constraints on the axion's coupling to the photon

Constraints on the axion's dimensionless coupling to electrons

Direct conversion in a magnetic field

Several experiments search for astrophysical axions by the Primakoff effect, which converts axions to photons and vice versa in electromagnetic fields.

The Axion Dark Matter Experiment (ADMX) at the University of Washington uses a strong magnetic field to detect the possible weak conversion of axions to microwaves. ADMX searches the galactic dark matter halo for axions resonant with a cold microwave cavity. ADMX has excluded optimistic axion models in the 1.9–3.53 μeV range. From 2013–2018 a series of upgrades were done and it is taking new data, including at 4.9–6.2 µeV. In December 2021 it excluded the 3.3–4.2 μeV range for the KSVZ model.

Other experiments of this type include DMRadio, HAYSTAC, CULTASK, and ORGAN. HAYSTAC recently completed the first scanning run of a haloscope above 20 µeV.

Polarized light in a magnetic field

The Italian PVLAS experiment searches for polarization changes of light propagating in a magnetic field. The concept was first put forward in 1986 by Luciano Maiani, Roberto Petronzio and Emilio Zavattini. A rotation claim in 2006 was excluded by an upgraded setup. An optimized search began in 2014.

Light shining through walls

Another technique is so called "light shining through walls", where light passes through an intense magnetic field to convert photons into axions, which then pass through metal and are reconstituted as photons by another magnetic field on the other side of the barrier. Experiments by BFRS and a team led by Rizzo ruled out an axion cause. GammeV saw no events, reported in a 2008 Physics Review Letter. ALPS I conducted similar runs, setting new constraints in 2010; ALPS II is currently being built in 2022. OSQAR found no signal, limiting coupling and will continue.

Astrophysical axion searches

Axion-like bosons could have a signature in astrophysical settings. In particular, several recent works have proposed axion-like particles as a solution to the apparent transparency of the Universe to TeV photons. It has also been demonstrated that, in the large magnetic fields threading the atmospheres of compact astrophysical objects (e.g., magnetars), photons will convert much more efficiently. This would in turn give rise to distinct absorption-like features in the spectra detectable by current telescopes. A new promising means is looking for quasi-particle refraction in systems with strong magnetic gradients. In particular, the refraction will lead to beam splitting in the radio light curves of highly magnetized pulsars and allow much greater sensitivities than currently achievable. The International Axion Observatory (IAXO) is a proposed fourth generation helioscope.

Axions can resonantly convert into photons in the magnetospheres of neutron stars. The emerging photons lie in the GHz frequency range and can be potentially picked up in radio detectors, leading to a sensitive probe of the axion parameter space. This strategy has been used to constrain the axion-photon coupling in the 5–11 μeV mass range, by re-analyzing existing data from the Green Bank Telescope and the Effelsberg 100 m Telescope. A novel, alternative strategy consists in detecting the transient signal from the encounter between a neutron star and an axion minicluster in the Milky Way.

Axions can be produced in the Sun's core when X-rays scatter in strong electric fields. The CAST solar telescope is underway, and has set limits on coupling to photons and electrons. Axions may be produced within neutron stars, by nucleon-nucleon bremsstrahlung. The subsequent decay of axions to gamma rays allows constraints on the axion mass to be placed from observations of neutron stars in gamma-rays using the Fermi LAT. From an analysis of four neutron stars, Berenji et al. (2016) obtained a 95% confidence interval upper limit on the axion mass of 0.079 eV. In 2021 it has been also suggested that a reported excess of hard X-ray emission from a system of neutron stars known as the magnificent seven could be explained as axion emission.

In 2016, a theoretical team from Massachusetts Institute of Technology devised a possible way of detecting axions using a strong magnetic field that need be no stronger than that produced in an MRI scanning machine. It would show variation, a slight wavering, that is linked to the mass of the axion. As of 2019, the experiment is being implemented by experimentalists at the university.

