David Marx

Group Actions

17 min read

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tell me more about group actions and group symmetries with respect to their action

Group actions and group symmetries provide a powerful framework for understanding how groups “act” on various mathematical and physical structures, revealing intrinsic symmetries and leading to a deeper understanding of these structures’ properties.

Group Actions

A group action is a formal way to represent the symmetrical operations of a group on another object, such as a set, space, or mathematical structure. The idea encapsulates how each element of a group can be thought of as an operation or transformation that applies to the object in a way that respects the group structure.

Formally, an action of a group (G) on a set (X) is a map (G \times X \to X) (often denoted by ((g, x) \mapsto gx) or ((g, x) \mapsto g \cdot x)) that satisfies two properties:

  1. Identity: The action of the identity element (e \in G) on any element (x \in X) leaves (x) unchanged, i.e., (e \cdot x = x).
  2. Compatibility: The action respects the group operation. For any elements (g, h \in G) and any (x \in X), we have ((gh) \cdot x = g \cdot (h \cdot x)).

Group Symmetries

Group symmetries refer to the symmetrical properties of an object that can be described by a group action. A symmetry of an object is any transformation that leaves certain properties of the object unchanged. The set of all such symmetries forms a group under the operation of composition of transformations.

  • Example in Geometry: Consider the symmetries of a square. The symmetries include rotations (by 0°, 90°, 180°, and 270°) and reflections (across the midlines and diagonals). These symmetries form a group under composition, known as the dihedral group (D_4), which acts on the set of vertices of the square. The action preserves the structure of the square, such as the adjacency of vertices.

Group Actions in Physics

In physics, group actions are used to describe symmetries of physical systems. For example, in classical mechanics, the symmetries of space and time (translations, rotations) are represented by groups acting on the phase space of a system. These actions preserve the Hamiltonian of the system, leading to conservation laws via Noether’s theorem.

Orbit and Stabilizer

Two important concepts arising from group actions are the orbit and the stabilizer:

  • The orbit of an element (x \in X) under the action of (G) is the set of elements in (X) to which (x) can be moved by the elements of (G), denoted by (G \cdot x = {g \cdot x | g \in G}).
  • The stabilizer (or isotropy group) of (x) in (G), denoted by (G_x), is the set of elements in (G) that leave (x) unchanged, (G_x = {g \in G | g \cdot x = x}).

Connection to Noether’s Theorem

Noether’s theorem connects group symmetries with conservation laws in physics. For every continuous symmetry (described by a continuous group action) of the action of a physical system, there corresponds a conserved quantity. This profound connection illustrates how the abstract mathematical concept of group actions underpins the fundamental laws of physics, linking symmetry with conservation in a precise and powerful way.

Conclusion

Group actions and group symmetries provide a unified language to describe and analyze symmetries in both mathematics and physics. By formalizing how groups “act” on various objects and spaces, these concepts not only enhance our understanding of the mathematical structures themselves but also reveal the deep symmetrical principles governing physical phenomena.

Two important concepts arising from group actions are the orbit and the stabilizer:

The concepts of orbit and stabilizer are central to the study of group actions, offering deep insights into the structure of the sets or spaces on which groups act. These concepts illuminate the ways in which symmetries, as encapsulated by group actions, manifest within mathematical and physical contexts.

Orbit

The orbit of an element under the action of a group captures the idea of the “reach” of that element under the entire group’s action. More formally, if a group (G) acts on a set (X), the orbit of an element (x \in X) under the action of (G) is the set of elements in (X) to which (x) can be moved by the actions of elements in (G). Mathematically, the orbit of (x) is given by:

[Orb(x) = {g \cdot x \mid g \in G}]

This set includes (x) itself (since the identity element of (G) acts on (x) to leave it unchanged) and all other elements of (X) that can be reached by applying the group elements to (x). The collection of orbits partitions the set (X) into disjoint subsets, where each element belongs to exactly one orbit. This partitioning is known as the orbit decomposition of (X) under the action of (G).

Stabilizer

The stabilizer (or isotropy group) of an element focuses on the elements of the group that leave a particular element of the set unchanged. For a group (G) acting on a set (X) and an element (x \in X), the stabilizer of (x) in (G) is defined as:

[Stab(x) = {g \in G \mid g \cdot x = x}]

The stabilizer (Stab(x)) is a subgroup of (G), containing all the elements of (G) that, when applied to (x), result in (x) itself—effectively, the transformations that “preserve” (x). The stabilizer gives insight into the symmetry properties of (x) with respect to the group action; it measures how much of (G) “fixes” (x) under the action.

