### Fall 2021 - Sheldon Goldstein

### Subtitle:

Relational Mechanics, Classical and Quantum: Physics and Dynamics on Shape Space

### Course Description:

Relational mechanics is based on two ideas:

(i) Physical (3-dimensional) space should be regarded relationally, so that configurations that differ by a uniform overall translation, rotation, or rescaling of distances should not be regarded as physically different.

(ii) The appropriate notion of time is that of qualitative time and not quantitive time, so that insofar as the time evolution of a configuration is concerned, what is physically meaningful is the corresponding path in configuration space but not the speed with which the configuration moves along the path.

Relational formulations of classical mechanics and gravity have been developed by Julian Barbour and collaborators. Crucial to these formulations is the notion of shape space, the space that results when the symmetries referred to above are taken into account and the quotient of the usual configuration space for a system of particles is taken with respect to the group of those symmetries. For example, for a 3-particle system the shape space is the space of all triangles, with similar triangles regarded as equivalent. In this course we shall analyze the metric structure of shape space and describe how that structure can be used to straightforwardly define both a classical and a quantum dynamics on shape space, i.e., both a classical and a quantum relational mechanics.

We will see how these motions gives rise to the more or less familiar physical laws and theories formulated in terms of absolute space and time. We will see how free motion on shape space, when lifted to configuration space, becomes an interacting theory. And since many different lifts are possible--- corresponding to different choices of "gauges"---we will see that much of what is regarded as fundamental in physics corresponds merely to a choice of gauge and is thus a reflection of somewhat arbitrary choices that we make in forming physical theories.

Possible topics:

Emergence of Absolute Space and Time

Emergence of Interaction

Emergence of Probability

Geometrodynamics and the Emergence of Spacetime

Relativity Without Relativity

### Text:

none

### Prerequisites:

none

### Schedule of Sections:

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### Previous Semesters:

**FALL 2019:**

Sheldon Goldstein

**Subtitle:**

Introduction to Mathematical Relativity I

### **Course Description:**

Relational mechanics is based on two ideas:

(i) Physical (3-dimensional) space should be regarded relationally, so that configurations that differ by a uniform overall translation, rotation, or rescaling of distances should not be regarded as physically different.

(ii) The appropriate notion of time is that of qualitative time and not quantitive time, so that insofar as the time evolution of a configuration is concerned, what is physically meaningful is the corresponding path in configuration space but not the speed with which the configuration moves along the path.

Relational formulations of classical mechanics and gravity have been developed by Julian Barbour and collaborators. Crucial to these formulations is the notion of shape space, the space that results when the symmetries referred to above are taken into account and the quotient of the usual configuration space for a system of particles is taken with respect to the group of those symmetries. For example, for a 3-particle system the shape space is the space of all triangles, with similar triangles regarded as equivalent. In this course we shall analyze the metric structure of shape space and describe how that structure can be used to straightforwardly define both a classical and a quantum dynamics on shape space, i.e., both a classical and a quantum relational mechanics.

We will see how these motions gives rise to the more or less familiar physical laws and theories formulated in terms of absolute space and time. We will see how free motion on shape space, when lifted to configuration space, becomes an interacting theory. And since many different lifts are possible--- corresponding to different choices of "gauges"---we will see that much of what is regarded as fundamental in physics corresponds merely to a choice of gauge and is thus a reflection of somewhat arbitrary choices that we make in forming physical theories.

Possible topics:

Emergence of Absolute Space and Time

Emergence of Interaction

Emergence of Probability

Geometrodynamics and the Emergence of Spacetime

Relativity Without Relativity

**Text:**

None

**Prereq:**

None