In 2022 the polarized light measurements of Messier 87* by the EHT were used to constrain the mass of the axion assuming that hypothetical clouds of axions could form around a black hole rejecting the ~${\displaystyle {10}^{-21}-{10}^{-20}}$ eV/c^2 range of mass values.

Searches for resonance effects

Resonance effects may be evident in Josephson junctions from a supposed high flux of axions from the galactic halo with mass of 110 µeV and density 0.05 GeV/cm³ compared to the implied dark matter density 0.3±0.1 GeV/cm³, indicating said axions would not have enough mass to be the sole component of dark matter. The ORGAN experiment plans to conduct a direct test of this result via the haloscope method.

Dark matter recoil searches

Dark matter cryogenic detectors have searched for electron recoils that would indicate axions. CDMS published in 2009 and EDELWEISS set coupling and mass limits in 2013. UORE and XMASS also set limits on solar axions in 2013. XENON100 used a 225 day run to set the best coupling limits to date and exclude some parameters.

Nuclear spin precession

While Schiff's theorem states that a static nuclear electric dipole moment (EDM) does not produce atomic and molecular EDMs, the axion induces an oscillating nuclear EDM which oscillates at the Larmor frequency. If this nuclear EDM oscillation frequency is in resonance with an external electric field, a precession in the nuclear spin rotation occurs. This precession can be measured using precession magnetometry and if detected, would be evidence for Axions.

An experiment using this technique is the Cosmic Axion Spin Precession Experiment (CASPEr).

Searches at particle colliders

Axions can also be produced at colliders, in particular in electron positron collisions as well as in ultra-peripheral heavy ion collisions at the LHC, reinterpreting the light-by-light scattering process. Those searches are sensitive for rather large axion masses between 100 MeV/c² and hundreds of GeV/c². Assuming a coupling of axions to the Higgs Boson, searches for anomalous Higgs boson decays into two axions can provide even stronger limits.

Possible detections

It was reported in 2014 that evidence for axions may have been detected as a seasonal variation in observed X-ray emission that would be expected from conversion in the Earth's magnetic field of axions streaming from the Sun. Studying 15 years of data by the European Space Agency's XMM-Newton observatory, a research group at Leicester University noticed a seasonal variation for which no conventional explanation could be found. One potential explanation for the variation, described as "plausible" by the senior author of the paper, is the known seasonal variation in visibility to XMM-Newton of the sunward magnetosphere in which X-rays may be produced by axions from the Sun's core.

This interpretation of the seasonal variation is disputed by two Italian researchers, who identify flaws in the arguments of the Leicester group that are said to rule out an interpretation in terms of axions. Most importantly, the scattering in angle assumed by the Leicester group to be caused by magnetic field gradients during the photon production, necessary to allow the X-rays to enter the detector that cannot point directly at the sun, would dissipate the flux so much that the probability of detection would be negligible.

In 2013, Christian Beck suggested that axions might be detectable in Josephson junctions; and in 2014, he argued that a signature, consistent with a mass ≈110 μeV, had in fact been observed in several preexisting experiments.

In 2020, the XENON1T experiment at the Gran Sasso National Laboratory in Italy reported a result suggesting the discovery of solar axions. The results are not yet significant at the 5-sigma level required for confirmation, and other explanations of the data are possible though less likely. New observations made in July 2022, after the observatory upgrade to XENONnT, discarded the excess.

Properties

Predictions

One theory of axions relevant to cosmology had predicted that they would have no electric charge, a very small mass in the range from 1 µeV/c² to 1 eV/c², and very low interaction cross-sections for strong and weak forces. Because of their properties, axions would interact only minimally with ordinary matter. Axions would also change to and from photons in magnetic fields.

Cosmological implications

Inflation suggests that if they exist, axions would be created abundantly during the Big Bang. Because of a unique coupling to the instanton field of the primordial universe (the "misalignment mechanism"), an effective dynamical friction is created during the acquisition of mass, following cosmic inflation. This robs all such primordial axions of their kinetic energy.