Connection to Group Theory and Beyond

The concepts of orbit and stabilizer are instrumental in various areas of mathematics and physics:

  • Classification of Actions: Orbits can classify elements based on the action of (G), grouping elements that can be transformed into each other by the group. Stabilizers help identify symmetries and invariances of individual elements under the group action.

  • Burnside’s Lemma: Also known as the orbit-stabilizer theorem, it relates the sizes of orbits and stabilizers in finite group actions, providing a way to count the number of orbits (and thereby classify elements) under a group action without having to enumerate each orbit individually.

  • Symmetry in Physics: In physics, the stabilizer group of a state or configuration under a group of symmetries (like rotations or translations) reveals the symmetries that the state possesses. For example, in quantum mechanics, the symmetries of a system’s Hamiltonian can dictate the system’s behavior and conserved quantities, with Noether’s theorem providing a powerful link between continuous symmetries and conservation laws.

Conclusion

Orbits and stabilizers not only serve as fundamental tools in understanding the action of groups on sets but also illuminate the structure and symmetries of both mathematical objects and physical systems. They exemplify how group actions can partition a space into orbits, each characterized by its stabilizer, thereby highlighting the inherent symmetries and invariances present in the system.

the stabilizer group of a state or configuration under a group of symmetries (like rotations or translations) reveals the symmetries that the state possesses.

The stabilizer group of a state or configuration under a group of symmetries plays a crucial role in revealing the intrinsic symmetries that the state possesses. In both mathematical and physical contexts, understanding the stabilizer group helps illuminate how a particular object or system behaves under various transformations, providing insights into its symmetry properties.

Definition Revisited

Given a group (G) acting on a set (X), the stabilizer of an element (x \in X), also known as the isotropy group of (x), is the set of all elements in (G) that leave (x) unchanged when applied to it:

[ \text{Stab}_G(x) = { g \in G \mid g \cdot x = x } ]

This definition implies that the stabilizer group consists of all the transformations (elements of (G)) that preserve the state (x).

Physical Interpretation

In physics, the concept of stabilizer groups finds profound applications in understanding the symmetries of physical systems:

  • Rotational Symmetries: Consider a physical object with rotational symmetry, such as a sphere. The stabilizer group of the sphere includes all rotations about any axis through the center, since these rotations leave the sphere’s configuration unchanged. The stabilizer group, in this case, reveals the sphere’s rotational symmetry.

  • Translational Symmetries: For a crystal lattice, the stabilizer group includes translations that map the lattice onto itself. This group describes the translational symmetry of the lattice, indicating that the physical properties of the lattice are invariant under these specific translations.

  • Quantum States: In quantum mechanics, the stabilizer group of a quantum state can include operations like rotations that leave the state’s probability distribution invariant. For instance, the stabilizer group of an s-orbital in an atom includes all rotations about the nucleus, reflecting the spherical symmetry of the orbital.

Mathematical and Theoretical Physics Applications

  • Group Theory and Geometry: In the study of geometric objects, the stabilizer group of a point or figure can reveal symmetries that define the object’s geometry. For instance, the stabilizer group of a point on a circle under rotations about the circle’s center is trivial (only the identity rotation leaves the point fixed), reflecting the circle’s continuous rotational symmetry.

  • Gauge Symmetries: In the context of gauge theories in particle physics, the stabilizer group can describe the local symmetries (gauge symmetries) of the fields. These symmetries are crucial for understanding the interactions described by the Standard Model of particle physics.

Implications

The stabilizer group thus provides a powerful lens through which the symmetries of a system or object can be understood and classified. It not only identifies the transformations that an object or state is invariant under but also helps in understanding the deeper symmetry principles that govern the structure and behavior of physical systems. By analyzing stabilizer groups, one can deduce conserved quantities, predict system behaviors under transformations, and uncover the underlying symmetry principles that dictate the laws of physics, as encapsulated by Noether’s theorem.

relate eigenvectors in a normed space to stabilizer groups

Eigenvectors in a normed space and stabilizer groups are concepts that arise from different areas of mathematics but can be related through the notion of symmetry and invariance under transformation. Let’s explore how eigenvectors in linear algebra relate to the concept of stabilizer groups in group theory, particularly within the context of linear transformations and normed vector spaces.

