The mind is built on the physical foundation of the brain, and the mind always has a corresponding physical state.
The brain forms the field where the mind is made and functioning. “Me,” the subject, is not only physically supported by the body and the brain, but it also includes the consciousness of me which thinks using various concepts. The mind has the idea and content created by the brain, like the image made in a reader's head who is reading a novel, but the mind is not the state of the brain which corresponds to the mind at that time. In that sense, the content of the mind is not something which physically exist. However, my mind can only exist based on the physical state of my brain, and while my mind is working, my mind has a working state of my brain corresponding to my mind. Therefore, the physical state of my brain physically exist and the content of my mind exists as the physical phenomenon based on the physical state.
The content corresponding to the mind exists in the physical world as a physical phenomenon.
As already said, the mind has the content as an idea, it is not the state of the brain itself, and the mind cannot be reduced to the state of the brain. Therefore, the content of the mind cannot be said to exist physically. However, since the mind is a phenomenon caused by the function of brain, i.e., physical being, it cannot be said to be a non-physical phenomenon unrelated to the physical world. It would be rather unnatural to think that the mind that appears from such a physical being cannot interact with the physical world. The mind, including its content, exists in a state of constant interaction with the state of the brain. The mind, including its content, is a phenomenon within the physical world, not outside the closed causal relation of the physical world.
Difference between spirit and computer information processing
If you compare a computer to the brain, the mind can be compared to the information processing of a computer. Computer information processing converts information into electrical signals, mechanically processes the electrical signals according to programmed procedures, and outputs the processed results. On the other hand, the main part of information processing in the brain progresses with qualia. In its information processing in human brain, the computer-program equivalent is executed by the personality accumulated and formed by the innate abilities and experiences during the growth process. Due to the differences of these, the output of the information processing results will differ from person to person.
Suppose we are going to use a computer to automate the process of sorting apples by humans in a factory. In this case, the machine will be made to perform the flow of detecting the size of apples, judging the sorting, and executing the sorting. Regardless of being done based on judgment of human or on the judgment of the computer, the sorting is performed according to the intenti of humans. The machine is designed to produce similar results for humans simply by letting the machine do it. Sorting apples into large, medium and small apples is done for human convenience, not a natural phenomenon. However, whether the sorting work into groups of large, medium and small apples is performed by humans or by computers, both are equivalent as long as the sorting work is performed according to the intention of humans who need the sorting work. In other words, in the simple work here, the worker's will to classify this apple to “big” or “small" and moving his hand to that effect are equivalent mechanisms. Here, there is no issue of the causal closure of the physical world in this the low-level spirit and physical world.
Why is it necessary for the mind must to supervene over the deterministic physical world?
If the human brain evolves to achieve a high-level mental functions from such low-level mental functions, what makes it necessary to lose the low-level functions such as apple size detection, sorting judgment, and sorting execution. There is no reason to lose it. The human brain just acquired a high level of mental function that is an extension which includes the low- level mental function interacting with the physical world.
It does not make sense to divide the mental functions such that the low-level functions belong to the realm of "closed causation in the physical world", and that the content of the advanced spirit merely supervenes over the deterministic physical world that just follows the laws of physics.
In the first place, it should not be possible to deny free will early and advocate determinism. Nowadays, there are many philosophers who argue deterministically, but I would like you to notice the ridiculousness.