Ultralight axion (ULA) with m ~ 10⁻²² eV is a kind of scalar field dark matter which seems to solve the small scale problems of CDM. A single ULA with a GUT scale decay constant provides the correct relic density without fine-tuning.

Axions would also have stopped interaction with normal matter at a different moment after the Big Bang than other more massive dark particles. The lingering effects of this difference could perhaps be calculated and observed astronomically.

If axions have low mass, thus preventing other decay modes (since there are no lighter particles to decay into), theories predict that the universe would be filled with a very cold Bose–Einstein condensate of primordial axions. Hence, axions could plausibly explain the dark matter problem of physical cosmology. Observational studies are underway, but they are not yet sufficiently sensitive to probe the mass regions if they are the solution to the dark matter problem with the fuzzy dark matter region starting to be probed via superradiance. High mass axions of the kind searched for by Jain and Singh (2007) would not persist in the modern universe. Moreover, if axions exist, scatterings with other particles in the thermal bath of the early universe unavoidably produce a population of hot axions.

Low mass axions could have additional structure at the galactic scale. If they continuously fall into galaxies from the intergalactic medium, they would be denser in "caustic" rings, just as the stream of water in a continuously-flowing fountain is thicker at its peak. The gravitational effects of these rings on galactic structure and rotation might then be observable. Other cold dark matter theoretical candidates, such as WIMPs and MACHOs, could also form such rings, but because such candidates are fermionic and thus experience friction or scattering among themselves, the rings would be less sharply defined.

João G. Rosa and Thomas W. Kephart suggested that axion clouds formed around unstable primordial black holes might initiate a chain of reactions that radiate electromagnetic waves, allowing their detection. When adjusting the mass of the axions to explain dark matter, the pair discovered that the value would also explain the luminosity and wavelength of fast radio bursts, being a possible origin for both phenomena. In 2022 a similar hypothesis was used to constrain the mass of the axion from data of M87*.

Supersymmetry

In supersymmetric theories the axion has both a scalar and a fermionic superpartner. The fermionic superpartner of the axion is called the axino, the scalar superpartner is called the saxion or dilaton. They are all bundled up in a chiral superfield.

The axino has been predicted to be the lightest supersymmetric particle in such a model. In part due to this property, it is considered a candidate for dark matter.

Hierarchy problem

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Hierarchy_problem

Beyond the Standard Model
Simulated Large Hadron Collider CMS particle detector data depicting a Higgs boson produced by colliding protons decaying into hadron jets and electrons

In theoretical physics, the hierarchy problem is the problem concerning the large discrepancy between aspects of the weak force and gravity. There is no scientific consensus on why, for example, the weak force is 10²⁴ times stronger than gravity.

Technical definition

A hierarchy problem occurs when the fundamental value of some physical parameter, such as a coupling constant or a mass, in some Lagrangian is vastly different from its effective value, which is the value that gets measured in an experiment. This happens because the effective value is related to the fundamental value by a prescription known as renormalization, which applies corrections to it. Typically the renormalized value of parameters are close to their fundamental values, but in some cases, it appears that there has been a delicate cancellation between the fundamental quantity and the quantum corrections. Hierarchy problems are related to fine-tuning problems and problems of naturalness. Over the past decade many scientists argued that the hierarchy problem is a specific application of Bayesian statistics.

Studying renormalization in hierarchy problems is difficult, because such quantum corrections are usually power-law divergent, which means that the shortest-distance physics are most important. Because we do not know the precise details of the shortest-distance theory of physics, we cannot even address how this delicate cancellation between two large terms occurs. Therefore, researchers are led to postulate new physical phenomena that resolve hierarchy problems without fine-tuning.