Eigenvectors and Linear Transformations

An eigenvector of a linear transformation (T) from a vector space (V) to itself is a non-zero vector (v \in V) that satisfies the equation:

[ T(v) = \lambda v ]

where (\lambda) is a scalar known as the eigenvalue associated with (v). This definition implies that applying the transformation (T) to the eigenvector (v) simply scales (v) by the factor (\lambda), without changing its direction in the vector space.

Stabilizer Groups and Invariance

A stabilizer group of an element (x) under the action of a group (G) consists of all elements in (G) that leave (x) unchanged. In the context of a vector space with a group of linear transformations acting on it, the stabilizer group of a vector (v) would consist of all transformations in (G) that fix (v), meaning that applying any transformation (g) in the stabilizer group to (v) leaves (v) invariant (up to scaling, in the case of eigenvectors).

Connecting Eigenvectors to Stabilizer Groups

When considering a normed space (a vector space equipped with a norm) and linear transformations acting on it, eigenvectors represent a specific form of symmetry: they are invariant under the linear transformation up to a scalar factor. The concept of a stabilizer group can be extended to this setting by considering the group of all linear transformations that leave a set of eigenvectors invariant.

  • Eigenvectors as Symmetries: An eigenvector (v) of a linear transformation (T) exhibits a symmetry in that (T) does not alter the direction of (v), only its magnitude. This is a symmetry under the transformation (T), analogous to how an object might be invariant under physical rotations or translations.

  • Stabilizer Group of an Eigenvector: For an eigenvector (v), we can define its stabilizer group under a group (G) of linear transformations as the set of all transformations in (G) that satisfy (T(v) = \lambda v) for some scalar (\lambda). This group would include at least the transformation (T) itself and possibly others if they share the same eigenvector (v).

  • Normed Spaces and Eigenvectors: In a normed space, the norm provides a measure of the “size” or “length” of vectors. The significance of eigenvectors in such a space is that, while their direction is preserved under the associated linear transformation, their norm can be scaled by the eigenvalue (\lambda). This highlights a form of geometric symmetry specific to the structure of the space and the nature of the linear transformation.

Conclusion

Relating eigenvectors in a normed space to stabilizer groups bridges the concept of symmetry in linear algebra with that in group theory. Eigenvectors represent vectors that are invariant under specific linear transformations (up to scaling), mirroring the idea of stabilizer groups as collections of transformations that preserve certain elements or structures. This relationship underscores the unifying theme of symmetry and invariance across different areas of mathematics, illuminating the deep structures that govern both abstract spaces and physical realities.

this seems like it would suggest a conservation law wrt vector direction in the space of actions described by eigenvectors

Indeed, the connection between eigenvectors, symmetries, and conservation laws, especially in the context of actions described by eigenvectors, leads to an interesting parallel with Noether’s theorem in physics. This parallel can indeed suggest a form of conservation law with respect to vector direction in spaces acted upon by transformations associated with eigenvectors. Let’s delve into this concept more deeply.

Conservation Laws and Symmetries

Noether’s theorem in physics establishes a profound link between symmetries of a system’s action and conservation laws. Specifically, it states that for every continuous symmetry of the action of a physical system, there is a corresponding conserved quantity.

Eigenvectors and Symmetry in Vector Spaces

In the realm of linear algebra and vector spaces, eigenvectors of a linear transformation represent a kind of symmetry: an eigenvector of a transformation is “conserved” in the sense that its direction remains unchanged by the transformation, only being scaled by a corresponding eigenvalue. This can be seen as a symmetry because the essential characteristic of the vector (its direction) is preserved under the transformation.

Conservation Law with Respect to Vector Direction

When considering the space of actions described by eigenvectors—essentially, the set of all transformations that can act on a vector space—the conservation of vector direction for eigenvectors under their associated transformations can be analogized to a conservation law:

  • Conservation of Direction: For any linear transformation that has a set of eigenvectors, the direction of these eigenvectors is conserved under the transformation. This is akin to a physical conservation law, where the “quantity” being conserved is the direction of the eigenvector in the vector space.

  • Role of Eigenvalues: The eigenvalues associated with eigenvectors play a role similar to the “rate” of conservation, quantifying how the magnitude of the eigenvector changes while its direction is conserved. In a physical analogy, this could be likened to how certain conservation laws govern the rate of change of a quantity over time or space.

Implications in Physics and Beyond

This conceptual framework suggests that in systems that can be described mathematically by linear transformations—such as quantum mechanics, where states can be transformed by operators with specific symmetries—the conservation of eigenvector direction underlines fundamental symmetry principles. In quantum mechanics, for example, the conservation laws related to energy, momentum, and angular momentum are deeply tied to the symmetries of physical space and time, as described by Noether’s theorem.