Overview

Suppose a physics model requires four parameters which allow it to produce a very high-quality working model, generating predictions of some aspect of our physical universe. Suppose we find through experiments that the parameters have values: 1.2, 1.31, 0.9 and 404,331,557,902,116,024,553,602,703,216.58 (roughly 4×10²⁹). Scientists might wonder how such figures arise. But in particular, might be especially curious about a theory where three values are close to one, and the fourth is so different; in other words, the huge disproportion we seem to find between the first three parameters and the fourth. We might also wonder if one force is so much weaker than the others that it needs a factor of 4×10²⁹ to allow it to be related to them in terms of effects, how did our universe come to be so exactly balanced when its forces emerged? In current particle physics, the differences between some parameters are much larger than this, so the question is even more noteworthy.

One answer given by philosophers is the anthropic principle. If the universe came to exist by chance, and perhaps vast numbers of other universes exist or have existed, then life capable of physics experiments only arose in universes that by chance had very balanced forces. All of the universes where the forces were not balanced didn't develop life capable of asking this question. So if lifeforms like human beings are aware and capable of asking such a question, humans must have arisen in a universe having balanced forces, however rare that might be.

A second possible answer is that there is a deeper understanding of physics that we currently do not possess. There might be parameters that we can derive physical constants from that have less unbalanced values, or there might be a model with fewer parameters.

Examples in particle physics

The Higgs mass

In particle physics, the most important hierarchy problem is the question that asks why the weak force is 10²⁴ times as strong as gravity. Both of these forces involve constants of nature, the Fermi constant for the weak force and the Newtonian constant of gravitation for gravity. Furthermore, if the Standard Model is used to calculate the quantum corrections to Fermi's constant, it appears that Fermi's constant is surprisingly large and is expected to be closer to Newton's constant unless there is a delicate cancellation between the bare value of Fermi's constant and the quantum corrections to it.

Cancellation of the Higgs boson quadratic mass renormalization between fermionic top quark loop and scalar stop squark tadpole Feynman diagrams in a supersymmetric extension of the Standard Model

More technically, the question is why the Higgs boson is so much lighter than the Planck mass (or the grand unification energy, or a heavy neutrino mass scale): one would expect that the large quantum contributions to the square of the Higgs boson mass would inevitably make the mass huge, comparable to the scale at which new physics appears unless there is an incredible fine-tuning cancellation between the quadratic radiative corrections and the bare mass.

The problem cannot even be formulated in the strict context of the Standard Model, for the Higgs mass cannot be calculated. In a sense, the problem amounts to the worry that a future theory of fundamental particles, in which the Higgs boson mass will be calculable, should not have excessive fine-tunings.

Theoretical solutions

There have been many proposed solutions by many physicists.

UV/IR mixing

In 2019, a pair of researchers proposed that IR/UV mixing resulting in the breakdown of the effective quantum field theory could resolve the hierarchy problem. In 2021, another group of researchers showed that UV/IR mixing could resolve the hierarchy problem in string theory.

Supersymmetry

Some physicists believe that one may solve the hierarchy problem via supersymmetry. Supersymmetry can explain how a tiny Higgs mass can be protected from quantum corrections. Supersymmetry removes the power-law divergences of the radiative corrections to the Higgs mass and solves the hierarchy problem as long as the supersymmetric particles are light enough to satisfy the Barbieri–Giudice criterion. This still leaves open the mu problem, however. The tenets of supersymmetry are being tested at the LHC, although no evidence has been found so far for supersymmetry.

Each particle that couples to the Higgs field has an associated Yukawa coupling λ_f. The coupling with the Higgs field for fermions gives an interaction term ${\displaystyle {\mathcal{L}}_{\mathrm{Y}\mathrm{u}\mathrm{k}\mathrm{a}\mathrm{w}\mathrm{a}}=-{\lambda }_f\overline{\psi }H\psi }$, with ${\displaystyle \psi }$ being the Dirac field and ${\displaystyle H}$ the Higgs field. Also, the mass of a fermion is proportional to its Yukawa coupling, meaning that the Higgs boson will couple most to the most massive particle. This means that the most significant corrections to the Higgs mass will originate from the heaviest particles, most prominently the top quark. By applying the Feynman rules, one gets the quantum corrections to the Higgs mass squared from a fermion to be:

${\displaystyle \mathrm{\Delta }{m}_{\mathrm{H}}^2=-\frac{{\left|{\lambda }_f\right|}^2}{8{\pi }^2}[{\mathrm{\Lambda }}_{\mathrm{U}\mathrm{V}}^2+...].}$

The ${\displaystyle {\mathrm{\Lambda }}_{\mathrm{U}\mathrm{V}}}$ is called the ultraviolet cutoff and is the scale up to which the Standard Model is valid. If we take this scale to be the Planck scale, then we have the quadratically diverging Lagrangian. However, suppose there existed two complex scalars (taken to be spin 0) such that:

${\displaystyle {\lambda }_S={\left|{\lambda }_f\right|}^2}$ (the couplings to the Higgs are exactly the same).

Then by the Feynman rules, the correction (from both scalars) is:

${\displaystyle \mathrm{\Delta }{m}_{\mathrm{H}}^2=2\times \frac{{\lambda }_S}{16{\pi }^2}[{\mathrm{\Lambda }}_{\mathrm{U}\mathrm{V}}^2+...].}$

(Note that the contribution here is positive. This is because of the spin-statistics theorem, which means that fermions will have a negative contribution and bosons a positive contribution. This fact is exploited.)

This gives a total contribution to the Higgs mass to be zero if we include both the fermionic and bosonic particles. Supersymmetry is an extension of this that creates 'superpartners' for all Standard Model particles.

Conformal

Without supersymmetry, a solution to the hierarchy problem has been proposed using just the Standard Model. The idea can be traced back to the fact that the term in the Higgs field that produces the uncontrolled quadratic correction upon renormalization is the quadratic one. If the Higgs field had no mass term, then no hierarchy problem arises. But by missing a quadratic term in the Higgs field, one must find a way to recover the breaking of electroweak symmetry through a non-null vacuum expectation value. This can be obtained using the Weinberg–Coleman mechanism with terms in the Higgs potential arising from quantum corrections. Mass obtained in this way is far too small with respect to what is seen in accelerator facilities and so a conformal Standard Model needs more than one Higgs particle. This proposal has been put forward in 2006 by Krzysztof Antoni Meissner and Hermann Nicolai and is currently under scrutiny. But if no further excitation is observed beyond the one seen so far at LHC, this model would have to be abandoned.

Extra dimensions

No experimental or observational evidence of extra dimensions has been officially reported. Analyses of results from the Large Hadron Collider severely constrain theories with large extra dimensions. However, extra dimensions could explain why the gravity force is so weak, and why the expansion of the universe is faster than expected.

If we live in a 3+1 dimensional world, then we calculate the gravitational force via Gauss's law for gravity:

${\displaystyle \mathbf{g}(\mathbf{r})=-Gm\frac{{\mathbf{e}}_{\mathbf{r}}}{r^2}}$ (1)

which is simply Newton's law of gravitation. Note that Newton's constant G can be rewritten in terms of the Planck mass.