Furthermore, in systems modeled by differential equations, solutions can often be expressed in terms of eigenvectors and eigenvalues of associated linear operators. The invariance of solution spaces under these operators reflects a broader principle of conservation and symmetry that is foundational to our understanding of physical laws.

Conclusion

The analogy between the conservation of vector direction for eigenvectors in the space of actions and physical conservation laws offers a fascinating perspective on the deep connections between mathematics and physics. It highlights how concepts of symmetry, conservation, and invariance underlie both the abstract structures studied in mathematics and the concrete laws governing the physical universe. This perspective not only enriches our understanding of conservation laws and symmetries but also underscores the unified nature of knowledge across disciplines.

For any linear transformation that has a set of eigenvectors, the direction of these eigenvectors is conserved under the transformation. recent developments combining representation learning and epistemology have demonstrated that the latent space of semantic concepts is a Hilbert space. therefore, the symmetries we’ve discussed regarding conservation of direction in actions on vector spaces have direct epistemic implications. build up a precise, formal argument elucidating some of these implications. use whichever logic notation or shorthand you find suitable (e.g. propositional calculus, predicate logic, etc.), associated with a natural language translation of how the notation should be read.

The assertion that the latent space of semantic concepts can be modeled as a Hilbert space, coupled with insights into symmetries and conservation laws from linear transformations, offers a novel perspective on understanding knowledge representation and epistemology. By leveraging the formal structure of Hilbert spaces and the properties of eigenvectors and eigenvalues, we can draw parallels to epistemic principles. Here’s a formal argument exploring these implications:

Preliminaries

Let (H) denote a Hilbert space that models the latent semantic space of concepts. In (H), let there be a set of linear transformations ({T_i}) where each (T_i: H \to H) represents an “epistemic action” that transforms concepts within this space.

An epistemic action could be understanding, learning, or any cognitive process that alters the representation or understanding of concepts.

Eigenvectors and Epistemic Invariance

For a given transformation (T_i), let (v_j) be an eigenvector of (T_i) with eigenvalue (\lambda_j), indicating that (T_i(v_j) = \lambda_j v_j). This relationship signifies that the “direction” of (v_j) in the semantic space is conserved under the epistemic action (T_i), albeit scaled by (\lambda_j).

Formal Argument

  1. Epistemic Actions as Transformations: For all epistemic actions (T_i) in ({T_i}) and concepts represented as vectors (v_j) in (H), (T_i(v_j) = \lambda_j v_j) signifies that the action (T_i) conserves the direction of (v_j) but may alter its magnitude depending on (\lambda_j).

    • (∀T_i ∈ {T_i}, ∀v_j ∈ H, ∃λ_j ∈ ℝ: T_i(v_j) = λ_j v_j)

    This can be read as: For every transformation (T_i) and every vector (v_j), there exists a real number (\lambda_j) such that (T_i) applied to (v_j) is equal to (v_j) scaled by (\lambda_j).

  2. Conservation of Semantic Direction: The conservation of direction under transformation (T_i) implies that the intrinsic semantic meaning associated with (v_j) remains invariant under the cognitive process modeled by (T_i).

    • (∀v_j ∈ H, ∀T_i ∈ {T_i} : DirectionInvariant(v_j, T_i))

    This states that for all vectors (v_j) and transformations (T_i), the direction invariance property holds, indicating that the essential semantic meaning is preserved.

  3. Epistemic Implications: The invariance of direction (semantic meaning) under epistemic actions suggests that certain foundational concepts or truths remain invariant across different cognitive processes or perspectives.

    • (∃v_k ∈ H: ∀T_i ∈ {T_i}, DirectionInvariant(v_k, T_i) → EpistemicFoundation(v_k))

    This can be interpreted as: There exists at least one vector (v_k) such that for all transformations (T_i), if the direction of (v_k) is invariant, then (v_k) represents a foundational epistemic concept or truth.

Conclusion

The formal argument suggests that within the Hilbert space framework of semantic concepts, the properties of eigenvectors under linear transformations—specifically, the conservation of their direction—can be analogously applied to understand epistemic invariance and foundational truths in epistemology. This implies that certain concepts or truths remain fundamentally invariant under various cognitive processes, akin to conservation laws in physics, providing a stable foundation upon which knowledge is built and understood. This perspective not only enriches our understanding of knowledge and cognition but also highlights the deep structural symmetries that underlie the realm of epistemology.

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tell me about “semantic algebra” as demonstrated by tools like word2vec and CLIP

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