${\displaystyle G=\frac{\hslash c}{M_{\mathrm{P}\mathrm{l}}^2}}$

If we extend this idea to ${\displaystyle \delta }$ extra dimensions, then we get:

${\displaystyle \mathbf{g}(\mathbf{r})=-m\frac{{\mathbf{e}}_{\mathbf{r}}}{M_{{\mathrm{P}\mathrm{l}}_{3+1+\delta }}^{2+\delta }{r}^{2+\delta }}}$ (2)

where ${\displaystyle M_{{\mathrm{P}\mathrm{l}}_{3+1+\delta }}}$ is the 3+1+${\displaystyle \delta }$ dimensional Planck mass. However, we are assuming that these extra dimensions are the same size as the normal 3+1 dimensions. Let us say that the extra dimensions are of size n ≪ than normal dimensions. If we let r'≪n, then we get (2). However, if we let r≫n, then we get our usual Newton's law. However, when r ≫ n, the flux in the extra dimensions becomes a constant, because there is no extra room for gravitational flux to flow through. Thus the flux will be proportional to ${\displaystyle n^{\delta }}$ because this is the flux in the extra dimensions. The formula is:

${\displaystyle \mathbf{g}(\mathbf{r})=-m\frac{{\mathbf{e}}_{\mathbf{r}}}{M_{{\mathrm{P}\mathrm{l}}_{3+1+\delta }}^{2+\delta }{r}^2{n}^{\delta }}}$

${\displaystyle -m\frac{{\mathbf{e}}_{\mathbf{r}}}{M_{\mathrm{P}\mathrm{l}}^2{r}^2}=-m\frac{{\mathbf{e}}_{\mathbf{r}}}{M_{{\mathrm{P}\mathrm{l}}_{3+1+\delta }}^{2+\delta }{r}^2{n}^{\delta }}}$

which gives:

${\displaystyle \frac{1}{M_{\mathrm{P}\mathrm{l}}^2{r}^2}=\frac{1}{M_{{\mathrm{P}\mathrm{l}}_{3+1+\delta }}^{2+\delta }{r}^2{n}^{\delta }}\Rightarrow }$

${\displaystyle M_{\mathrm{P}\mathrm{l}}^2=M_{{\mathrm{P}\mathrm{l}}_{3+1+\delta }}^{2+\delta }{n}^{\delta }.}$

Thus the fundamental Planck mass (the extra-dimensional one) could actually be small, meaning that gravity is actually strong, but this must be compensated by the number of the extra dimensions and their size. Physically, this means that gravity is weak because there is a loss of flux to the extra dimensions.

This section is adapted from "Quantum Field Theory in a Nutshell" by A. Zee.

Braneworld models

Main article: Brane cosmology

In 1998 Nima Arkani-Hamed, Savas Dimopoulos, and Gia Dvali proposed the ADD model, also known as the model with large extra dimensions, an alternative scenario to explain the weakness of gravity relative to the other forces. This theory requires that the fields of the Standard Model are confined to a four-dimensional membrane, while gravity propagates in several additional spatial dimensions that are large compared to the Planck scale.

In 1998–99 Merab Gogberashvili published on arXiv (and subsequently in peer-reviewed journals) a number of articles where he showed that if the Universe is considered as a thin shell (a mathematical synonym for "brane") expanding in 5-dimensional space then it is possible to obtain one scale for particle theory corresponding to the 5-dimensional cosmological constant and Universe thickness, and thus to solve the hierarchy problem. It was also shown that four-dimensionality of the Universe is the result of stability requirement since the extra component of the Einstein field equations giving the localized solution for matter fields coincides with one of the conditions of stability.

Subsequently, there were proposed the closely related Randall–Sundrum scenarios which offered their solution to the hierarchy problem.

The cosmological constant

Main article: Cosmological constant problem

In physical cosmology, current observations in favor of an accelerating universe imply the existence of a tiny, but nonzero cosmological constant. This problem, called the cosmological constant problem, is a hierarchy problem very similar to that of the Higgs boson mass problem, since the cosmological constant is also very sensitive to quantum corrections, but it is complicated by the necessary involvement of general relativity in the problem. Proposed solutions to the cosmological constant problem include modifying and/or extending gravity, adding matter with unvanishing pressure, and UV/IR mixing in the Standard Model and gravity. Some physicists have resorted to anthropic reasoning to solve the cosmological constant problem, but it is disputed whether anthropic reasoning is scientific.